arXiv:1004.4572v1 [math.RA] 26 Apr 2010

COENDOMORPHISM BIALGEBROID AND CHAIN COMPLEXES

arXiv:1004.4572v1 [math.RA] 26 Apr 2010

A. ARDIZZONI, L. EL KAOUTIT, AND C. MENINI Abstract. Inspired by the works of B. Pareigis and D. Tambara, we give a new approach of comparing the categories of chain complexes of left modules and left comodules over left coendomorphism bialgebroids. Roughly speaking, given an associative and unital ring R, there is a left coendomorphism R-bialgebroid L such that the category of chain complexes of left R-modules is equivalent to the category of left comodules over an epimorphic image of L . Such an equivalence is monoidal, whenever R is commutative. Basically, these are immediate conclusions after combining several key outcomes of this paper. Our approach relies heavily on the non commutative theory of Tannaka reconstruction, and the generalized faithfully flat descent for small additive categories, or rings with enough orthogonal idempotents.

Contents 1. Introduction 1.1. Methodology and motivation overviews. 1.2. A brief description of the main results. 1.3. Basic notions and notations. 2. Monoidal Results. 2.1. Coequalizers in the category of monoids. 2.2. An adjunction between the categories of monoids. 2.3. General examples. 3. The construction of coendomorphism bialgebroids. 3.1. An adjunction between Re -bimodules and R-bimodules. 3.2. The bi-functor (− ×R −) on Re -bimodules. 3.3. The ×R -bialgebra Lm (A). 3.4. The monoidal structure of left Lm (A)l -comodules. 3.5. Examples of coendomorphism bialgebroids. 4. Categories of comodules and chain complexes of modules. 4.1. The complex of left L -comodules Q• . 4.2. The infinite comatrix bialgebroid induced by Q• . 4.3. The isomorphism between comatrices and coendomorphisms bialgebroids. 4.4. Monoidal equivalence between chain complexes of k-modules and left L -comodules. 4.5. Equivalence between chain complexes of left R-modules and left L -comodules. 4.6. Conditions under which QC is faithfully flat. 4.7. The commutative case. 4.8. The main example. References

2 2 4 5 7 7 10 14 15 15 17 20 26 27 30 30 33 38 42 44 48 51 52 53

Date: April 27, 2010. 1991 Mathematics Subject Classification. Primary 18D10; Secondary 16W30. Key words and phrases. Monoidal categories. Chain complexes. Ring extension. Bialgebroids. Tannakian categories. This paper was written while the first and the last authors were members of G.N.S.A.G.A. with partial financial support from M.I.U.R. within the National Research Project PRIN 2007. The second author was supported by grant MTM2007-61673 from the Ministerio de Educación y Ciencia of Spain. His stay, as a visiting professor at University of Ferrara, was supported by I.N.D.A.M.. 1

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A. ARDIZZONI, L. EL KAOUTIT, AND C. MENINI

1. Introduction 1.1. Methodology and motivation overviews. The starting point of this paper is a result due to B. Pareigis [30, Theorem 18] which asserts that the category of unbounded complexes of vector spaces is monoidally equivalent to the category of left comodules over a certain Hopf algebra which is neither commutative nor cocommutative. Later on, in [37, Theorem 4.4], D. Tambara associated to every finite dimensional algebra A over a field k, a bialgebra Lm (A) (termed coendomorphism bialgebra) such that the category of left comodules Lm (A) Comod is monoidally equivalent to the category Ch + (k) of chain complexes of k-vector spaces. The Hopf algebra considered by B. Pareigis is recovered by choosing A = k ⊕ kt, i.e. the trivial extension of k (t2 = 0), and localizing the bialgebra Lm (A) using a multiplicative set generated by a single grouplike element. The equivalence of categories established by Tambara relies on the use of a slightly variant of the equivalence between simplicial k-vector spaces and chain complexes of k-vector spaces, provided by the normalization functor, due to Dold and Kan, see [12, Theorem 1.9, Corollary 1.12] or [22, Theorem 2.4]. The functor that provides such equivalence is given, in some sense, by tensoring chain complexes with the augmented cochain complex Q• constructed using the Amitsur cosimplicial vector space attached to the k-algebra A. Note that Q• is given by Q0 = k, Q1 = A and Qn = K ⊗A · · · ⊗A K, (n − 1)times for n ≥ 2, where K is the kernel of the multiplication of A. The construction of this functor will be clarified in Section 4 (see also the end of this subsection). A different approach to Pareigis’s result, using Tannaka reconstruction for several-objects coalgebras, was also given by Paddy McCrudden in [28, Examples 6.6, 6.9], where the same coendomorphisms bialgebra Lm (A) was constructed for a commutative base ring k instated of a field. A monoidal equivalence between categories of chain complexes of (left) modules and left comodules over bialgebroids, allows one freely to transfer at least the model structure of chain complexes, as was described in [18, §2.3], to left comodules over bialgebroids. This in fact suggests that the categories of comodules over certain bialgebroids could be endowed within a (monoidal) model structure. This indeed is our main motivation for further investigating the relationship between categories of chain complexes of modules and left comodules over bialgebroids. Let R be an algebra over a commutative ring k. The purpose of this paper is to provide a comprehensive treatment of the relationship between categories of chain complexes of left R-modules and categories of left comodules over certain left R-bialgebroids, constructed hereby. Tambara’s results, and in particular Pareigis’s one, are then immediate consequences of the general theory we developed. It is noteworthy that our methods can be seen as new and more conceptual even for the case of vector spaces. Our approach makes use of the ”non commutative” Tannakian categories theory following the spirit of [11, 8] and [17], as well as of the generalized faithfully flat descent for rings with enough orthogonal idempotents stated in [15]. We mean that all the (left) bialgebroids that arise here come from the non commutative version of Tannaka reconstruction process which involves rings with enough orthogonal idempotents. In the classical Tannaka theory a commutative bialgebroid, more precise a Hopf algebroid [31, Appendix I], appears as follows. It is well known that a commutative Hopf algebroid can be regarded as a presheaf of groupoids in affine schemes, the opposite category of commutative rings. Let S be a scheme over a field k. Following the terminology of the first chapter of Exposé V in SGA 3 [33] (see also [11, §1.6]), a k-groupoid action on S is a k-scheme G endowed with two morphisms s, t : G → S, called target and source, and a morphism of (S × S := S × S)-schemes ◦ : G × G → G, called composition law (here G is an Spec(k)

s, t

(S × S)-scheme via (t, s)). These are subject to the following condition. For each k-scheme T , let G(T ) and S(T ) denote the morphisms of k-schemes from T to G and S, respectively. So s and t induce maps sT and tT from G(T ) to S(T ), and ◦ induces, up to a canonical bijection, a composition law ◦T on G(T ). / / S(T ), ◦T ) form a groupoid, that is, a small category We ask for each T that the data ( sT , tT : G(T ) whose morphisms are isomorphisms. This leads in fact to a contravariant functor from k-schemes to the category of groupoids. In this way, if G = Spec(C) is a k-scheme acting on an affine k-scheme S = Spec(R)


3

and G is itself affine over S × S, then there is a structure of an (R ⊗k R)-ring on C. In other words, a morphism R ⊗k R → C of commutative k-algebras. The rest of groupoid axioms say then that the pair (R, C) is actually a commutative Hopf algebroid. Conversely, every commutative Hopf algebroid leads to a presheaf of groupoids in affine schemes. Furthermore, through this correspondence, a representation of G (i.e. a quasi-coherent sheaf endowed with an action of G) is equivalent to a right C-comodule. A k-groupoid acts transitively on S when the pair (t, s) is a cover in the fpqc topology. In this direction, an important result due to P. Deligne [11, Théorème 1.12] says that there is a dictionary between tensorial categories over k 1 with a fiber functor over a k-scheme S 2, and k-groupoids acting transitively over S which are affine over S × S. As was shown in [11, 2.6, 2.7], each fiber functor takes values in the category of locally free sheaves of finite rank over S. A Tannakian category is then a tensor category together with a fiber functor over S 6= ∅. Based on Deligne’s result, A. Buguières showed in [8, Théorème 8.2] that a k-groupoid G = Spec(C) acts transitively on S = Spec(R) (R 6= 0) if and only if the underlying R-coring C (i.e. k-cogébröde de base R) is geometrically transitive. The later means that C is projective as an R-bimodule, the category comodC of right C-comodules which are finitely generated as right R-modules is locally finite over k, and the coinvariant subring w.r.t the grouplike element 1C coincides with the base field k. Therefore, if the canonical map is bijective (see below), then R becomes a principal Galois C-comodule in the sense of [4] and [14]. Conversely, assuming that R CR is projective and R is Galois comodule whose coinvariant subring is the base field k, then one can easily deduce from [16, Theorem 4.4] that C is a simple cosemisimple coring. This means that the representations of G = Spec(C) form an abelian semisimple category with only one class of simples whose representative has k as ring of endomorphism. In this case, C is obviously geometrically transitive, and so G = Spec(C) acts transitively on S = Spec(R). The treatment of (right semi)transitive coring with non necessary commutative base ring as well as a non commutative version of Tannaka-Krein duality can be found in [8] and [17]. Notice that the construction performed in these references, especially the coring one, coincides with that given in [15]. We should also mention here that the definition of semi-transitivity given in [17, Definition page 215] has a redundant condition. Namely, condition (ii) in that definition is deduced from condition (i) by using the theory of rational modules developed in [16], see also [1] and [6]. For lack of space and time, we will not investigate the (right semi)-transitivity property of the bialgebroids constructed hereby. In the non commutative setting, one basically starts with a small k-linear monoidal category (A, ⊗, 1) and a faithful monoidal functor 3 from A to the category of R-bimodules, ω : A → R ModR (the fiber functor), valued in the category finitely generated and projective left R-modules (i.e. locally free sheaves of finite rank). There are several objects under consideration: Σ(ω) = ⊕ ω(p), ∨ Σ(ω) = ⊕ ∗ ω(p), G (A) = ⊕ HomAo p, p′ . p∈A

p∈A

p, p′ ∈A

Here the second is the right R-module direct sum of the left duals while the third is Gabriel’s ring with enough orthogonal idempotents, introduced in [19], attached to the opposite category. Using the canonical actions, we consider L(ω) := Σ(ω) ⊗G (A) ∨ Σ(ω) as an Re -bimodule, where Re := R ⊗k Ro . A well known argument in small additive categories says that the object L(ω) solves the following universal problems in R-bimodules Nat ω, − ⊗R ω ∼ = HomR−R L(ω), − , ∼ e Nat ω ⊗R ω, − ⊗R (ω ⊗R ω) = HomR−R L(ω) ⊗R L(ω), − , 1These

are abelian closed symmetric monoidal categories with the endomorphism ring of the unit object isomorphic to k. is, an exact k-linear and symmetric monoidal functor valued in the category of quasi-coherent sheaves over S. 3Our setting requires an isomorphism only at the level of unit. That is, R ∼ ω(1), while ω(− ⊗ −) → ω(−) ⊗ ω(−) is not = R necessarily a natural isomorphism. 2That

4


where the R-bimodule structures of L(ω) have been chosen properly. It is indeed this solution which allows us to construct a left R-bialgebroid (or a Hopf bialgebroid if desired). Of course there is an obvious (monoidal) functor connecting left unital G (A)-modules and left L(ω)-comodule, namely Σ(ω) ⊗G (A) − : G (A) Mod −→ L(ω) Comod. In the case when each of the left R-modules ω(p) is endowed with a structure of left C-comodule for some R-coring C (or certain left R-bialgebroid), there is a map of R-corings, known as a canonical map, canG (A) : L(ω) −→ C defined by using the left C-coaction of the ω(p)’s. The associated coinduction functor leads to the following composition of functors Σ(ω)⊗G (A) −

G (A) Mod

/

(−)canG (A)

L(ω) Comod

/

C Comod.

Indeed this is the conceptual framework that allows us to compare certain categories of k-linear functors with the categories of comodules over some corings (or left bialgebroids). For instance, take R = k to be a field and A a finite dimensional k-algebra. Consider the cochain complex Q• mentioned above and the monoidal category k-linear category k(N) generated by the natural number N. There is a fiber functor χ : k(N) → Modk defined by χ(n) = Qn on objects and sending the morphism n 7→ n + 1 to the differential ∂ : Qn → Qn+1 , for every n ∈ N. Using the previous arguments and notations, we then arrive to the following composition Ch + (k)

O ∼ =

/

Q⊗G (k(N)) −

G (k(N)) Mod

/

L(χ) Comod

(−)canG (k(N))

/

Lm (A) Comod

where O is the canonical equivalence between chain complexes of k-vector spaces and left unital G (k(N))modules. This in fact is exactly the functor used by D. Tambara in the proof of [37, Theorem 4.4]. 1.2. A brief description of the main results. Let k be a commutative base ring with 1. Fix a morphism of k-algebras R → A. Assume that R A is finitely generated and projective left R-module with a finite dual basis {ei , ∗ ei }i . We consider the monoidal functor − ×R A : Re ModRe → R ModR , where − ×R − is the Sweedler-Takeuchi’s product [34, 36]. Applying the general theory described in Section 2, we show that the restriction of this functor to the category of Re -rings (i.e. the category of monoids in Re ModRe ) admits a left adjoint which we denoted by Lm : R-Rings → Re -Rings. We then show in Proposition 3.3.5 and Corollary 3.3.6, that the image of A, Lm (A) admits a structure of left R-bialgebroid (termed a coendomorphism bialgebroid ) such that A is a ring left Lm (A)-comodule 4. Lm (A) is given by the following quotient of the tensor Re -ring of A ⊗k ∗ A: ∗ TRe A ⊗ A E (1) Lm (A) := D P . ′ ∗ ′ o e (a ⊗ ei ) − (aa ⊗ ϕ), (1 ⊗ ϕ) − 1 ⊗ ϕ(1) (a ⊗ e ϕ) ⊗ i R i {a, a′ ∈A, ϕ ∈∗ A}

Then we consider the augmented cochain complex of the universal differential graded algebra: (2)

Q• : R

1

/

A

∂

/

K

∂2

/

K ⊗A K

∂3

/

K ⊗A K ⊗A K /

······

where K denotes the kernel of A ⊗R A → A the multiplication of A. We check that this is in fact a cochain complex of left Lm (A)-comodules whose components are finitely generated and projective left R-modules. This leads to a fiber functor χ : k(N) → R ModR defined in the obvious way, as well as to a canonical map 4Note

that, from categorical point of view, one can expect that this is rather a trivial result. However, this is far from being a direct or immediate verification, since the handled categories have a very complicated monoidal structure. This is due to the fact that we are dealing with multi-modules over R rather than fixed bimodules and that the product − ×R − is not associative, see Section 3.


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canB : Q ⊗B ∨ Q → Lm (A), where B = k(N) ⊕ k(N) is the ring with enough orthogonal idempotents attached to the small category k(N). We show that canB is an isomorphism of left R-bialgebroids. In this way we arrive to our first main result stated below as Theorem 4.4.18: Theorem A. Let R → A be a k-algebra map with A finitely generated and projective as left R-module. Consider the associated left R-bialgebroid Lm (A) (see equation (1) above) and the cochain complex Q• of equation (2) with its canonical right unital B-action and left Lm (A)-coaction, where B = k(N) ⊕ k(N) . Then the following statements are equivalent (1) The right R-module 1⊗k Ro Lm (A) is flat and the functor Q ⊗B − : B Mod −→ Lm (A) Comod is an equivalence of monoidal categories; (2) QB is a faithfully flat module. Since the category of left unital B-module B Mod is isomorphic to the category of chain complexes of kmodules, Theorem A suggests that certain categories of left comodules over coendomorphism bialgebroids can be equipped with a (possibly monoidal) model structure. This is indeed one of the main motivations of this paper. Clearly the unit k → R map can be extended to a morphism of rings with the same set of orthogonal idempotents: B = k(N) ⊕ k(N) → R(N) ⊕ R(N) = C. This enables us to consider the usual adjunction between the scalars-restriction functor and the tensor product functor and, in particular, to define a canonical map canC with codomain a suitable quotient of Lm (A). Thus one can try to extend Theorem A to left unital C-modules. In this way we arrive to our second main theorem which is stated below as Theorem 4.5.24: Theorem B. Let R → A be a k-algebra map with A finitely generated and projective as left R-module. Consider Lm (A) the associated left R-bialgebroid (see equation (1) above) and J the coideal of Lm (A) generated by the set of elements {1Lm (A) (r ⊗ 1o − 1 ⊗ r o )}r∈R ; denote by Lm (A) = Lm (A)/J the corresponding quotient R-coring. Consider the cochain complex Q• of equation (2) with its structures of right unital C-module and left Lm (A)-comodule. Then the following statements are equivalent (1) The right R-module 1⊗k Ro Lm (A) is flat and the functor Q ⊗C − : equivalence of categories; (2) QC is a faithfully flat module.

C Mod

−→

Lm (A) Comod

is an

The problem of obtaining an equivalence of categories as above, is then closely linked to the faithfully flat condition on the right unital module Q. This is in fact not at all easy to check. Our third main result, which is a combination of Theorem 4.6.25 and Proposition 4.8.28, gives some homological conditions under which Q becomes flat (or faithfully flat). Theorem C. The notations and assumptions are that of Theorem B. Assume further that AR is finitely generated and projective, and the cochain complex Q• is exact and splits, in the sense that, for every m ≥ 1, Qm = ∂Qm−1 ⊕ Qm = Ker(∂) ⊕ Qm as right R-modules, for some right R-module Qm . Then QC is a flat module. In particular, QC is faithfully flat in either one of the following cases. (1) A = R ⊕ Rt (t2 = 0), the trivial extension of R. (2) k is a field and R is a division k-algebra. As a consequence of Theorems B and C, we get that for every k-algebra R, there is a left R-bialgebroid L such that the category of chain complexes of left R-modules is equivalent to the category of left comodules over an epimorphic image of L . In particular, if R is commutative, then this equivalence is in fact a monoidal equivalence. 1.3. Basic notions and notations. Given any Hom-set category C, the notation X ∈ C means that X is an object of C. The identity morphism of X will be denoted by X itself. The set of all morphisms f : X → X ′ in C is denoted by HomC (X, X ′ ). The identity functor of C is denoted by idC . We denote the

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dual (or opposite) category of C by C o . The class of all natural transformations between two functors F and G is denoted by Nat(F, G). For any pair of morphisms f, g : X → Y in C, we denote by CoeqC (f, g) the coequalizer of f and g in C, whenever it exists. We work over a ground commutative ring with 1 denoted by k. Up to Section 4, all rings under consideration are k-algebras, and morphisms of rings are morphisms of k-algebras. Modules are assumed to be unital modules and bimodules are assumed to be central k-bimodules. For every ring R, these categories are denoted by R Mod (left modules), ModR (right modules) and R ModR (bimodules) respectively. The tensor product over R, is denoted as usual by − ⊗R −. We denote by Ch(R) the category of chain complexes of left R-modules. That is, complexes of left modules of the form: (M• , d• ) : · · ·

Mn /

dn

/

··· /

M2

d2

/

M1

d1

/

M0

d0

/

M−1

··· /

M−n /

d−n

/

··· .

Let Ch + (R) denote the full subcategory of Ch(R) consisting of positive chain complexes i.e. of complexes of the form: (M• , d• ) : · · ·

Mn /

dn

··· /

M2 /

d2

M1 /

d1

M0 . /

Similarly one defines the category of cochain complexes Coch(R) consisting of complexes of the form: (M• , d• ) : · · · /

M−n

d−n

/

··· /

M2

d2

/

M0

d0

/

M1

d1

/

M2

d2

/

··· /

Mn

dn

/

··· .

and its full subcategory of positive cochain complexes Coch + (R) consisting of complexes of the form: d0

(M• , d• ) : M0

/

M1

d1

/

M2

d2

/

··· /

Mn

dn

/

··· .

From now on, chain complex of left R-modules will stand for an object of the category Ch + (R). When R is commutative (i.e. commutative k-algebra), we will considered this category in a standard way as a monoidal category with unit object the chain complex R[0]• , where R[0]0 = R, and R[0]n = 0, for n > 0. Given an R-bimodule X, its k-submodule of R-invariant elements is denoted by n o X R := x ∈ X| xr = rx, ∀ r ∈ R .

This in fact defines a functor (−)R : R ModR → ModZ(R) , where Z(R) is the centre of R. As usual, we use the symbols HomR− (−, −), Hom−R (−, −) and HomR−R (−, −) to denote the Hom-functor of left R-linear maps, right R-linear maps and R-bilinear maps, respectively. All maps are acting on the left of their arguments. In this way, each right R-module M is considered as an (EndR (M), R)-bimodule, while each left R-module N is considered as (R, EndR (N))-bimodule. Thus the multiplication of the endomorphism ring of a left module is considered to be the opposite of composition. For two bimodules R PS and R QS over rings R and S, we will consider the k-modules of R-linear maps HomR− (P, Q) as an S-bimodule with actions: sf : p 7→ f (ps), and f s′ : p 7→ f (p)s′ , for every f ∈ HomR− (P, Q), s, s′ ∈ S, and p ∈ P. Similarly, Hom−S (P, Q) is considered as an R-bimodule with actions: rg : p 7→ rg(p), and gr ′ : p 7→ g(r ′p), for every g ∈ Hom−S (P, Q), r, r ′ ∈ R, and p ∈ P. Under these considerations, the left dual ∗ X = HomR− (X, R) of an R-bimodule X, is an R-bimodule, as well as its right dual X ∗ = Hom−R (X, R). Given modules XR , R Y , S V , and US , we consider in the obvious way X ⊗k U (resp. Y ⊗k V ) as right (resp. as left) (R ⊗k S)-module. Then we have a natural isomorphism (3) (X ⊗R Y ) ⊗k (U ⊗S V ) ∼ = (X ⊗k U) ⊗ (Y ⊗k V ). R⊗k S


7

Consider another bimodule S ZR , using the above actions, there is a map / HomS− Z ⊗R Y, V HomR− Y, R ⊗R HomS− Z, V (4) h i / z ⊗R y 7→ g(zf (y)) f ⊗R g

which becomes bijective whenever R Y is finitely generated and projective. For a fixed ring R, we denote by R-Rings the category of R-rings. This is the comma category over R in the category of all k-algebras. That is, objects are morphisms of rings R → A and morphisms are commutative triangles. Obviously, this category is identified with the category of monoids of the monoidal category of bimodules R ModR . Dually, one can define R-corings [35]. Thus, an R-coring is a comonoid in R ModR , which is by definition a three-tuple (C, ∆, ε) consisting of R-bimodule C and two R-bilinear maps ∆ : C → C ⊗R C (comultiplication), ε : C → R (counit) satisfying the usual coassociativity and counitary constraints. In contrast with coalgebras, corings admit several convolution rings. For instance, the right convolution of an R-coring C, is the right dual R-bimodule C∗ whose multiplication is defined by σ . σ ′ = σ ◦ (σ ′ ⊗R C) ◦ ∆, for all σ, σ ′ ∈ C∗ , and its unit is the counit ε of C. A morphism of R-corings is an R-bilinear map φ : C → C′ such that ∆′ ◦ φ = (φ ⊗R φ) ◦ ∆ and ε′ ◦ φ = ε. A left C-comodule is pair (N, λN ) consisting of left R-module N and left R-linear map λN : N → C⊗R N (coaction) compatible in the canonical way with comultiplication and counit. A morphism of left C-comodules is a left R-linear map which is compatible with coactions. We denote by C Comod the category of left C-comodules. Right comodules are similarly defined. Given any morphism of R-corings φ : C → C′ one can define, in the obvious way, a functor (−)φ : C Comod → C′ Comod refereed to as the coinduction functor. For more information on comodules as well as the definitions of bicomodules and cotensor product over corings, the reader is referred to [6]. For the notions of bialgebroids and their basic properties, the reader is referred to [7]. In Section 4, we will consider rings with enough orthogonal idempotents. These are central k-modules B with internal multiplication which admit a decomposition of k-modules B = ⊕p ∈P B1p = ⊕p ∈P 1p B, where {1p }p ∈P ( B is a set of orthogonal idempotents. Module over a ring with enough orthogonal idempotents stands for k-central and unital module. Recall that M is a left unital B-module provided that M has an associative left B-action which satisfies M = ⊕p ∈P 1p M. We denote by B Mod the category of left unital B-modules. 2. Monoidal Results. The constructions performed in the next section, as well as the results proved there, need some monoidal results which we recall with full details in this section. Given a monoidal functor R : C → B with a left adjoint L, we will construct a left adjoint Lm to the functor Rm which lifts R to the categories of monoids in the monoidal categories B, C. The construction of this left adjoint is a well known process, at least when the existence of inductive limits is guaranteed and the tensor product preserves them. However, we did not find in the literature a precise reference suitable for our needs. For sake of completeness, we include a detailed proof. 2.1. Coequalizers in the category of monoids. Recall that (see [25, Chap. XI]) a monoidal category is a category M endowed with an object 1 ∈ M (called unit), a functor ⊗ : M × M → M (called tensor product), and functorial isomorphisms aX,Y,Z : (X ⊗ Y ) ⊗ Z → X ⊗ (Y ⊗ Z), lX : 1 ⊗ X → X, rX : X ⊗ 1 → X, for every X, Y, Z in M. The functorial morphism a is called the associativity constraint and satisfies the Pentagon Axiom, that is the following relation (U ⊗ aV,W,X ) ◦ aU,V ⊗W,X ◦ (aU,V,W ⊗ X) = aU,V,W ⊗X ◦ aU ⊗V,W,X

8


holds true, for every U, V, W, X in M. The morphisms l and r are called the unit constraints and obey the Triangle Axiom, that is (V ⊗ lW ) ◦ aV,1,W = rV ⊗ W , for every V, W in M. It is well known that the Pentagon Axiom completely solves the consistency problem arising out of the possibility of going from ((U ⊗ V ) ⊗ W ) ⊗ X to U ⊗ (V ⊗ (W ⊗ X)) in two different ways (see [29, page 420]). This allows the notation X1 ⊗· · ·⊗Xn forgetting the brackets for any object obtained from X1 , · · · Xn using ⊗. Also, as a consequence of the coherence theorem, the constraints take care of themselves and can then be omitted in any computation involving morphisms in M. Thus, for sake of simplicity, we will omit in this subsection the associativity constraints. Recall a monoidal category is called strict whenever the associativity and unitary constraints are identities. A monoid in a monoidal category (M, ⊗, 1) is a three-tuple (A, mA , uA ) where A is an object of M and mA : A ⊗ A → A ( multiplication), uA : 1 → A (unit) are morphisms in M satisfying the usual associativity and unitary constraints. A morphism of monoids is a morphism in M which is compatible in the obvious way with multiplications and units. The category of monoids in M will be denoted by Mm . Dually, one can define the category of comonoids which we denote by Mc . Let (A, mA , uA ) be a monoid in a monoidal category (M, ⊗, 1) with coequalizers. For every morphism α : X → A in M, we set Λα := mA ◦ (mA ⊗ A) ◦ (A ⊗ α ⊗ A) : A ⊗ X ⊗ A −→ A. We say that ⊗ preserves coequalizers provided that, for every object Y ∈ M, the functors − ⊗ Y and Y ⊗ − preserve them. Lemma 2.1.1. Let (A, mA , uA ) be a monoid in a monoidal category (M, ⊗, 1) with coequalizers. Let α, β : X → A be morphisms in M and consider the coequalizer A⊗X ⊗A

Λα Λβ

/

/

π

A

/

B

in M. Assume that ⊗ preserve coequalizers. Then B carries a unique monoid structure such that π is a homomorphism of monoids. Proof. Consider the following diagram. A⊗A⊗X ⊗A

A⊗Λα A⊗Λβ

/ /

A⊗A

A⊗π

/

A⊗B mlB

mA

mA ⊗X⊗A

A⊗X ⊗A

Λα Λβ

/

/

A

π

/

B

The horizontal rows form two coequalizers. In fact the first row is obtained from the second one by applying the functor A ⊗ (−). We have mA ◦ (A ⊗ Λα ) = = = =

mA ◦ (A ⊗ mA ) ◦ (A ⊗ mA ⊗ A) ◦ (A ⊗ A ⊗ α ⊗ A) mA ◦ (A ⊗ mA ) ◦ (mA ⊗ A ⊗ A) ◦ (A ⊗ A ⊗ α ⊗ A) mA ◦ (A ⊗ mA ) ◦ (A ⊗ α ⊗ A) ◦ (mA ⊗ X ⊗ A) Λα ◦ (mA ⊗ X ⊗ A)

so that (5)

mA ◦ (A ⊗ Λα ) = Λα ◦ (mA ⊗ X ⊗ A)

and similarly (6)

mA ◦ (A ⊗ Λβ ) = Λβ ◦ (mA ⊗ X ⊗ A) .


9

Thus by the universal property of coequalizers, there exists a unique morphism mlB such that π ◦ mA = mlB ◦ (A ⊗ π) . Consider the following coequalizer. A⊗X ⊗A⊗B

Λα ⊗B

/

Λβ ⊗B

/

A⊗B mlB

w

π⊗B

/

B⊗B

mB

B Using a right-hand version of (5) and (6), we get

mlB ◦ (Λα ⊗ B) ◦ (A ⊗ X ⊗ A ⊗ π) = mlB ◦ (A ⊗ π) ◦ (Λα ⊗ A) = π ◦ mA ◦ (Λα ⊗ A) = π ◦ mA ◦ (Λβ ⊗ A) = mlB ◦ (A ⊗ π) ◦ (Λβ ⊗ A) = mlB ◦ (Λβ ⊗ B) ◦ (A ⊗ X ⊗ A ⊗ π) . Since A ⊗ X ⊗ A ⊗ π is an epimorphism, we obtain mlB ◦ (Λα ⊗ B) = mlB ◦ (Λβ ⊗ B) . Thus there is a unique morphism mB : B ⊗ B → B such that mB ◦ (π ⊗ B) = mlB . Then mB ◦ (π ⊗ π) = mB ◦ (π ⊗ B) ◦ (A ⊗ π) = mlB ◦ (A ⊗ π) = π ◦ mA . Since π is an epimorphism and A is a monoid, one easily checks that (B, mB , uB ) is a monoid, where uB := π ◦ uA . This is the unique monoid structure that makes π a homomorphism of monoids. Lemma 2.1.2. Let (A, mA , uA ) be a monoid in a monoidal category (M, ⊗, 1) with coequalizers. Let α : X → A be morphism in M. Then, identifying 1 ⊗ X ⊗ 1 with X, we have (7)

α = Λα ◦ (uA ⊗ X ⊗ uA )

and

τ ◦ Λα = Λτ ◦α ◦ (τ ⊗ X ⊗ τ )

for every monoid homomorphism τ : A → L in M. In particular, if β : X → A is a morphism in M, then τ ◦ Λα = τ ◦ Λβ implies τ ◦ α = τ ◦ β for every morphism τ : A → L in M. The converse is true whenever τ is a homomorphism of monoids. Proof. The left-hand side of (7) is trivial. Moreover for every homomorphism of monoids τ : A → L in M, we have τ ◦ Λα = τ ◦ mA ◦ (mA ⊗ A) ◦ (A ⊗ α ⊗ A) = mL ◦ (mL ⊗ L) ◦ (τ ⊗ (τ ◦ α) ⊗ τ ) = Λτ ◦α ◦ (τ ⊗ X ⊗ τ ) . The last part of the statement follows by (7).

Lemma 2.1.3. Let (A, mA , uA ) be a monoid in a monoidal category (M, ⊗, 1) with coequalizers. Assume that ⊗ preserves coequalizers. Let α, β : X → A and f, g : Y → A be morphisms in M. Assume that (8)

τ ◦ α = τ ◦ β ⇐⇒ τ ◦ f = τ ◦ g, for every monoid homomorphism τ : A → L in M.

Then CoeqM (Λα , Λβ ) = CoeqM (Λf , Λg ). In particular the monoid structures that these objects carry in view of Lemma 2.1.1 coincide. Proof. Let (E1 , π1 ) := CoeqM (Λα , Λβ ) and (E2 , π2 ) := CoeqM (Λf , Λg ).

10


By Lemma 2.1.1, E1 carries a unique monoid structure such that π1 is a monoid homomorphism and E2 carries a unique monoid structure such that π2 is a monoid homomorphism. Now, since π1 is a homomorphism of monoids we get π1 ◦ Λα = π1 ◦ Λβ ks +3

π1 ◦ α = π1 ◦ β ks

π1 ◦ f = π1 ◦ g ks +3

+3

π1 ◦ Λf = π1 ◦ Λg ,

where the first and the third equivalences are deduced from Lemma 2.1.2 while the second one is obtained from equation (8). By the universal property of (E2 , π2 ) , there is a unique morphism u : E2 → E1 in M such that u◦π2 = π1 . In a similar way there is a unique morphism v : E1 → E2 in M such that v ◦ π1 = π2 . Since π1 and π2 are epimorphisms in M, one gets that u and v are mutual inverses. Notation 2.1.4. Let (M, ⊗, 1) be a monoidal category. Then the categories of monoids and comonoids in M will be denoted by Mm and Mc respectively. Proposition 2.1.5. Let (M, ⊗, 1) be a monoidal category with coequalizers. Assume that ⊗ preserves coequalizers. Then the category Mm has coequalizers too. Explicitly, let α, β : E → A be homomorphisms of monoids in the category M. Then the coequalizer Λα

A⊗E⊗A

/

π

A

Λβ

/

/

B

(B, π) of (Λα , Λβ ) in M carries a unique monoid structure such that (B, π) is the coequalizer of (α, β) in the category Mm . Proof. Let α, β : E → A be homomorphisms in the category Mm , and consider the coequalizer Λα

A⊗E⊗A

Λβ

/

/

π

A

/

B

in M. By Lemma 2.1.1, B carries a unique monoid structure such that π is a monoid homomorphism. Let us prove that E

α β

/ /

A

π

/

B

is a coequalizer in Mm . First, by Lemma 2.1.2, we have (9)

τ ◦ α = τ ◦ β ⇐⇒ τ ◦ Λα = τ ◦ Λβ , for every homomorphism of monoids τ : A → L.

By applying (9) to the case τ = π we get π ◦ α = π ◦ β. Let τ : A → L be a monoid homomorphism such that τ ◦ α = τ ◦ β. By (9) we obtain τ ◦ Λα = τ ◦ Λβ so that there exists a unique morphism τ : B → L in M such that τ ◦ π = τ . Since π is an epimorphism in M and both π and τ are homomorphisms of monoids, then τ is also a homomorphism of monoids. Proposition 2.1.6. Let (M, ⊗, 1) be a monoidal category with equalizers. Assume that ⊗ preserves equalizers. Then the category Mc has equalizers too. Proof. Apply Proposition 2.1.5 to the dual category Mo .

2.2. An adjunction between the categories of monoids. Let (B, ⊗B , 1B ) and (C, ⊗C , 1C ) be monoidal categories. A monoidal functor from C to B is a triple (F, Φ2 , Φ0 ) where F : C → B is a functor, Φ0 : 1B → F (1C ) is a morphism and Φ2(−, −) : F (−) ⊗B F (−) → F (− ⊗C −) is a natural transformation defined by a family of morphisms Φ2(U, V ) : F (U) ⊗B F (V ) → F (U ⊗C V ) , for every U, V ∈ C


11

such that F (U )⊗Φ2(V, W )

F (U) ⊗B (F (V ) ⊗B F (W )) (10)

F (U) ⊗B F (V ⊗C W )

(F (U) ⊗B F (V )) ⊗B F (W )

Φ2(U, V ⊗W )

Φ2(U, V ) ⊗F (W )

F (U ⊗C (V ⊗C W ))

F (U ⊗C V ) ⊗B F (W )

(11)

/

r9 ∼ = rrrr rr rrr

1B ⊗B F (U)

Φ0 ⊗F (U )

F (1C ) ⊗B F (U) /

Φ2(1

∼ =

F (U)

/

∼ =

/

Φ2(U ⊗V, W )

9 rrr r r rr∼ rrr =

F ((U ⊗C V ) ⊗C W )

F (U) ⊗B 1B

F (U )⊗Φ0

/

F (U) ⊗B F (1C ) Φ2(U, 1

∼ =

C, U)

F (1C ⊗C U)

F (U)

∼ =

/

C)

F (U ⊗C 1C )

are commutative diagrams. A comonoidal functor from B to C is a monoidal functor from B to C o . The following lemma is a well known fact in monoidal categories. Some of the steps in its proof will be used in the sequel. So it is convenient to sketch the proof here. / Lemma 2.2.7. Let (B, ⊗B , 1B ) and (C, ⊗C , 1C ) be monoidal categories with adjunction L : B o C:R , where L is a left adjoint to R (notation L ⊣ R). Then L is comonoidal if and only if R is monoidal.

Proof. As explained above, there is no loss of generality if we assume that both B and C are strict monoidal categories. Henceforth, the isomorphisms in diagrams (10) and (11) can be assumed to be identities. Throughout this proof, both tensor functors will be denoted by ⊗. (⇐) Let Φ2(−, −) : R(−) ⊗ R(−) −→ R(− ⊗ −) and Φ0 : 1B −→ R(1C ) be the structure morphisms of the monoidal functor R. Let us denote by η− : idB −→ RL ,

ξ− : L R −→ idC

the unit and the counit of the stated adjunction. We set Ψ2

(X, Y ) L (X ⊗ Y ) _ _ _ _ _ _ _ _ _ _ _ _ _ _/ L (X) ⊗ L (Y )

(12)

O

ξL (X)⊗L (Y )

L (ηX ⊗ηY )

L RL (X) ⊗ RL (Y )

L (Φ2(L (X), L (Y )) )

/ L R L (X) ⊗ L (Y )

0

Ψ L (1B ) _ _ _ _ _q8/ 1C q L (Φ0 )

qq qqq q q ξ qqq 1C

L R(1C )

for every pair of objects X and Y in B. A direct computations show that (L , Ψ2 , Ψ0 ) is a comonoidal functor. (⇒) Apply the previous implication to the dual categories. The following result was announced by D. Tambara in [37, Remark 1.5] with no proof. Since its applications in the forthcoming section are crucial, we will give here a detailed proof. Theorem 2.2.8. Let (B, ⊗B , 1B ) and (C, ⊗C , 1C ) be a monoidal categories. Let L ⊣ R be an adjunction where R : C → B is a monoidal functor with structure morphisms Φ2(−,−) and Φ0 . Then R induces a functor Rm : Cm → Bm between the associated categories of monoids.

12


Assume that C has inductive limits and that the tensor product preserves them. Then Rm has a left adjoint Lm : Bm → Cm . Proof. It is straightforward to prove that any monoidal functor can be restricted to the respective categories of monoids. Thus R can be restricted to Rm . Now, observe that, by [27, Theorem 2, page 172], the forgetful functor H : Cm → C has a left adjoint T : C → C m where T (X) = 1C ⊕ X ⊕ (X ⊗C X) ⊕ ((X ⊗C X) ⊗C X) ⊕ · · · is the tensor monoid of X in the category C. Let X ⊗C t be defined recursively by setting X ⊗C 0 := 1C and ⊗C t → T (X) the canonical monomorphism. X ⊗C t := X ⊗C (t−1) ⊗C X, when t > 0. Denote by iX t : X Let (B, mB , uB ) be an object in Bm . Let α1 , β1 : L (1B ) → T L (B) and α2 , β2 : L (B ⊗ B) → T L (B) be defined by L (B)

α1 = α1B := i0 α2 = α2B :=

L (B) i2

◦ Ψ0

L (B)

β1 = β1B := i1

and

◦ Ψ2(B,B)

β2 = β2B :=

and

◦ L (uB )

L (B) i1

◦ L (mB )

respectively, where Ψ2(−,−) and Ψ0 are the structure morphisms of L defined in the proof of Lemma 2.2.7, see equation (12) . By the universal property of the tensor monoid there are unique homomorphisms of monoids f1 , g1 : T L (1B ) → T L (B) and f2 , g2 : T L (B ⊗ B) → T L (B) such that L (1B )

f1 ◦ i1

= α1 ,

L (1B )

g1 ◦ i1

= β1

L (B⊗B)

f2 ◦ i1

and

= α2 ,

L (B⊗B)

g2 ◦ i1

= β2 .

Since C has coequalizers, by Proposition 2.1.5, the pair (f1 , g1 ) admits a coequalizer (γ1 : T L (B) → E1 ) in Cm . Note that, by definition (13)

(E1 , γ1) = CoeqC (Λf1 , Λg1 ) = CoeqC (Λα1 , Λβ1 ) ,

where the second equality is obtained by Lemma 2.1.3 (in fact (8) holds in this case in view of the universal property of the tensor monoid). By Proposition 2.1.5, the pair (γ1 ◦ f2 , γ1 ◦ g2 ) admits a coequalizer (γ2 : E1 → E2 ) in Cm . We set (EB , πB : T L (B) → E) := (E2 , γ2 ◦ γ1 ), so we have the following commutative diagram of coequalizers in the category of monoids Cm πB

T L (1B )

f1 g1

/

/

γ1

T L (B) O O

f2

g2

/

EJ 1 J

γ2

/

#

E2 := EB

γ1 ◦f2 γ1 ◦g2

T L (B ⊗ B) Now, by definition, we have (13)

(E2 , γ2) = CoeqC (Λγ1 ◦f2 , Λγ1 ◦g2 ) = CoeqC (Λγ1 ◦α2 , Λγ1 ◦β2 ) = CoeqC (γ1 ◦ Λα2 , γ1 ◦ Λβ2 ) , where the last equality holds as γ1 ◦ Λα2 = Λγ1 ◦α2 ◦ (γ1 ⊗ L (1B ) ⊗ γ1 ) ,

γ1 ◦ Λβ2 = Λγ1 ◦β2 ◦ (γ1 ⊗ L (1B ) ⊗ γ1 )

and γ1 ⊗ L (1B ) ⊗ γ1 is an epimorphism in C. We have so proved that (E1 , γ1 ) = CoeqC (Λα1 , Λβ1 )

and

(E2 , γ2 ) = CoeqC (γ1 ◦ Λα2 , γ1 ◦ Λβ2 ) .

From these equalities one can prove that (EB , πB ) is the universal object coequalizing in C at the same time both the pairs (Λα1 , Λβ1 ) and (Λα2 , Λβ2 ). Define Lm : Bm → Cm by setting Lm (B) := EB . By Lemma 2.1.1 EB admits a structure of monoid for which πB is a morphism of monoids. For every morphism h : B → B ′


13

in Bm , Lm (h) is defined to be the unique monoid homomorphism such that Lm (h) ◦ πB = πB′ ◦ T L (h). Such a morphism exists once proved that πB′ ◦ T L (h) equalizes both the pairs (ΛαB1 , Λβ1B ) and (ΛαB2 , Λβ2B ). In view of Lemma 2.1.2, this amounts to prove that πB′ ◦ T L (h) equalizes both the pairs α1B , β1B and α2B , β2B . Now, we have ◦ Ψ0 = i0

L (B)

◦ L (uB ) = i1

L (B)

◦ Ψ2(B,B) = i2

T L (h) ◦ β1B = T L (h) ◦ i1

T L (h) ◦ α2B = T L (h) ◦ i2 L (B ′ )

= i2

L (B ′ )

L (B)

T L (h) ◦ α1B = T L (h) ◦ i0

′

◦ Ψ0 = α1B ,

L (B ′ )

L (B ′ )

L (B ′ )

◦ L (h) ◦ L (uB ) = i1

′

◦ L (uB′ ) = β1B ,

◦ (L (h) ⊗C L (h)) ◦ Ψ2(B,B) ′

◦ Ψ2(B′ ,B′ ) ◦ L (h ⊗B h) = α2B ◦ L (h ⊗B h) , L (B)

T L (h) ◦ β2B = T L (h) ◦ i1

L (B ′ )

◦ L (mB ) = i1

L (B ′ )

◦ L (h) ◦ L (mB ) ′

◦ L (mB′ ) ◦ L (h ⊗B h) = β2B ◦ L (h ⊗B h) . ′ ′ ′ ′ Using these equalities and since πB′ equalizes both α1B , β1B and α2B , β2B , we obtain = i1

′

′

πB′ ◦ T L (h) ◦ α1B = πB′ ◦ α1B = πB′ ◦ β1B = πB′ ◦ T L (h) ◦ β1B , ′

′

πB′ ◦ T L (h) ◦ α2B = πB′ ◦ α2B ◦ L (h ⊗B h) = πB′ ◦ β2B ◦ L (h ⊗B h) = πB′ ◦ T L (h) ◦ β2B . Thus, we obtain that πB′ ◦ T L (h) equalizes both pairs α1B , β1B and α2B , β2B so that there is a unique morphism Lm (h) such that Lm (h) ◦ πB = πB′ ◦ T L (h) . Since πB′ ◦ T L (h) and πB are monoid homomorphisms and πB is an epimorphism in C, one easily obtains that Lm (h) is a monoid homomorphism too. Let us check that Lm is a left adjoint of Rm . So let (C, mC , uC ) be an object in Cm . Denote by ξ and η the counit and the unit of the adjunction (L , R) respectively. Denote by ϕC : T (C) → C the unique monoid homomorphism that restricted to C gives the identity. Let us check that ϕC ◦ T (ξC ) ◦ ΛαR(C) = ϕC ◦ T (ξC ) ◦ Λβ R(C) 1

1

and

ϕC ◦ T (ξC ) ◦ ΛαR(C) = ϕC ◦ T (ξC ) ◦ Λβ R(C) . 2

2

Since ϕC ◦ T (ξC ) is homomorphism of monoids, by Lemma 2.1.2, we have to prove that (14) (15)

R(C)

ϕC ◦ T (ξC ) ◦ α1 ϕC ◦ T (ξC ) ◦

R(C) α2

R(C)

= ϕC ◦ T (ξC ) ◦ β1 = ϕC ◦ T (ξC ) ◦

and

R(C) β2 .

Equalities (14) and (15) are easily derived from definitions. By the universal property of ER(C) , πR(C) , there is a unique morphism

(16)

ξCm : ER(C) = Lm Rm (C) −→ C such that ξCm ◦ πR(C) = ϕC ◦ T (ξC ) .

Clearly ξCm is a homomorphism of monoids. For every (B, mB , uB ) ∈ Bm define L (B) (17) ηBm : B → Rm Lm (B) = R (EB ) by ηBm := R (πB ) ◦ R i1 ◦ ηB .

A routine verifications show that ηBm is a homomorphism of monoids. m m : idBm → Rm Lm are natural transformations. We leave to By definitions ξ− : Lm Rm → idCm and η− the reader to check that they satisfy the triangles equalities which make (Lm , Rm ) an adjunction. Remark 2.2.9. The construction of the left adjoint functor performed in the proof of Theorem 2.2.8 can be simplified when B has also inductive limits, tensor product preserves them and L commutes with direct sums. Effectively, for every monoid (B, mB , uB ) in B, this left adjoint is defined to be the coequalizer in the following diagram T L T (B) /

/

T L (B) /

Lm (B, mB , uB )

14


where T denotes the left adjoint of the forgetful functor (i.e. the free monoid functor) both for both C and B. One of the coequalized maps is given by the extension T (B) → B of the identity map B → B. The other one is constructed as follows from the natural transformation Ψ2−,− of diagram (12) L (T (B)) ∼ = ⊕n≥0 L (B ⊗n )

ˆ n,B ⊕n≥0 Ψ

⊕n≥0 L (B)⊗n = T (L (B)) /

ˆ n,B : L (B ⊗n ) → L (B)⊗n denotes the n-iteration of Ψ2 . where, for every n ≥ 0, Ψ B,B 2.3. General examples. Apart from the main construction, which we will discuss in the forthcoming section, we present here another simple application of Theorem 2.2.8. Example 2.3.10. Let A and B be two Grothendieck categories. We denote by Funct (A, B) the setcategory of continuous additive functors from A to B (i.e. functors which commute with inductive limits, or equivalently, which are right exact and commute with direct sums). The category Funct (A, A) is a strict monoidal category where the unit is the identity functor on A and the tensor product is the composition of functors. / Assume that there is an adjunction F : A o B : G with F ⊣ G, and F ∈ Funct (A, B), G ∈ Funct (B, A). Let θ : F G → idB and η : idA → GF be, respectively, the counit and unit of this adjunction. One can easily check that the following functor R

Funct (B, B)

/ GHF h i / GσF : GHF → GH ′ F

H i h σ : H → H′

is a monoidal functor with structure maps Φ2H, H ′ : R(H)R(H ′)

GHθH ′ F

Funct (A, A) /

/

R(HH ′) ,

Φ0 : idA

η

/

GF = R(idB )

and that the functor L

Funct (A, A)

/

Funct (B, B)

/ FTG h i / F αG : F T G → F T ′ G

T i h α : T → T′

is left adjoint to R. Since Funct (B, B) has cokernels and direct sums, it has inductive limits, and of course they are preserved by the tensor product. Therefore, we can assert, using Theorem 2.2.8, that the adjunction L ⊣ R gives rise to a new adjunction between the categories of monoids Funct (B, B)m and Funct (A, A)m . These are the categories of continuous endo-monads, respectively, on B and A. For instance, let A be the category of right A-modules ModA , B the category of right B-modules ModB , F = −⊗A M and G = − ⊗B M ∗ (here M ∗ stands for the dual module of MB ), for some (A, B)-bimodule M which is finitely generated and projective as right B-module. Then, by Eilenberg-Watts Theorem, we get the following (probably well known) adjunction between the categories of A-rings and B-rings (i.e. the categories of ring extensions) / TB (M ∗ ⊗A − ⊗A M)/I− : A-Rings o B-Rings : M ⊗B − ⊗B M ∗ where the left hand functor sends any A-ring C (i.e. an algebra map A → C) to the quotient of the tensor B-ring TB (M ∗ ⊗A C ⊗A M) by the two-sided ideal IC =

where

*

′

ϕ ⊗A 1C ⊗A m − 1B ϕ(m); ϕ ⊗A cc ⊗A m −

{mi , m∗i }i

X i

denotes the dual basis of MB .

(ϕ ⊗A c

⊗A mi ) ⊗B (m∗i

′

+

⊗A c ⊗A m)

ϕ ∈M ∗ , m ∈M, c,c′ ∈C


15

/ Example 2.3.11. Let B and C two monoidal categories, with adjunction L : B o C : R (L ⊣ R) S S such that R is a monoidal functor. We denote by B and C , the functors categories with domain a small category S and value, respectively, in B and C. One can easily check that this adjunction induces an adjunction on the functors categories

L S : BS o

/

BS : R S

with L S ⊣ R S and where this functors are obviously defined using composition of functors. On the other hand, we endow this categories, in the canonical way, with a monoidal structure. That is, for f ∈ BS and g ∈ BS , we set (18) f ⊗BS g (c) := f(c) ⊗B g(c), and f ⊗BS g (j) := f(j) ⊗B g(j)

for every arrow j and object c in S. In the same way we consider C S as monoidal category. Assume now that C has inductive limits and that the tensor product preserves them. Then one can show that C S inherits the same properties. Since R is a monoidal functor, R S is monoidal too. Therefore, by Theorem 2.2.8 we can construct the left adjoint of Rm : Cm → Bm and of (R S )m : (C S )m → (BS )m . This left adjoint functors are related as follows. If Lm is the left adjoint of Rm , then (L S )m acts on objects by sending any monoid (B, m, u) ∈ (BS )m to the monoid functor which sends any object c ∈ S to the monoid Lm (B(c), mc , uc ) (recall that here m and u are natural transformations). Thus, we have (L S )m = (Lm )S . 3. The construction of coendomorphism bialgebroids. In this section we construct the coendomorphisms left bialgebroid and give several examples. Let R → A be a ring extension and assume that R A is finitely generated and projective. We first show that the monoidal functor − ×R A : Re ModRe → R ModR has a left adjoint functor, where Re = R ⊗ Ro is the enveloping ring and − ×R − is the Sweedler-Takeuchi product [34, 36]. This allows us to apply the theory developed in Section 2. Thus, we can construct using Theorem 2.2.8, a functor Lm : R-Rings → Re -Rings between the categories of ring extensions which is left adjoint to − ×R A : Re -Rings → R-Rings. We then prove that the image of A under this functor, i.e. Lm (A), admits a structure of left R-bialgebroid such that A becomes a left Lm (A)-comodule. We also clarify the monoidal structure of the category of left comodules of the underlying R-coring of Lm (A). Each one of these results will be crucial in proving our main theorems in the next section. From now on, the unadorned symbol ⊗ stands for the tensor product over the ground ring k. 3.1. An adjunction between Re -bimodules and R-bimodules. Let R be a ring. For any r ∈ R, we denote by r o the same element regarded as an element in the opposite ring Ro . Let Re := R ⊗ Ro be the enveloping ring of R. Given an Re -bimodule M, the underlying k-module M admits several structures of R-bimodule. Among them, we will select the following two ones. The first structure is that of the opposite bimodule 1⊗Ro M1⊗Ro which we denote by M o . That is, the R-biaction on M o is given by o o o o o o (19) r m = m (1 ⊗ r ) , m s = (1 ⊗ s ) m , mo ∈ M o , r, s ∈ R. Notice, that this construction defines in fact a functor (−)o : Re ModRe → R ModR . The second structure is defined by the left Re -module Re M. That is, the R-bimodule M l = R⊗1o MR whose R-biaction is defined by l l (20) r ml = (r ⊗ 1o )m , ml s = (1 ⊗ so )m , ml ∈ M l , r, s ∈ R.

16


This also defines a functor, namely, the right Re -action forgetful functor (−)l : easily observes that there is a commutative diagram: (21)

(−)l

Re ModRe

Re ModRe

→ R ModR . One

R ModR

/

(−)o

(−)R

R ModR

ModR , /

(−)R

where (−)R is the left R-action forgetful functor. Another Re -bimodule derived from M, which will be used in the sequel, is M † . The underlying k-module of M † is M and an element m ∈ M is denoted by m† when it is viewed in M † . The Re -biaction on M † is given by (22) (p ⊗ q o ) m† (r ⊗ so ) = (p ⊗ r o ) m (q ⊗ so ) † , m† ∈ M † , p, r ∈ R, q o , so ∈ Ro . Here also we have a functor (−)† : Re ModRe → Re ModRe which has the following properties † † † † e e e e e e e e and Hom = Hom M = (M ) M , U M, U , R −R R −R R R R R for every pair of Re -bimodules U and M. Furthermore, there is a commutative diagram

(23)

(−)o

Re ModRe

R ModR

/

(−)†

Re ModRe

/

(−)Re

ModRe ,

where as before (−)Re denotes the left Re -action forgetful functor. It is clear that the left module Re M † induces the already existing R-bimodule structure of R⊗1o MR⊗1o . Now, let N be another R-bimodule, and consider the tensor product M o ⊗R N. The additive k-submodule of invariant elements ( ) X X X (M o ⊗R N)R = rmoi ⊗R ni = moi ⊗R ni r, for all r ∈ R moi ⊗R ni | i

i

i

admits a structure of an R-bimodule given by the actions: ! o X X r⇀ moi ⊗R ni = (r ⊗ 1o ) mi ⊗R ni , (24) i

(25) P

X

moi ⊗R ni

i

o i mi

!

o

i

↼s =

X

mi (s ⊗ 1o )

i

o

⊗R ni ,

for every set of elements ⊗R ni ∈ M ⊗R N and r, s ∈ R. In this way, to each R-bimodule N one associates two functors: R ∗ † o / − ⊗ N : R ModR , Mod e e (−) ⊗R N : R ModR R R

/

Re ModRe ,

where, for each R-bimodule X, we consider X ⊗ ∗ N as an Re -bimodule with the following actions ! X X o (p ⊗ q ) xi ⊗ ϕi (r ⊗ so ) = (p xi q) ⊗ (s ϕi r),

for every element

P

i

i

∗

i

xi ⊗ ϕi ∈ X ⊗ N, p, q, r, s ∈ R. These functors are related as follows.


17

Lemma 3.1.1. Let N be an R-bimodule such that R N is finitely generated and projective module with left dual basis {(ej , ∗ ej )}1≤j≤m ⊂ N × ∗ N. There is a natural isomorphism / HomRe −Re (X ⊗ ∗ N )† , M h i / (x ⊗ ϕ)† 7−→ (M o ⊗R ϕ) ◦ σ(x)

HomR−R X, (M o ⊗R N )R σ

i o h P x 7−→ j α (x ⊗ ∗ ej )† ⊗R ej o

α

for every R-bimodule X and Re -bimodule M. Equivalently, the functor (− ⊗ ∗ N)† is left adjoint to the functor ((−)o ⊗R N)R . Proof. By the isomorphism (M o ⊗R N)R ∼ = Hom−Re (R, M o ⊗R N) of k-modules, the right hand object e inherits a structure of left R -module coming from the actions ⇀, ↼ defined in (24) and (25). This left Re -action is explicitly given by the formula: (p ⊗ q o )α (1) = p ⇀ α(1) ↼ q,

for every p, q ∈ R, and α ∈ Hom−Re (R, M o ⊗R N). Since R N is finitely generated and projective, we have a k-linear isomorphism ∼ h i = f : Hom−Re R, M o ⊗R N −→ Hom−Re ∗ N, M † , α 7−→ ϕ 7→ (M o ⊗R ϕ) ◦ α(1) , P with inverse map f −1 (σ)(1) = j σ(∗ ej )o ⊗R ej , for every σ ∈ Hom−Re (∗ N, M † ) (recall that the underlying right Re -module of the Re -bimodule M † is M o , see diagram (23)). One can show that f is left Re -linear, where Hom−Re (∗ N, M † ) is left Re -module by the Re -bimodule structure of M † . We then obtain the following chain of natural isomorphisms HomRe − X, (M o ⊗R N )R

∼ =

/ HomRe − X, Hom−Re R, M o ⊗R N

∼ =

/ HomRe − X, Hom−Re ∗ N, M † ∼ =

∼ o R _ _ _ _ _ _ _ _ _ _ _ _=_ _ _ _ _ _ _ _ _ _ _ _/ HomR−R X, (M ⊗R N ) HomRe −Re (X ⊗ ∗ N ), M † ,

where the right vertical isomorphism is the usual Tensor-Hom adjunction. Since the functor (−)† is self adjoint, the right hand term in the second row becomes HomRe −Re (X ⊗ ∗ N)† , M so that we get the desired natural isomorphism. 3.2. The bi-functor (− ×R −) on Re -bimodules. As we have seen previously in Subsection 3.1, there is a bi-functor R − ×R − := (−)o ⊗R − : Re ModRe × R ModR −→ R ModR .

This is Sweedler-Takeuchi’s product of bimodules [34], [36], which can be also redefined using the notion of ends (limits) and coends (colimits), see [27, pages 222 and 226].P Given an Re -bimodule M and an R-bimodule N, an element i moi ⊗R ni which belongs to M ×R N will P be denoted by i mi ×R ni . Thus, for every r ∈ R and m ×R n ∈ M ×R N, we have o o (26) m (1 ⊗ r ) ×R n = m ×R nr, and (1 ⊗ r ) m ×R n = m ×R rn. With this notation the left Re -action on M ×R N defined in (24) and (25) can be written as follows: ! X X (27) (r ⊗ so ) mi ×R ni = (r ⊗ 1o ) mi (s ⊗ 1o ) ×R ni ,

for every elements

P

i

i

i

mi ×R ni ∈ M ×R N and r, s ∈ R.

18


Next, we want to restrict the bi-functor (− ×R −) to Re ModRe × Re ModRe , the product category of Re bimodules. As one can realize there are many ways to do that. That is, if N is an Re -bimodule, then there are several structures of R-bimodules on N over which one can construct M ×R N. Here we define M ×R N by using the R-bimodule R⊗1o NR⊗1o . In this way, M ×R N admits a structure of Re -bimodule: Using the above left Re -action (27), we obtain an Re -biaction ! X X (28) (r ⊗ so ) mi ×R ni (p ⊗ q o ) = (r ⊗ 1o ) mi (s ⊗ 1o ) ×R (1 ⊗ po ) ni (1 ⊗ q o ) , i

i

P

for every elements i mi ×R ni ∈ M ×R N and r, s, p, q ∈ R. Whence the Re -biaction on (M ×R N)† is given by the formula: ! ! X X † (29) (r ⊗ so ) mi ×R ni † (p ⊗ q o ) = (r ⊗ 1o ) mi (p ⊗ 1o ) ×R (1 ⊗ so ) ni (1 ⊗ q o ) . i

i

From now on, the restriction of the bi-functor (− ×R −) to Re ModRe × Re ModRe will be understood as the following compositions of functors: Re ModRe

(−)o

× Re ModWRe

W W W W W W W W W W W W R W−× W RW−W ⊗ R⊗1o (−)R⊗1o W W W R W W W W W W W W W + / e Mod e R

R

(−)†

Re ModRe ,

and this will be our definition for ×R -product of Re -bimodules. That is, for two bimodules Re MRe and Re NRe , we set R † M ×R N := M ⊗R N , o where R MR = 1⊗Ro M1⊗Ro and R NR = R⊗1o NR⊗1o . Thus, (26) reads as (30) m (1 ⊗ r o ) ×R n = m ×R n (r ⊗ 1o ), and (1 ⊗ r o ) m ×R n = m ×R (r ⊗ 1o ) n, and (29) as (31)

(p ⊗ q o ) m ×R n (r ⊗ so ) =

(p ⊗ 1o ) m (r ⊗ 1o ) ×R (1 ⊗ q o ) n (1 ⊗ so )) ,

for every r, s, p, q ∈ R and m ×R n ∈ M ×R N. On the other hand, since we have MRo = MRl for every Re -bimodule M, there is a canonical natural transformation (injective at least as k-linear map) ΘM, N : M ×R N

(32)

/

M l ⊗R N l .

Now, given another Re -bimodule W , there are three Re -bimodules under consideration. Namely, M ×R (N ×R U), (M ×R N) ×R U, and M ×R N ×R W . The later is constructed as follows: First we consider the underlying left Re -module of N, that is, N l = Re N which we consider obviously as an R-bimodule, see diagram (21). Secondly, we construct the k-module M o ⊗R N l ⊗R W using the left R-module R⊗1o W . This is an Re -bimodule with actions ! X X (33) (r ⊗ to ) moi ⊗R nli ⊗R wi (p ⊗ q o ) = rmoi ⊗R (ni (p ⊗ q o ))l ⊗R wi (t ⊗ 1o ), i

i


19

P

for every elements i moi ⊗R nli ⊗R wi ∈ M o ⊗R N l ⊗R W and p, q, r, t ∈ R. Lastly, M ×R N ×R W is defined to be the Re -invariant submodule with respect to the Re -biaction (33), that is, Re = M ×R N ×R W = M o ⊗R N l ⊗R W ( ) X X X moi ⊗R nli ⊗R wi | rmoi ⊗R nli ⊗R w(s ⊗ 1o ) = moi ⊗R (ni (r ⊗ so ))l ⊗R w, for all r, s ∈ R . i

i

i

The k-module M ×R N ×R W admits a structure of an Re -bimodule given by ! X X (r ⊗ so ) mi ×R ni ×R wi (p ⊗ q o ) = (r ⊗ 1o )mi (p ⊗ 1o ) ×R ni ×R (1 ⊗ so )wi (1 ⊗ q o ) , i

i

P

for every elements i mi ×R ni ×R wi ∈ M ×R N ×R W and r, s, p, q ∈ R. As before there is a canonical natural transformation (injective at least as k-linear map) ΞM, N, W : M ×R N ×R W

(34)

/

M l ⊗R N l ⊗R W l .

The bi-functor − ×R − is not associative. However, the are natural Re -bilinear maps ! ! X X X αl : (M ×R N) ×R W −→ M ×R N ×R W, mij ×R nij ×R wi 7−→ mij ×R nij ×R wi , i

X

αr : M ×R (N ×R W ) −→ M ×R N ×R W,

j

i,j

mi ×R

X

i

nij ×R wij

j

The following lemma will be used in the sequel.

!

7−→

X

mi ×R nij ×R wij

i,j

!

.

Lemma 3.2.2. Let N be an R-bimodule such that R N is finitely generated and projective with left dual basis {(ej , ∗ ej )}1≤j≤m ⊂ N × ∗ N. Consider the bimodule Re NRe = (N ⊗ ∗ N)† . Then there is a well defined map ! X N −→ N ×R N ×R N, n 7−→ (n ⊗ ∗ ej )† ×R (ej ⊗ ∗ ei )† ×R ei . i,j

Proof. Straightforward.

Another useful natural transformation of Re -bimodules is given as follows, see [32, p. 206]: For every R -bimodules M, M ′ , N, N ′ , we have an Re -bilinear map: e

(35)

(M ×R M ′ ) ⊗Re (N ×R N ′ ) P P ′ ′ e m × m ⊗ n × n i R R R i j i j j

τ

(M ⊗Re N) ×R (M ′ ⊗Re N ′ ) P ′ ′ / i,j (mi ⊗Re nj ) ×R (mi ⊗Re nj ). /

In this way, S ×R T is an Re -ring whenever S and T are. Precisely, the multiplication of S ×R T is defined using the map τ of equation (35), and explicitly given by ! ! X X X xi ×R yi uj ×R vj = xi uj ×R yi vj , i

P

j

P

i,j

for every pair of elements i xi ×R yi and j uj ×R vj in S ×R T . The unit is the map Re −→ S ×R T which sends p ⊗ q o 7−→ ((p ⊗ 1o ) 1S ) ×R (1T (1 ⊗ q o )).

20


It is clear that the k-linear endomorphisms ring Endk (R) is an Re -ring via the map ̺ : Re → Endk (R) which sends p ⊗ q o 7→ [r 7→ p r q]. Given a pair of bimodules Ro MRo and R NR , there are two bilinear maps, see [36, §2] θr : M ×R Endk (R) P i mi ×R fi

/

/

P

i

θl : Endk (R) ×R N P j gj ×R nj

M,

fi (1)o mi

/

P

/

N j gj (1) nj .

If M and N are two Re -bimodules, then θr and θl are defined using the underlying bimodules and R⊗1o NR⊗1o , and both maps are Re -bilinear. That is, ! ! X X θr mi ×R fi = (1 ⊗ fi (1)o )mi , and θl gj ×R nj = (gj (1) ⊗ 1o ) nj . i

1⊗Ro M1⊗Ro

j

Recall from [36, §4, Definition 4.5] (see also [5] and [32]) the definition of ×R -bialgebra. A ×R -coalgebra is an Re -bimodule C together with two Re -bilinear maps ∆ : C → C ×R C (comultiplication) and ε : C → Endk (R) (counit) such that the diagrams C ×R C

C

6 nnn ∆ nnnn nn nnn nnn PPP PPP PPP P ∆ PPPPP (

∆×R C

/

(C ×R C) ×R C

UUUU UUUUαl UUUU UUUU U*

C ×R C ×R C

C ×R C

C×R ∆

C ×R C o

/

i4 iiii i i i i iiiiαr iiii

C ×R (C ×R C)

∆

∆

C

/

C ×R C C×R ε

ε×R C

Endk (R) ×R C

θl

/

Co

θr

C ×R Endk (R)

are commutative. A ×R -coalgebra C is said to be an ×R -bialgebra provided that comultiplication and counit are morphisms of Re -rings. A left ×R -C-comodule, is a pair (X, λX ) consisting of an R-bimodule X and an R-bilinear map λX : X → C ×R X satisfying, in the sense of the previous diagrams, the coassociativity and counitary axioms. Morphism between left ×R -C-comodules are R-bilinear maps compatible in the obvious way with the left ×R -C-coactions. This leads to the definition of the category of left ×R -C-comodules. When C is a ×R bialgebra, this category becomes a monoidal category [32, Proposition 5.6], and the forgetful functor to the category of R-bimodules is a monoidal functor. There is a strong relation between the category of left ×R -comodules over an ×R -bialgebra and the category of left comodules over the underlying R-coring whose structure maps are C

/

C ×R C

ΘC,C

/

Cl ⊗R Cl ,

C

ε(−)(1R )

/

R,

where Θ−,− is the natural transformation of (32). We will analyze this relation in more detail in Subsection 3.4. 3.3. The ×R -bialgebra Lm (A). Let A be an R-ring. Using the bifunctor of 3.2, we get a functor −×R A : e Re ModRe → R ModR . Now, for every pair of R -bimodules M and N, we have well defined and R-bilinear


21

maps: (36)

Φ2(M, N)

(M ×R A) ⊗R (N ×R A)

(M ⊗Re N) ×R A,

(m ×R a) ⊗R (n ×R a′ )

/ /

R r

(m ⊗Re n) ×R aa′

Φ0

Re ×R A / (r ⊗ 1o ) ×R 1A , /

where Φ2(−,−) is obviously a natural transformation. Lemma 3.3.3. Let A be an R-ring. Then − ×R A : Re ModRe → R ModR is a monoidal functor. Proof. One need to show that the maps Φ2(−,−) and Φ0 of (36) satisfy the commutativity of diagrams (10) and (11). These are routine verifications. From now on, we assume that our R-ring A is finitely generated and projective as left R-module. We fix a left dual basis {(ej , ∗ ej )}1≤j≤n ⊂ A × ∗ A. By Lemma 3.1.1, R = − ×R A : Re ModRe −→ R ModR is a right adjoint to the functor L = (− ⊗ ∗ A)† : R ModR −→ Re ModRe . The unit and counit of this adjunction are explicitly given as follows. For any R-bimodule X and any Re -bimodule U, the unit at the object X is given by (37)

ηX

X x

while the counit at U is given by

/

RL (X) = (X ⊗ ∗ A)† ×R A, P ∗ † / j (x ⊗ ej ) ×R ej ,

L R(U) = (U ×R A) ⊗ ∗ A †

(38)

((u ×R a) ⊗ ϕ)†

By Lemma 2.2.7, the functor L : R ModR → using (12), (37) and (38), are given by (X ⊗R Y ) ⊗ ∗ A † (x ⊗R y) ⊗ ϕ †

Ψ2(X, Y )

/

/

Re ModRe

ξU

/

/

U

(1 ⊗ ϕ(a)o )u.

is a comonoidal functor whose structures maps,

(X ⊗ ∗ A)† ⊗Re (Y ⊗ ∗ A)† ,

P

∗ † † e i (x ⊗ ej ϕ) ⊗R (y ⊗ ej )

(R ⊗ ∗ A)† (r ⊗ ϕ)†

Ψ0

/

/

Re

r ⊗ ϕ(1A )o ,

for every pair of R-bimodules X and Y . By Theorem 2.2.8 and Lemma 3.3.3, the adjunction L ⊣ R restricts to the categories of ring extension. That is, we have an adjunction (39)

Lm : R-Rings o

/

Re -Rings : Rm .

For a given R-ring C, i.e. a k-algebra map R → C, the Re -ring Lm (C) is defined, as seen in the proof of Theorem 2.2.8, by the quotient algebra (40) Lm (C) = TRe L (C) /IL (C)

22


where TRe L (C)

=

L

n

L (C)⊗Re is the tensor algebra of the Re -bimodule L (C) = (C ⊗ ∗ A)† and

n∈N

where IL (C) is the two-sided ideal generated by the set ) ( X (41) (c ⊗ ei ϕ)† ⊗Re (c′ ⊗ ∗ ei )† − (cc′ ⊗ ϕ)† ; 1R ⊗ ϕ(1A )o − (1C ⊗ ϕ)† i

c, c′ ∈ C,

. ϕ∈ ∗ A

We denote by πC : TRe (L (C)) → Lm (C) the canonical projection. From now on, given a homogeneous elements (c ⊗ ϕ)† ∈ TRe (C) of degree one, we denote by πC (c ⊗ ϕ) its image in the Re -ring Lm (C). That is, throughout this section we will drop the symbol dag in the upper indices, and consider C ⊗ ∗ A as an Re -bimodule with its dag biaction, see (22). Now, using the proof of Theorem 2.2.8, precisely (16) and (17), the unit and counit of the adjunction (39), can be written as follows: (42)

m ηC

C c

/

Rm Lm (C) = Lm (C) ×R A, P ∗ / j πC (c ⊗ ej ) ×R ej

Lm Rm (B) = (B ×R A) ⊗ ∗ A πLm (B) (b ×R a) ⊗ ϕ

(43)

m ξB

/

/

B

(1 ⊗ ϕ(a)o )b,

for every R-ring C and Re -ring B. Notice that ξ m is defined by the universal property of the tensor algebra, see the argument before (16). Next, we proceed to show that Lm (A) is an ×R -bialgebra. The structure of an Re -ring, is given by the following composition of algebra maps Re

ι0

/

TRe (L (A))

πA

/

Lm (A),

where ιn denotes the canonical Re -bilinear injection in degree n ≥ 0. Lemma 3.3.4. Let A be an R-ring which is finitely generated and projective as left R-module with dual basis {(∗ ei , ei )}i . The following maps ! X ∗ ∗ δ : A −→ Lm (A) ×R Lm (A) ×R A, a 7−→ πA (a ⊗ ej ) ×R πA (ej ⊗ ei ) ×R ei j,i

ω : A −→ Endk (R) ×R A,

a 7−→

X j

are morphisms of R-rings.

∗

ej (a •) ×R ej

!

, where

h

∗

ej (a•) : r 7→ ∗ ej (a r)

i

Proof. We only prove that δ is a morphism of R-rings. Similar arguments are used to show that ω is also a morphism of R-rings. The map δ is in fact the composition of the following two maps (π ◦ι × π ◦ι )× A / L (A) ×R L (A) ×R A A 1 R A 1 R / Lm (A) ×R Lm (A) ×R A, δ:A

where the first one is defined via Lemma 3.2.2. Thus δ is a well defined map. Now, let us show that δ is a morphism of R-rings. The unit is preserved by δ, since we have X δ(1A ) = πA (1A ⊗ ∗ ej ) ×R πA (ej ⊗ ∗ ei ) ×R ei , (πA (1A ⊗ ∗ ej ) = πA (1R ⊗ ∗ ej (1A )o )) j,i


=

23

X πA (1R ⊗ ∗ ej (1A )o ) ×R πA (ej ⊗ ∗ ei ) ×R ei j,i

=

X (1 ⊗ ∗ ej (1A )o ).πA (1Re ) ×R πA (ej ⊗ ∗ ei ) ×R ei j,i

=

X πA (1Re ) ×R ∗ ej (1A ).πA (ej ⊗ ∗ ei ) ×R ei j,i

=

X ∗ ∗ e πA (1R ) ×R πA ( ej (1A )ej ⊗ ei ) ×R ei j,i

=

X πA (1Re ) ×R πA (1A ⊗ ∗ ei ) ×R ei i

=

Xh

=

=

X πA (1Re ) ×R πA (1Re ) ×R ei ∗ ei (1A )

o

i

πA (1Re ) ×R πA (1Re ) (1 ⊗ ei (1A ) ) ×R ei

i

(26)

∗

i

πA (1 ) ×R πA (1 ) ×R 1A . Re

Re

For any a, a′ ∈ A, we have δ(aa′ ) = i Xh = πA (aa′ ⊗ ∗ ej ) ×R πA (ej ⊗ ∗ ei ) ×R ei j,i

=

i Xh πA (a ⊗ ek ∗ ej ) ⊗Re (a′ ⊗ ∗ ek ) ×R πA (ej ⊗ ∗ ei ) ×R ei j,i,k

=

i Xh ′ ∗ ∗ ∗ ∗ e πA (a ⊗ el ej (el ek )) ⊗R (a ⊗ ek ) ×R πA (ej ⊗ ei ) ×R ei

j,i,k,l

=

X h

j,i,k,l

=

i πA (a ⊗ ∗ el ) ⊗Re (a′ ⊗ ∗ ek ) ∗ ej (el ek ) ×R πA (ej ⊗ ∗ ei ) ×R ei

i Xh πA (a ⊗ ∗ el ) ⊗Re (a′ ⊗ ∗ ek ) ×R (∗ ej (el ek ) ⊗ 1oR ) πA (ej ⊗ ∗ ei ) ×R ei

j,i,k,l

=

i Xh πA (a ⊗ ∗ el ) ⊗Re (a′ ⊗ ∗ ek ) ×R πA (∗ ej (el ek )ej ⊗ ∗ ei ) ×R ei

j,i,k,l

=

i Xh ′ ∗ ∗ ∗ πA (a ⊗ el ) ⊗Re (a ⊗ ek ) ×R πA (el ek ⊗ ei ) ×R ei i,k,l

=

i X h πA (a ⊗ ∗ el ) ⊗Re (a′ ⊗ ∗ ek ) ×R πA (el ⊗ em ∗ ei ) ⊗Re (ek ⊗ ∗ em ) ×R ei

i,k,l,m

=

i X h πA (a ⊗ ∗ el ) ⊗Re (a′ ⊗ ∗ ek ) ×R πA (el ⊗ ∗ en ∗ ei (en em )) ⊗Re (ek ⊗ ∗ em ) ×R ei

i,k,l,m,n

=

i X h ×R ei πA (a ⊗ ∗ el ) ⊗Re (a′ ⊗ ∗ ek ) ×R (1 ⊗ ∗ ei (en em )o ) πA (el ⊗ ∗ en ) ⊗Re (ek ⊗ ∗ em )

i,k,l,m,n (31)

=

ii h X h ×R ei (1 ⊗ ∗ ei (en em )o ) πA (a ⊗ ∗ el ) ⊗Re (a′ ⊗ ∗ ek ) ×R πA (el ⊗ ∗ en ) ⊗Re (ek ⊗ ∗ em )

i,k,l,m,n

24


i X h πA (a ⊗ ∗ el ) ⊗Re (a′ ⊗ ∗ ek ) ×R πA (el ⊗ ∗ en ) ⊗Re (ek ⊗ ∗ em ) ×R ∗ ei (en em )ei

=

i,k,l,m,n

=

i X h πA (a ⊗ ∗ el ) ⊗Re (a′ ⊗ ∗ ek ) ×R πA (el ⊗ ∗ en ) ⊗Re (ek ⊗ ∗ em ) ×R en em

k,l,m,n

=

X h

πA (a ⊗ ∗ el ) ×R πA (el ⊗ ∗ en )

k,l,m,n

= =

"

πA (a′ ⊗ ∗ ek ) ×R πA (ek ⊗ ∗ em )

X ∗ ∗ πA (a ⊗ el ) ×R πA (el ⊗ en ) ×R en l,n

′

δ(a)δ(a ),

#"

i

×R en em

X ′ ∗ ∗ πA (a ⊗ ek ) ×R πA (ek ⊗ em ) ×R em k,m

#

hence δ(aa′ ) = δ(a)δ(a′ ), which shows that δ is multiplicative. Lastly, a similar computation shows that δ satisfies δ(r1A ) = πA (r1Re ) ×R πA (1Re ) ×R 1A = r1Lm (A)×R Lm (A) ×R 1A ,

for every r ∈ R,

which means that the diagram / R OOO oo A o OOO o oo OOO oooδ OOO o o O' woo (Lm ×R Lm ) ×R A

is commutative, and this finishes the proof.

Part of the following proposition was first stated by D. Tambara in [37, Remark 1.7] with no proof. As one can realize, this can not be immediately deduced. Proposition 3.3.5. Let A be an R-ring which is finitely generated and projective as left R-module with dual basis {(∗ ei , ei )}i . Then Lm (A) is a ×R -bialgebra with structure maps Lm (A) πA (a ⊗ ϕ)

∆

/

P

j

/

Lm (A) ×R Lm (A),

πA (a ⊗ ∗ ej ) ×R πA (ej ⊗ ϕ)

Lm (A) πA (a ⊗ ϕ)

ε

/ Endk (R) h i / r→ 7 ϕ(ar) .

Proof. Both ∆ and ε are defined via the adjunction Lm ⊣ Rm . In fact, we have m ∆ = ξL ◦ Lm (δ), m (A)×R Lm (A) m where δ is the morphism of R-rings defined in Lemma 3.3.4, and ξ− is the counit of the adjunction Lm ⊣ Rm . e Therefore, it is immediate that ∆ is a morphism of R -rings. To show that ∆ is coassociative it suffices to


25

check that the following diagram is commutative A QQQQ

QQQ QQQ QQ

m ηA QQ

QQQ QQQ QQQ Q(

Rm (∆)

Rm (Lm (A))

/

Rm Lm (A) ×R Lm (A)

Rm (∆)

Rm (∆×R Lm (A))

Rm Lm (A) ×R Lm (A)

Rm Lm (A) ×R Lm (A) ×R Lm (A)

Rm (Lm (A)×R ∆)

Rm (αl )

Rm Lm (A) ×R Lm (A) ×R Lm (A)

Rm (αr )

/ Rm Lm (A) ×R Lm (A) ×R Lm (A) ,

and this follows from routine computations. m We also have ε = ξEnd ◦ Lm (ω), where ω : A → Endk (R) ×R A is the morphism of R-rings defined in k (R) Lemma 3.3.4. Hence ε is clearly a morphism of Re -rings. Furthermore, it satisfies the following equality θl ◦ (ε ×R Lm (A)) ◦ ∆ = Lm (A) = θr ◦ (Lm (A) ×R ε) ◦ ∆, which is the counitarity.

Next we provide the relation between the R-ring structure of A and the ×R -bialgebra structure of Lm (A). Corollary 3.3.6. Let A be an R-ring such that R A is finitely generated and projective and Lm (A) the associated ×R -bialgebra defined in Proposition 3.3.5. Then A is a left ×R -Lm (A)-comodule R-ring, that is, A admits a left ×R -Lm (A)-coaction λA : A → Lm (A) ×R A which is also a morphism of R-rings. Proof. The unit of the adjunction given in (39) at A ηAm : A −→ Lm (A) ×R A,

a 7−→

X

πA (a ⊗ ∗ ei ) ×R ei

i

!

is by definition a morphism of R-rings. Let us check that it is a left Lm (A)-coaction. It remains to show that the following diagrams are commutative A

m ηA

/

m ηA

∆×R A

Lm (A) ×R Lm (A) ×R A

Lm (A) ×R A m Lm (A)×R ηA

Lm (A) ×R Lm (A) ×R A

Lm (A) ×R A

αl

αr

/

Lm (A) ×R Lm (A) ×R A,

26

A. ARDIZZONI, L. EL KAOUTIT, AND C. MENINI m ηA

A

/

Lm (A) ×R A ε×R A

A

o

θl

For every element a ∈ A, we have αl ◦ (∆ ×R A) ◦ ηAm (a) =

X i,j

=

X i,j

=

Endk (R) ×R A

X i,j

αl

πA (a ⊗ ∗ ei ) ×R πA (ei ⊗∗ ej ) ×R ej

πA (a ⊗ ∗ ei ) ×R πA (ei ⊗∗ ej ) ×R ej αr πA (a ⊗∗ ej ) ×R πA (ej ⊗ ∗ ei ) ×R ei

= αr ◦ Lm (A) ×R ηAm ◦

X

πA (a ⊗∗ ei ) ×R ei

i

= αr ◦ Lm (A) ×R ηAm ◦ ηAm (a),

and also we have

θl ◦ ε ×R A ◦ ηAm (a) = θl ◦ ε ×R A =

X

θl

i

=

X

∗

∗

X i

ei (a•) ×R ei

πA (a ⊗∗ ei ) ×R ei

!

!

ei (a)ei = a.

i

This proves the commutativity of the above diagrams and establishes the corollary.

The ×R -bialgebra Lm (A) constructed in Proposition 3.3.5 is refereed to as coendomorphism R-bialgebroid since, by [5, Theorem 3.1], Lm (A) is in fact a (left) bialgebroid whose structure of Re -ring is the map πA ◦ ι0 : Re −→ Lm (A), and its structure of R-coring is given as follows. The underlying R-bimodule is Lm (A)l = comultiplication and counit are given by (44)

∆ : Lm (A)l −→ Lm (A)l ⊗R Lm (A)l ,

πA (a ⊗ ϕ) 7−→

X

ε : Lm (A)l −→ R,

the

!

πA (a ⊗ ∗ ei ) ⊗R πA (ei ⊗ ϕ) ,

i

(45)

Re Lm (A),

πA (a ⊗ ϕ) 7−→ ϕ(a) .

3.4. The monoidal structure of left Lm (A)l -comodules. In what follows we will denote by L := Lm (A) the coendomorphism R-bialgebroid. In [32] it was shown that the category of left ×R -comodules over an ×R -bialgebra is a monoidal category such that the forgetful functor to the category of R-bimodules is a monoidal functor. What we will need in the sequel is a monoidal structure on the category of left L -comodules where L is viewed as an R-coring with structure maps (44) and (45). Next we will use a Schauenburg’s result [32, Proposition 5.6] to give a monoidal structure on L Comod with the same property as for left ×R -comodules over L , see also [7, 3.6]. To do so, we should first clarify how to construct a


27

canonical R-biaction on each left L l -comodule. Since L is an Re -bimodule and the comultiplication is Re -bilinear, there is an algebra map (46)

/

R r /

(L l )∗

[z 7→ ε(z (r ⊗ 1o )]

with codomain the right convolution ring of the underlying R-coring L l , see [6, 35]. It is well known that each left L l -comodule admits a canonical right (L l )∗ -action, given as follows: take a left L l -comodule P l l (X, λX ), λX : X → L ⊗R X, x 7→ (x) x(−1) ⊗R x(0) , we have X xσ = σ(x(−1) )x(0) , (x)

for every x ∈ X and σ ∈ (L l )∗ . Therefore, the restriction scalars functor associated to the map (46), gives the following right R-action X (47) xr = ε x(−1) (r ⊗ 1o ) x(0) , for every r ∈ R, (x)

which clearly induces an R-bimodule structure on X. Analogously, given a left ×R -L -comodule (Z, λ′Z ), i.e. an R-bimodule with R-bilinear map λ′Z : Z can recover the right R-action from this P→ L ×R Z, one o coaction. That is, we have the equality zr = z ε(z(−1) (r ⊗ 1 ))z(0) , for every elements z ∈ Z and r ∈ R. At the level of comodule we have Lemma 3.4.7. Let L be any left R-bialgebroid. Then the category of left ×R -L -comodule is isomorphic to the category of left L l -comodules over the underlying R-coring L l . In particular, the category of left L l -comodules inherits a monoidal structure with unit object (R, R → L l ) and the left forgetful functor U : L l Comod → R Mod factors throughout a monoidal functor into the category of R-bimodules. Thus, we have a commutative diagram L l Comod M

U

M

M

M

M&

/ Mod R t9 t t tt tt tt

R ModR

where the dashed arrow is a monoidal functor. Proof. The stated isomorphism of categories is constructed using a slightly variant of the natural transformations Θ−,− and Ξ−,− defined in (32) and (34). The monoidal structure as well as the monoidal forgetful functor property are then consequences of [32, Proposition 5.6]. Summing up, given two left L l -comodules (X, λX ) and (Y, λY ), using Lemma 3.4.7, we can consider (X ⊗R Y, λX⊗R Y ) as a left L l -comodule with coaction   X (48) λX⊗R Y : X ⊗R Y → L l ⊗R X ⊗R Y, x ⊗R y 7−→ (x(−1) y(−1) )l ⊗R (x(0) ⊗R y(0) ),  (x),(y)

where we have considered X as R-bimodule with the right R-action given by (47).

3.5. Examples of coendomorphism bialgebroids. In this subsection we give some examples of coendomorphism bialgebroids. Example 3.5.8. Assume that R = k is a field with characteristic not equal to 2. Let A be the Hamilton quaternion k-algebra associated to the pair (−1, −1). That is, A = k⊕ki⊕kj⊕kij with relation i2 = −1 = j2

28


and ij = −ji. Then one can prove, using Proposition 3.3.5, that Lm (A) is a k-bialgebra, which is generated as an k-algebra by elements {xk , yk , zk , uk }1≤k≤3 subject to the relations 1 + x2k x1 x2 + x2 x1 x1 y1 u1 y1 z1 y1 x3 y3 z3 u3 u2 x1 u2 z1 u2 y1 x2 y2 x2 z2 x2 u2

= = = = = = = = = = = = = = =

yk2 + zk2 + u2k , for all k = 1, 2, 3, y2 y1 + y1 y2 + u2 u1 + u1 u2 + z2 z1 + z1 z2 , −y1 x1 + z1 u1 − u1 z1 , y1 u1 + z1 x1 + x1 z1 , y1 z1 − x1 u1 − u1 x1 , x1 x2 − y1 y2 − z1 z2 − u1 u2 , x1 y2 + y1 x2 − z1 u2 + u1 z2 , x1 z2 + y1 u2 + z1 x2 − u1 y2 , x1 u2 − y1 z2 + z1 y2 + u1 x2 , −x2 u1 − y2 z1 − z2 y1 , −x2 y1 − y2 x1 − z2 u1 + y3 , x2 z1 + y2 u1 + z2 x1 + z3 , −y2 x2 + z2 u2 − u2 z2 , −y2 u2 − z2 x2 + u2 y2 , −y2 z2 − z2 y2 − u2 x2 .

The k-coalgebra structure is given as follows: ∆(xk ) ε(xk ) ∆(yk ) ε(yk ) ∆(zk ) ε(zk ) ∆(uk ) ε(uk )

= = = = = = = =

xk ⊗ 1 + yk ⊗ x1 + uk ⊗ x2 + zk ⊗ x3 , 0, k = 1, 2, 3; yk ⊗ y1 + uk ⊗ y2 + zk ⊗ y3 , k = 1, 2, 3; 0, k = 2, 3, ε(y1) = 1. yk ⊗ z1 + uk ⊗ z2 + zk ⊗ z3 , k = 1, 2, 3; 0, k = 1, 2, ε(z3 ) = 1 yk ⊗ u1 + uk ⊗ u2 + zk ⊗ u3 , k = 1, 2, 3; 0, k = 1, 3, ε(u2) = 1.

A is a left Lm (A)-comodule algebra with coaction λ : A → Lm (A) ⊗ A sending λ(1A ) = 1L (A) ⊗ 1A , λ(i) = x1 ⊗ 1 + y1 ⊗ i + u1 ⊗ j + z1 ⊗ ij, λ(j) = x2 ⊗ 1 + y2 ⊗ i + u2 ⊗ j + z2 ⊗ ij. Of course, we have λ(ij) = λ(i)λ(j) = x3 ⊗ 1 + y3 ⊗ i + u3 ⊗ j + z3 ⊗ ij. Example 3.5.9. Assume that A = Rn , the obvious R-ring attached to the free R-module of rank n. One can easily check, using (40) and Proposition 3.3.5, that Lm (A) is an R-bialgebroid generated as ring by the image of Re and a set of Re -invariant elements {xij }1≤i, j≤n with relation x2ii = xii , for all i = 1, 2, · · · , n. xji xki = 0, for all j 6= k, and i, j, k = 1, 2, · · · , n. n X xij = 1, for all j = 1, 2, · · · , n. i=1


29

Its structure of R-coring is given by the following comultiplication and counit ∆(xij ) =

n X

xik ⊗R xkj ,

for all i, j = 1, 2, · · · , n;

k=1

ε(xij ) = δij ,

(Kronecker delta) for all i, j = 1, 2, · · · , n.

Let us denote by {ei }1≤i≤n the canonical basis of R A. Then A is a left Lm (A)-comodule ring by the coaction: λ : A → Lm (A) ⊗R A defined by λ(ei ) =

n X

xij ⊗R ej ,

∀i = 1, · · · , n.

j=1

Example 3.5.10. Let A = R ⊕ Rt be the trivial generalized R-ring i.e. the R-ring which is free as left R-module with basis 1 = (1, 0) and t = (0, t) such that t2 = 0. Using (40) and Proposition 3.3.5, we can easily check that Lm (A) is an R-bialgebroid generated by the image of Re and two Re -invariant elements {x, y} subject to the relations xy + yx = 0, x2 = 0. The comultiplication and counit of it underlying R-coring are given by ∆(x) = x ⊗R 1 + y ⊗R x, ε(x) = 0 ∆(y) = y ⊗R y, ε(y) = 1. A is a left Lm (A)-comodule ring with coaction: λ : A → Lm (A) ⊗R A sending λ(1A ) = 1Lm (A) ⊗R 1A ,

λ(t) = x ⊗R 1A + y ⊗R t.

Example 3.5.11. Let A be the trivial crossed product of R by the cyclic group Gn of order n. We know that R A is the left free module with basis Gn . It is easily checked, using (40) and Proposition 3.3.5, that if n = 2, then Lm (A) is an R-bialgebroid generated as an Re -ring by two Re -invariant elements x, y subject to the relations xy + yx = 0 and 1 = x2 + y 2. The comultiplication and counit of the underlying R-coring structure are given by ∆(x) = x ⊗R 1 + y ⊗R x,

∆(y) = y ⊗R y,

ε(x) = 0,

ε(y) = 1.

For n > 2, we can prove that Lm (A) is an Re -ring generated by the Re -invariant elements x(k, l) with (k, l) ∈ (Zn \ {0}) × Zn subject to the following relations: x(k, l) =

n−1 X

x(t, l−s) x(k−t, s) , ∀ (k, l) ∈ (Zn \ {0, 1}) × Zn , ∀ t ∈ Zn \ {0} with t < k,

s=0

x(1, l) =

n−1 X

x(n−t, l−s) x(n−t′ , s) , ∀ l ∈ Zn , ∀ t, t′ ∈ Zn \ {0}, with t + t′ = n − 1,

s=0

and

1 =

n−1 X

x(t, n−s) x(t′ , s) , ∀ t, t′ ∈ Zn \ {0}, with t + t′ = 0,

s=0

where the ring Zn is endowed with the canonical ordering 0 < 1 < · · · < n − 1. The comultiplication and counit of its underlying R-coring structure are given by ∆(x(k, l) ) =

n−1 X s=0

x(k, s) ⊗R x(s, l) ,

ε(x(k, l) ) = δk,l ,

∀ (k, l) ∈ (Zn \ {0}) × Zn .

30


The left comodule ring structure of A is given by the following coaction. Consider {gk }0≤k≤n−1 the basis of the free module R A, where g denotes the generating element of Gn which we identify with its image using the canonical injection. The coaction is then given by λ : A → Lm (A) ⊗R A sending n−1 X

k

λ(1A ) = 1Lm (A) ⊗R 1A ,

λ(g ) =

x(k, l) ⊗R gl , ∀k ∈ (Zn \ {0}).

l=0

4. Categories of comodules and chain complexes of modules. This section contains our main results, namely Theorems 4.4.18, 4.5.24 and 4.6.25. As a consequence, we obtain that the category of chain complexes of left R-modules is always equivalent to the category of left comodules over a quotient R-coring of the left R-bialgebroids Lm (A) constructed in Example 3.5.10. When R is commutative, this quotient inherits a left R-bialgebroid structure from Lm (A), and the stated equivalence is actually a monoidal equivalence. Fix a ring R, and consider an R-ring A which is finitely generated and projective as left R-module. From now on, we fix a dual basis for R A, {(ei , ∗ ei )}i ∈ A × ∗ A, and we denote by L = Lm (A) the associated bialgebroid constructed in Proposition 3.3.5, with the canonical projection coendomorphism π : TRe (A ⊗ ∗ A)† → L . The underlying R-coring will be also denoted by L . The structure maps of this coring are given in (44) and (45).

4.1. The complex of left L -comodules Q• . Let us denote by µ / K = Ker A ⊗R A A

the kernel of the multiplication µ of A with canonical derivation ∂

A a /

/

K

∂a = 1 ⊗R a − a ⊗R 1 .

The associated cochain complex is denoted by Q• : R

∂0 =1

/

A

∂1 =∂

/

∂2

K

/

K ⊗A K

∂3

/

K ⊗A K ⊗A K /

······

where ∂n : Qn → Qn+1 sends a0 ∂a1 ⊗A · · · ⊗A ∂an−1 to ∂a0 ⊗A ∂a1 ⊗A · · · ⊗A ∂an−1 , n ≥ 2. The following lemma, which will play a key role in Subsection 4.6, characterizes a split ring extension R → A (in ModR ) in terms of the cochain complex Q• . Lemma 4.1.1. Let A be any R-ring. Then the following conditions are equivalent. (i) The unit u : R → A is a split monomorphism in ModR . (ii) The cochain complex Q• is exact and splits, in the sense that, for every m ≥ 1, Qm = ∂Qm−1 ⊕ Qm = Ker(∂) ⊕ Qm , as right R-modules, for some right R-module Qm . Proof. (ii) ⇒ (i) It is trivial. (i) ⇒ (ii). Let us denote by uc : A → A the cokernel of u : R → A in R ModR . Put Ω0 := R, Ω1 := A, and Ωn := A ⊗R A ⊗R · · · ⊗R A, (n − 1)-fold A, for n ≥ 2. Consider now the following split exact sequences of right R-modules 0 ⊗R n

where γn = u ⊗R A

/

γn

⊗R n

A

/

⊗R n

/

A ⊗R A

⊗R n+1

/

A

0,

, for n ≥ 1. In view of this, we have a split exact cochain complex of right R-modules Ω• : Ω0

d0

/

Ω1

d1

/

Ω2

d2

/

Ω3 /

··· ,


31

⊗R n−1

with differential d0 = u, d1 = γ1 ◦ uc , dn = γn ◦ (uc ⊗R A ), for n ≥ 2. Since Ω2 is the cokernel of the map A ⊗R u, and the later split by mA the multiplication of A, we obtain the following split exact sequence of R-bimodules /

0 This gives the split exact sequence

A⊗R u

A /

0

Ω2

/

A ⊗R A /

A ⊗R A

A⊗R uc

ma

/

/

Ω2

A

/

/

0.

0.

Thus we have an R-bilinear isomorphism ω2 : Ω2 → Q2 = K. Henceforth, there is an unique A-bimodule structure on Ω2 which renders ω2 an A-bilinear isomorphism, namely a · (x ⊗R y) · b = ax ⊗R yb − axy ⊗R b for every a, x, y, b ∈ A, wherein the notation uc (z) = z, for every z ∈ A, have been used. Define iteratively ωn : Ωn → Qn , for all n ≥ 3, as the composition ωn−1 ⊗A ω2 ⊗A n−1 /Q Ωn = Ωn−1 ⊗R A ∼ = Qn . = Ωn−1 ⊗A A ⊗R A = Ωn−1 ⊗A Ω2 n−1 ⊗A K = K

By construction, ω• : (Ω• , d• ) → (Q• , ∂• ) is a morphism of complexes of R-bimodules. We leave to the reader to check that ω• is in fact an isomorphism of cochain complexes. Now, since (Ω• , d• ) is split exact in right R-modules, then so is (Q• , ∂• ). Remark 4.1.2. In the finitely generated and projective case, the left version of condition (i) in Lemma 4.1.1 implies that R A is in fact faithfully flat module (see, for example [3, Chap. I, Proposition 9, page 51]). In this case, one can easily show that Q ⊗R A is homotopically trivial which by [20, Théorème 2.4.1] gives condition (ii). In this way, Lemma 4.1.1 can be seen as a generalization of [2, Propositions 6.1, 6.2]. The convolution product on the left dual chain complex of Q• is given as follows: For every ϕ ∈ ∗ Qn and ψ ∈ ∗ Qm with n, m ≥ 1, we have a left R-linear map

(49)

ϕ ⋆ ψ : Qn+m x ⊗A ∂(a) ⊗A y

/R / ϕ xψ(ay) − ϕ xaψ(y) ,

where x ∈ Qn , y ∈ Qm , and a ∈ A. The convolution product with zero degree element is just the left and right R-actions of ∗ Qn , for every n ≥ 1, namely (50)

r ⋆ ϕ : Qn x

/

R / ϕ(x r),

ϕ ⋆ s : Qn x

/

R / ϕ(x) s,

for every elements r, s ∈ R and ϕ ∈ ∗ Qn . Remark 4.1.3. The convolution product defined in (49) and (50) derives from the structure of comonoid of the cochain complex Q• viewed as an object in the monoidal category of cochain complexes of R-bimodules. Precisely, the identity map A ⊗R · · · ⊗R A = A⊗R n = A⊗R p ⊗R A⊗R q , for p + q = n, rereads as a map Qn → Qp ⊗R Qq sending x ⊗A ∂a ⊗A y 7→ x ⊗R ay − xa ⊗R y, for every x ∈ Qp , a ∈ A and y ∈ Qq . Thus, Q = ⊕n≥0 Qn has a structure of differential R-coring in the sense of [10, pages 6, 7]. Since each Qn is finitely generated an projective left R-module (see Lemma 4.1.4 below), the comultiplication of Q is transferred to the graded left dual ∨ Q = ⊕n≥0 ∗ Qn which gives a multiplication defined explicitly by (49) and (50). A comonoidal structure on Q• could also be obtained by transferring some comonoidal structure of the Amitsur cosimplicial object of R-bimodules induced by A [2], using the normalization functor and it structure of comonoidal functor obtained from Eilenberg-Zilber Theorem, see [26, Theorem 8.1, Exercise 4. p. 244] (of course in their dual form). It seems that Tambara’s approach [37] runs in this direction. Anyway this approach uses a slightly variant of the category of cosimplicial groups endowed with some monoidal

32


structure which is not the usual one. Since our methods run in a different way, we will not make use of the normalization process here. Using the dual basis of R A, one can check that R Q2 = R K is finitely generated and projective module whose dual basis is given by the set {(ei ∂ej , ∗ ei ⋆ ∗ ej )}i, j . Moreover, we have Lemma 4.1.4. Each Qn , n ≥ 0, is finitely generated and projective as left R-module. Furthermore, if {(ωn,α, ∗ ωn,α )}α is a dual basis for Qn with n ≥ 1, then {(ωn,α ⊗A ∂ωm,β , ∗ ωn,α ⋆ ∗ ωm,β )}α, β is a dual basis for Qn+m , while {(ωn,α ⊗A ωm,β , ∗ ωn,α ⋆ ∂ ∗ ωm,β )}α, β is a dual basis for Qn+m−1 when m ≥ 2. Proof. Straightforward.

Proposition 4.1.5. The cochain complex Q• is a complex of left L -comodules. For n = 0, the coaction is given by (R → L , r 7→ π(r ⊗ 1o )) and, for n ≥ 1, by λn : Qn → L ⊗R Qn defined by (51)

a0 ∂a1 ⊗A · · · ⊗A ∂an−1 /

X

i0 , i1 , ··· , in−1

π(a0 ⊗ ei0 ) · · · π(an−1 ⊗ ein−1 ) ⊗R ei0 ∂ei1 ⊗A · · · ⊗A ∂ein−1 . ∗

∗

Proof. The statement is trivial for n = 0. For n ≥ 1, the coassociativity of λn is deduced using that {(ei0 ∂ei1 ⊗A · · · ⊗A ∂ein−1 , ∗ ei0 ⋆ · · · ⋆ ∗ ein−1 }i0 , i1 , ··· ,in−1 is a dual basis for Qn , see Lemma 4.1.4. Here each ∗ ei0 ⋆ · · · ⋆ ∗ ein−1 is the n-fold convolution product defined in (49). The counitary property is clear since the counit sends . π(a0 ⊗ ∗ ei0 ) · · · π(an−1 ⊗ ∗ ein−1 ) 7−→ ∗ ei0 a0 ∗ ei1 a1 ∗ ei2 · · · ∗ ein−1 (an−1 ) Let us show that the differential of the complex Q• consists of left L -colinear maps. Take an element u ∈ Qn of the form u = a0 ∂a1 ⊗A · · · ⊗A ∂ain−1 , so we have X π(a0 ⊗ ∗ ei0 ) · · · π(an−1 ⊗ ∗ ejn−1 ) ⊗R ∂ei0 ⊗A · · · ⊗A ∂ein−1 λn+1 ◦ ∂n (u) = i0 , i1 , ··· ,in−1



= (L ⊗R ∂n ) 

X

π(a0 ⊗ ∗ ei0 ) · · · π(an−1 ⊗ ∗ ejn−1 ) ⊗R

i0 , i1 , ··· ,in−1

 ei0 ∂ei1 ⊗A · · · ⊗A ∂ein−1 

= (L ⊗R ∂n ) ◦ λn (u),

where in the first equality we have used, the fact that, each coaction λn , n ≥ 1, satisfies the equality (52) λn ∂b1 ⊗A · · · ⊗A ∂bn−1 = X π(b1 ⊗ ∗ ei1 ) · · · π(bn−1 ⊗ ∗ ein−1 ) ⊗R ∂ei1 ⊗A · · · ⊗A ∂ein−1 i1 , ··· , in−1

which can be proved by using the underlying structure of R-bimodule of the R-coring L , i.e. the equality π(1 ⊗ ϕ) = (1 ⊗ ϕ(1)o ).1L , which holds for every ϕ ∈ ∗ A.

Re L ,

and

The following lemma will be used in the sequel. Lemma 4.1.6. Given two elements un = a0 ∂a1 ⊗A · · · ⊗A ∂an−1 ∈ Qn and um = b0 ∂b1 ⊗A · · · ⊗A ∂bn−1 ∈ Qm with n, m ≥ 1. Then X π(a0 ⊗ ∗ ei0 ) · · · π(an−1 ⊗ ∗ ein−1 )π(b0 ⊗ ∗ ej0 ) · · · π(bm−1 ⊗ ∗ eim−1 ) λn+m−1 (un ⊗A um ) = i0 ,··· , in−1 , j0 ,··· ,jn−1

⊗R ei0 ∂ei1 ⊗A · · · ⊗A ∂ein−1 ⊗A ej0 ∂ej1 ⊗A · · · ⊗A ∂ein−1 .


33

Furthermore, for every u ∈ Qn , n ≥ 1 and v ∈ Qm , m ≥ 1, we have X X λn+m−1 (u⊗A v) = u(−1) v(−1) ⊗R (u(0) ⊗A v(0) ), and λn+m (u⊗A ∂v) = u(−1) v(−1) ⊗R (u(0) ⊗A ∂v(0) ), where Sweedler’s notations for coactions have been used.

Proof. The proof of the first claim is based upon the observation that the coaction of any Qk = K ⊗A · · ·⊗A K ((k −1)-times), with k ≥ 2, is induced from that of A⊗R · · ·⊗R A (k-times). The later is a left L -comodule, by Corollary 3.3.6 and Lemma 3.4.7, using the coactions described in (48). The last statement is deduced from the first one by left R-linearity. 4.2. The infinite comatrix bialgebroid induced by Q• . Let Q• be the cochain complex of L -comodules considered in Proposition 4.1.5. In this subsection we will construct a left bialgebroid associated to Q• and a canonical map from this left bialgebroid to L . First we recall from [14, 15] the notion of infinite comatrix coring and the canonical map. A different approach to this notion can be found in [38], [9] and [21]. We should mention here that this object coincides with the one already constructed in the context of TannakaKrein duality over fields or commutative rings, see [11], [8], [23] and [17], see also [28]. However, the description given in [15] in terms of tensor product over a ring with enough orthogonal idempotents, seems to easier to handle from a computational point of view. Let A be a small full sub-category of an additive category. Following [19, page 346], we can associate to A the ring with enough orthogonal idempotents S = ⊕p, p′ ∈A HomAo (p, p′ ), where Ao is the opposite category of A. The category of left unital S-module is denoted by S Mod. Let us denote by add(R R) the full sub-category of R Mod consisting of all finitely generated and projective left R-modules. Let χ : A → add(R R) be a faithful functor, refereed to as fiber functor. We denote by pχ the image of p ∈ A under χ or by p itself if no confusion arises. Consider the left R-module direct sum of the p’s: Σ = ⊕p∈A p (i.e. Σ = ⊕p∈A pχ ) and the right R-module direct sum of their duals: ∨ Σ = ⊕p∈A ∗ p. It is clear that ∨ Σ is a left unital S-module while Σ is a right unital S-module. In this way Σ becomes an (R, S)-bimodule and ∨ Σ an (S, R)-bimodule. Then Σ ⊗S ∨ Σ is now an R-bimodule whose elements are described as a finite sum of diagonal ones, i.e. of the form ιp (up ) ⊗S ι∗ p (ϕp ) where (up , ϕp ) ∈ pχ × (∗ pχ ) and ι− are the canonical injections in ∨ Σ and Σ. From now on, we will use up ⊗S ϕp instate of ιp (up ) ⊗S ι∗ p (ϕp ) to denote a generic element of Σ ⊗S ∨ Σ. This bimodule admits a structure of an R-coring given by the following comultiplication (53)

∆ : Σ ⊗S ∨ Σ up ⊗S ϕp

(Σ ⊗S ∨ Σ) ⊗R (Σ ⊗S ∨ Σ) P ∗ / i up ⊗S up, i ⊗R up, i ⊗S ϕp , /

where, for a fixed p ∈ A, the finite set {(up, i , ∗ up, i )}i ⊂ p × ∗ p is a left dual basis of the left R-module p. The counit is just the evaluating map. Note that this comultiplication is independent from the chosen bases. With this structure Σ ⊗S ∨ Σ is refereed to as the infinite comatrix coring associated to the small category A and the fiber functor χ. On the other hand, each of the left R-modules pχ is actually a left Σ ⊗S ∨ Σ-comodule with coaction, using the above notation is given by ! X ep : p −→ Σ ⊗S ∨ Σ ⊗R p, (54) λ u 7−→ u ⊗S ∗ up, i ⊗R up, i . i

Another description of the infinite comatrices is given in [15, Proposition 5.2] which establishes an isomorphism of R-bimodules ⊕ p ⊗Tp ∗ p p∈ A (55) Σ ⊗B ∨ Σ ∼ =

ut ⊗Tq ϕ − u ⊗Tp tϕ {u ∈ p, ϕ ∈ ∗ q, t ∈ Tq,p } where Tp := EndAo (p) and Tp, q := HomAo p, q , for every objects p, q in A.

34


Now, let C be an R-coring and let Q be a small full sub-category of the category of comodules C Comod whose underlying left R-modules are finitely generated and projective. Denote by λq the coaction of q ∈ Q. Then one can directly apply the above constructions, by putting χ(q) = U(q), where U : C Comod → R Mod is the left forgetful functor. In this case, the left C-coaction of Σ = ⊕q∈Q q is right S-linear, while the right C-coaction of ∨ Σ is left S-linear. Moreover, there is a canonical morphism of R-corings defined by (56)

canS : Σ ⊗S ∨ Σ uq ⊗S ϕq /

/

C (C ⊗R ϕq ) ◦ λq (uq ).

Here S is the induced ring from the category Q, that is, S = ⊕q, p∈Q HomC q, p .

(57)

However, the construction of the infinite comatrix coring, as well as the canonical map can, can be also performed for any sub-ring of S with the same set of orthogonal idempotents (i.e. the q’s identities). Let us consider the k-linear category k(N) whose objects are the natural numbers N, and homomorphisms sets are defined by   / {n, n + 1} 0, if m ∈ Homk(N) n, m = k.1n , if n = m  k.n+1 , if m = n + 1. n The last two terms are free k-modules of rank one. The induced ring with enough orthogonal idempotents is the free k-module B = k(N) ⊕ k(N) generated by the set {hn , vn }n∈N , where hn and vn corresponds to 1n and jnn+1 respectively, subject to the following relations: hn hm = δn, m hn , ∀m, n ∈ N (Kronecker delta) vn vm = vm vn = 0, ∀m, n ∈ N vn hn+1 = vn = hn vn , ∀m, n ∈ N. In other words B is the sub-ring of the ring of N × N-matrices over k of the form   k k 0 0  0 k k 0    0 0 k k   .. .. .. (58)   . . .     0 k k   .. .. .. . . .

consisting of matrices with only possibly two non-zero entries in each row: (i, i) and (i, i + 1). It is clear that the category of unital left B-modules is isomorphic to the category Ch + (k) of chain complexes of k-modules. Precisely, this isomorphism functor O sends every chain complex (V• , ∂ V ) to its associated differential graded k-module O(V• ) = ⊕n≥0 Vn with the following left B-action X X hn . vi = vn , and vn . vi = ∂ V (vn+1 ) n≥0

n≥0

and acts in the obvious way on morphisms of chain complexes. The inverse functor is clear. By Proposition 4.1.5, we have a faithful functor χ : k(N) → L Comod sending n → Qn , whose composition with the left forgetful functor gives rise to a fiber functor χ : k(N) → add(R R). Therefore, we can apply the above process to construct an infinite comatrix R-coring Q ⊗B ∨ Q where Q = ⊕n∈N Qn and ∨ Q = ⊕n∈N ∗ Qn are given by the cochain complex of Subsection 4.1.


35

Since each of the Qn ’s has a structure of R-bimodule for which the differential ∂• is R-bilinear, we deduce that Q ⊗B ∨ Q is an Re -bimodule with actions (r ⊗ so ) (un ⊗B ϕn ) (p ⊗ q o ) = (run s) ⊗B (qϕn p),

(59)

for every p, q, r, s ∈ R and un ∈ Qn and ϕn ∈ Qn . In view of this Re -biaction, the infinite comatrix R-coring has Re (Q ⊗B ∨ Q)† as its underlying R-bimodule. The following lemma will be used in the sequel. Lemma 4.2.7. Let {ωn,α , ∗ ωn,α)}α be a dual basis for R Qn with n > 0. Then, for every element un ∈ Qn , um ∈ Qm , and ϕn ∈ ∗ Qn , ϕm ∈ ∗ Qm , we have i h i Xh ∗ ∗ (un ⊗A ∂um ) ⊗B ( ωn,α ⋆ ωm,β ) ×R (ωn,α ⊗A ωm,β ) ⊗B (ϕn ⋆ ∂ϕm ) = 0 α, β

and

i h i Xh (un ⊗A um ) ⊗B (∗ ωn,α ⋆ ∂ ∗ ωm,β ) ×R (ωn,α ⊗A ∂ωm,β ) ⊗B (ϕn ⋆ ϕm ) = 0 α, β

as elements in (Q ⊗B ∨ Q)† ×R (Q ⊗B ∨ Q)† . Proof. Straightforward.

Next we will construct an Re -ring structure on the Re -bimodule (Q ⊗B ∨ Q)† . We need the following general Lemma which can be found, under a slightly different form, in [11], [8], and [17]. We adopt the following general notations: For any small k-linear category C, we denote by Functf (C, add(R R)) the category of k-linear faithful functors valued in add(R R), i.e. that of fiber functors on C. For any object χ : C → add(R R), we denote by L(χ) the associated infinite comatrix R-coring stated above, see (55). Lastly, we consider Σ : Functf (C, add(R R)) → ModS(C) the canonical functor to the category of right unital S(C)-modules (recall that S(C) is the induced ring of C o ). That is, Σ(χ) := ⊕ cχ ,

(60)

c∈C

Σ(γ) := ⊕ γc c∈C

for every fiber functor χ and natural transformation γ between fibred functors. Lemma 4.2.8. Let A be a small k-linear category and let χ1 , χ2 : A → R ModR be two functors with images in add(R R). Define (χ1 ⊗R χ2 ) : A×A → R ModR by setting (χ1 ⊗R χ2 )(p, q) = χ1 (p)⊗R χ2 (q), for p, q ∈ A. Then (i) There is a left Re -linear isomorphism L(χ1 ⊗R χ2 ) ∼ = L(χ1 ) ⊗Re L(χ2 ) (ii) For every R-bimodule M, there is a natural isomorphism Nat (χ1 ⊗R χ2 ), M ⊗R (χ1 ⊗R χ2 ) σ

where σ(p, q) (u ⊗R v) = (p, q) ∈ A × A.

P

i

/ HomR−R L(χ1 ) ⊗Re L(χ2 ), M

i h P / (u ⊗S ϕ) ⊗Re (v ⊗S ψ) 7→ i mi ϕ(pi ψ(qi ))

mi ⊗R pi ⊗R qi ∈ M ⊗R p ⊗R q, for every u ∈ p, ϕ ∈ ∗ p, v ∈ q, ψ ∈ ∗ q and

Proof. (i) The stated isomorphism follows from the isomorphism given in (55) and the following R-bilinear ’local’ epimorphism (p ⊗R q) ⊗

Tp ×Tq

∗

(p ⊗R q)

//

(p ⊗Tp ∗ p) ⊗ (q ⊗Tq ∗ q), Re

36


for every p, q ∈ A, which can be easily checked using the maps defined in (3) and (4). (ii) Using the functor Σ given by (60), we can show that there is an isomorphism ∼ Nat χ1 ⊗R χ2 , M ⊗R χ1 ⊗R χ2 = HomR−S(A×A) Σ(χ1 ⊗R χ2 ), M ⊗R Σ(χ1 ⊗R χ2 ) ,

where the term of right-hand side stands for the set of (R, S(A × A))-bilinear maps. Now, applying [13, Proposition 5.1], we obtain a chain of isomorphisms ∼ HomR−S(A×A) Σ(χ1 ⊗R χ2 ), M ⊗R Σ(χ1 ⊗R χ2 ) Nat (χ1 ⊗R χ2 ), M ⊗R (χ1 ⊗R χ2 ) = ∨ ∼ HomR−R Σ(χ1 ⊗R χ2 ) ⊗ Σ(χ1 ⊗R χ2 ), M = S(A×A) = HomR−R L(χ1 ⊗R χ2 ), M item(i) = HomR−R L(χ1 ) ⊗Re L(χ2 ), M ,

whose composition gives exactly the stated isomorphism.

Let us come back to our situation. We are considering the functor χ : k(N) → L Comod sending n → Qn . On the one hand, we already observed that the composition of χ with the left forgetful functor gives rise to a fiber functor k(N) → add(R R). On the other hand, we can consider also the fiber functor χ : k(N) → R ModR obtained by composing the functor χ : k(N) → L Comod with the functor L Comod → R ModR introduced in Lemma 3.4.7. Note that Σ(χ) = Q = ⊕n≥0 Qn . It is clear from Lemma 4.2.8, that any multiplication on L(χ) = (Q ⊗B ∨ Q)† comes from a natural transformation (χ ⊗R χ) → L(χ) ⊗R (χ ⊗R χ). The later can be constructed using the left L(χ)-coaction on the Qn ’s defined in (54). Thus we obtain the following result. Lemma 4.2.9. Let Q• be the cochain complex of Subsection 4.1, and (Q ⊗B ∨ Q)† the associated R-coring. en,m : Qn ⊗R Qm → Then there is a natural transformation (χ ⊗R χ) → L(χ) ⊗R (χ ⊗R χ) given by: λ ∨ † (Q ⊗B Q) ⊗R (Qn ⊗R Qm ) un ⊗R um − 7 → i Xh (un ⊗A um ) ⊗B (∗ ωn, α ⋆ ∂ ∗ ωm, β ) + (un ⊗A ∂um ) ⊗B (∗ ωn, α ⋆ ∗ ωm, β ) ⊗R ωn, α ⊗R ωm β α, β

P for every n, m ≥ 1, and by e λ0, n = e λn, 0 : Qn → (Q ⊗B ∨ Q)† ⊗R Qn , un 7−→ α (un ⊗B ∗ ωn, α ) ⊗R ωn, α , where {(ωn, α , ∗ ωn, α )} is a dual basis for R Qn , n ≥ 1.

Proof. This is a routine computation using definitions and dual bases notions.

We then arrive to the Re -ring structure of (Q ⊗B ∨ Q)† . Proposition 4.2.10. There is a structure of Re -ring on D := (Q ⊗B ∨ Q)† given by the extension of rings Re → D sending r ⊗ so 7→ (r ⊗B s)† (i.e. ι0 (r) ⊗B ι0 (s)), where the multiplication of D is defined by the following rules: for every pair of generic elements (un ⊗B ϕn )† and (um ⊗B ϕm )† of D with n, m > 0, we set † † † † (un ⊗B ϕn ) . (um ⊗B ϕm ) = (un ⊗A ∂um ) ⊗B (ϕn ⋆ ϕm ) + (un ⊗A um ) ⊗B (ϕn ⋆ ∂ϕm ) and

(un ⊗B ϕn )† . (r ⊗B s)† = (un r ⊗B sϕn )† ,

(r ⊗B s)† . (un ⊗B ϕn )† = (run ⊗B ϕn s), ∀r, s ∈ R.

Proof. Using Lemmas 4.1.4 and 4.2.7, one can shows that each of the maps e λn,m given in Lemma 4.2.9 is ∨ coassociative with respect to the comultiplication of Q ⊗B Q. Hence, its image by the natural isomorphism of Lemma 4.2.8 leads to the stated associative multiplication. The unitary property is clear.


37

Remark 4.2.11. As we have seen, the construction of an Re -ring structure on D is not an immediate task. This is probably due to the fact that, although the category k(N) is a monoidal category, the fiber functor χ : k(N) → R ModR given by the complex Q• is not strong monoidal since the local ”comultiplication” maps Qn+m → Qn ⊗R Qm , m, n ≥ 1, see Remark 4.1.3, do not necessary form a natural isomorphisms. Of course, this has prevented us from directly using the results already existing in the literature, for example [17]. Proposition 4.2.12. Set D := Re (Q ⊗B ∨ Q)† Re , where Q• is the cochain complex defined in Subsection 4.1. Then D has a structure of left R-bialgebroid. Proof. Let us show that ∆(D) ⊆ D ×R D, where ∆ is given by (53). To this end, fix a dual basis {(ωn,α, ∗ ωn,α )}α for each Qn , n ≥ 0. Let (un ⊗B ϕn )† be a generic element in D. Then for every element r ∈ R, we have † X X ∗ † o ∗ †o † r. (un ⊗B ωn,α ) ⊗R (ωn,α ⊗B ϕn ) = (un ⊗B ωn,α) (1 ⊗ r ) ⊗R ωn,α ⊗B ϕn α

α

† † X ∗ = un ⊗B r ωn,α ⊗R ωn,α ⊗B ϕn α

† † X ∗ = un ⊗B ωn,α ⊗R ωn,α r ⊗B ϕn α

† † X = un ⊗B ∗ ωn,α ⊗R ωn,α ⊗B ϕn r. α

This shows that ∆((un ⊗B ϕn )† ) ∈ D ×R D, for any un ∈ Qn and ϕn ∈ ∗ Qn . Hence ∆(D) ⊆ D ×R D. Using the natural transformations given in (32) and (34), we can show that the structure of infinite comatrix R-coring of D l induces a structure of ×R -coalgebra over D with structures maps: ∆

D (un ⊗B ϕn )†

P /

/ α (un

D ×R D

⊗B ∗ ωn,α )† ×R (ωn,α ⊗B ϕn )† ,

D (un ⊗B ϕn )†

ε

Endk (R) h i / r 7→ ϕn (un r) . /

Let us now check that ∆ is a multiplicative map. In the forthcoming steps we will not use arguments concerning the Re -biaction of D, so we will drop the dag upper-script when writing elements of D. Thus, for every pair of generic elements (un ⊗B ϕn ) and (um ⊗B ϕm ) in D with n, m > 0, we have X (un ⊗B ∗ ωn,α)(um ⊗B ∗ ωm,β ) ×R (ωn,α ⊗B ϕn )(ωm,β ⊗B ϕm ∆(un ⊗B ϕn )∆(um ⊗B ϕm ) = α, β

i Xh (un ⊗A ∂um ) ⊗B (∗ ωn,α ⋆ ∗ ωm,β ) + (un ⊗A um ) ⊗B (∗ ωn,α ⋆ ∂ ∗ ωm,β ) = α, β

h i ×R (ωn,α ⊗A ωm,β ) ⊗B (ϕn ⋆ ∂ϕm ) + (ωn,α ⊗A ∂ωm,β ) ⊗B (ϕn ⋆ ϕm ) .

That is ∆(un ⊗B ϕn )∆(um ⊗B ϕm ) =

i h i Xh (un ⊗A ∂um ) ⊗B (∗ ωn,α ⋆ ∗ ωm,β ) ×R (ωn,α ⊗A ωm,β ) ⊗B (ϕn ⋆ ∂ϕm ) α, β

i h i Xh (un ⊗A ∂um ) ⊗B (∗ ωn,α ⋆ ∗ ωm,β ) ×R (ωn,α ⊗A ∂ωm,β ) ⊗B (ϕn ⋆ ϕm ) + α, β

i h i Xh (un ⊗A um ) ⊗B (∗ ωn,α ⋆ ∂ ∗ ωm,β ) ×R (ωn,α ⊗A ωm,β ) ⊗B (ϕn ⋆ ∂ϕm ) + α, β

38


+

i h i Xh (un ⊗A um ) ⊗B (∗ ωn,α ⋆ ∂ ∗ ωm,β ) ×R (ωn,α ⊗A ∂ωm,β ) ⊗B (ϕn ⋆ ϕm ) . α, β

Since the first and the last terms vanish by Lemma 4.2.7, we then get using Lemma 4.1.4 that ∆(un ⊗B ϕn )∆(um ⊗B ϕm ) = ∆ (un ⊗A um ) ⊗B (ϕn ⋆ ∂ϕm ) + ∆ (un ⊗A ∂um ) ⊗B (ϕn ⋆ ϕm )

whence ∆(un ⊗B ϕn )∆(um ⊗B ϕm ) = ∆((un ⊗B ϕn )(um ⊗B ϕm )) which implies that ∆ is multiplicative. Therefore, ∆ is a morphism of Re -rings since ∆(1D ) = 1D ×R 1D . We need to check that ε is also multiplicative. By definition we have ε(1R ⊗B 1R ) = 1Endk (R) . Take two generic elements un ⊗B ϕn and um ⊗B ϕm in D with n, m > 0, un = a0 ∂a1 ⊗A · · · ⊗A ∂an−1 ∈ Qn and um = b0 ∂b1 ⊗A · · · ⊗A ∂bm−1 ∈ Qm . Then, for every element r ∈ R, we have on the one hand that ε(un ⊗R ϕn )ε(um ⊗B ϕm )(r) = ϕn un ϕm (um r) . On the other hand, we have ε (un ⊗B ϕn ) . (um ⊗B ϕm ) (r) = ε (un ⊗A um ) ⊗B (ϕn ⋆ ∂ϕm ) + (un ⊗A ∂um ) ⊗B (ϕn ⋆ ϕm ) (r) = (ϕn ⋆ ∂ϕm )(un ⊗A um r) + (ϕn ⋆ ϕm )(un ⊗A ∂um r)

An easy computation shows that the first summand is (ϕn ⋆ ∂ϕm )(un ⊗A um r) = ϕn (un b0 ϕm (∂b1 ⊗A · · · ⊗A ∂bm−1 r)), while the second one is (ϕn ⋆ ϕm )(un ⊗A ∂um r) = ϕn (un ϕ(um r)) − ϕn (un b0 ϕm (∂b1 ⊗A · · · ⊗A ∂bm−1 )). Therefore,

ε(un ⊗B ϕn )ε(um ⊗B ϕm )(r) = ε un ⊗B ϕn . um ⊗B ϕm (r),

for every element r ∈ R. Thus ε is now a morphism of Re -rings, and this completes the proof.

4.3. The isomorphism between comatrices and coendomorphisms bialgebroids. Now, we come back to the canonical map. As was mentioned in the preamble of the previous subsection, there is a canonical map given explicitly by (56). Thus, using the L -coactions of Proposition 4.1.5, we have a morphism of R-corings canB : D l −→ L l sending X π(a0 ⊗ ∗ ei0 ) · · · π(an−1 ⊗ ∗ ein−1 )ϕn ei0 ∂ei1 ⊗A · · · ⊗A ∂ein−1 , (61) (un ⊗B ϕn ) 7−→ i0 , i1 ,··· , in−1

where un = a0 ∂a1 ⊗A · · · ⊗A ∂an−1 ∈ Qn , and canB (r ⊗B s) = π(r ⊗ so ), for r, s ∈ R. Our next goal is to show that canB is an isomorphism of left R-bialgebroids. To this end, we will need the following proposition. Proposition 4.3.13. For every n ≥ 1, un = a0 ∂a1 ⊗A · · · ⊗A ∂an−1 ∈ Qn and ϕn ∈ ∗ Qn , we have the following equality i† X h (a0 ⊗B ∗ ei0 ).(a1 ⊗B ∗ ei1 ) · · · (an−1 ⊗B ∗ ein−1 ) ϕn (ei0 ∂ei1 ⊗A · · · ⊗A ∂ein−1 ) (un ⊗B ϕn )† = i0 ,i1 ,··· , in−1

viewed as elements in the left Re -module D l . In particular, D is generated, as an Re -ring, by the image of Re and the set of elements {(ei ⊗B ∗ ej )}i, j (recall that {(ei , ∗ ei )}i is a dual basis of R A).


39

Proof. We proceed by induction on n. For n = 1, we have, for every element a ∈ A and ϕ ∈ ∗ A, X X (a ⊗B ∗ ei )† ϕ(ei ) = (1 ⊗ ϕ(ei )o )(a ⊗B ∗ ei )† i

i

=

X

(a ⊗B ∗ ei )(ϕ(ei ) ⊗ 1o )

i

=

X

(a ⊗B ∗ ei ϕ(ei ))†

†

i

= (a ⊗B ϕ)† . Now consider un+1 = a0 ∂a1 ⊗A · · · ⊗A ∂an ∈ Qn+1 and ϕn+1 ∈ ∗ Qn+1 . For α = (i0 , i1 , · · · , in−1 ) we set ωn,α = ei0 ∂ei1 ⊗A · · · ⊗A ∂ein−1 and ∗ ωn,α = ∗ ei0 ⋆ ∗ ei1 ⋆ · · · ⋆ ∗ ein−1 , see (49). By Lemma 4.1.4, we know that {(ωn,α, ∗ ωn,α }α is a dual basis for Qn . Hence {(ωn,α ⊗A ∂ein , ∗ ωn,α ⋆ ∗ ein )}α, in is a dual basis for Qn+1 still by Lemma 4.1.4. Thus i† h i† Xh un+1 ⊗B (∗ ωn,α ⋆ ∗ ein )ϕn+1 (ωα,n ⊗A ∂ein ) un+1 ⊗B ϕn+1 = α,in

i† Xh (un ⊗A ∂an ) ⊗B (∗ ωn,α ⋆ ∗ ein )ϕn+1 (ωα,n ⊗A ∂ein ) = α,in

=

X α,in

=

X α,in

−

1 ⊗ (ϕn+1 (ωα,n ⊗A ∂ein ))o

i† h (un ⊗A ∂an ) ⊗B (∗ ωn,α ⋆ ∗ ein )

i† h ∗ ∗ (un ⊗B ωn,α ).(an ⊗B ein ) 1 ⊗ (ϕn+1 (ωα,n ⊗A ∂ein )) o

i† h X 1 ⊗ (ϕn+1 (ωα,n ⊗A ∂ein ))o (un an ⊗B (∗ ωn,α ⋆ ∂ ∗ ein )) α,in

An easy argument using the star product (i.e the convolution product of (49)) shows that the second summand in the last equality vanishes. Henceforth, h i† X un+1 ⊗B ϕn+1 = 1 ⊗ (ϕn+1 (ωα,n ⊗A ∂ein ))o (un ⊗B ∗ ωn,α)† .(an ⊗B ∗ ein )† . α,in

Using induction we then obtain i† h un+1 ⊗B ϕn+1 = i† h X ∗ o ∗ o ∗ (a0 ⊗B ej0 ) · · · (an−1 ⊗B ejn−1 ) .(an ⊗B ∗ ein )† , 1 ⊗ ωn,α(ωn,α′ ) 1 ⊗ (ϕn+1 (ωn,α ⊗A ∂ein ))

α,in ,α′

α′ = (j0 , j1 , · · · , jn−1 ) i† h X ∗ o ∗ ∗ (a0 ⊗B ej0 ) · · · (an−1 ⊗B ejn−1 ) .(an ⊗B ∗ ein )† 1 ⊗ (ϕn+1 ( ωn,α (ωn,α′ )ωn,α ⊗A ∂ein )) = where

α,in ,α′

=

i† h X 1 ⊗ (ϕn+1 (ωn,α′ ⊗A ∂ein ))o (a0 ⊗B ∗ ej0 ) · · · (an−1 ⊗B ∗ ejn−1 ) .(an ⊗B ∗ ein )†

in ,α′

40


which says that X

(un+1 ⊗B ϕn+1 )† =

i0 ,i1 ,··· , in−1 ,in

i† h (a0 ⊗B ∗ ei0 ) · · · (an ⊗B ∗ ein ) ϕn+1 (ei0 ∂ei1 ⊗A · · · ⊗A ∂ein−1 ⊗A ∂ein ).

and this establishes the stated equality. The last part of the statement is an easy consequence of this equality. Theorem 4.3.14. The canonical map canB : D → L of (61) is an isomorphism of left R-bialgebroids. Proof. First we will show that canB is a multiplicative map. By Proposition 4.3.13 this is equivalent to show that (62) canB (a ⊗B ϕ) canB (un ⊗B ϕn ) = canB (a ⊗B ϕ) (un ⊗B ϕn ) ,

for every a ∈ A, ϕ ∈ ∗ A, un ∈ Qn , ϕn ∈ ∗ Qn with n ≥ 1. Let un be of the form un = b0 ∂b1 ⊗A · · · ⊗A ∂bn−1 . We have canB aun ⊗B (ϕ ⋆ ∂ϕn ) X π(ab0 ⊗ ∗ ej0 )π(b1 ⊗ ∗ ei1 ) · · · π(bn−1 ⊗ ∗ ein−1 )(ϕ ⋆ ∂ϕn )(ej0 ∂ei1 ⊗A · · · ⊗A ∂ein−1 ) = j0 , i1 ,··· , in−1

X

=

π(a ⊗ ei0 ∗ ej0 )π(b0 ⊗ ∗ ei0 ) · · · π(bn−1 ⊗ ∗ ein−1 )(ϕ ⋆ ∂ϕn )(ej0 ∂ei1 ⊗A · · · ⊗A ∂ein−1 )

i0 , j0 , i1 ,··· , in−1 (49)

=

X

i0 , j0 , i1 ,··· , in−1

X

=

k0 , i0 , j0 , i1 ,··· , in−1

X

=

k0 , i0 , j0 , i1 ,··· , in−1

X

=

π(a ⊗ ei0 ∗ ej0 )π(b0 ⊗ ∗ ei0 ) · · · π(bn−1 ⊗ ∗ ein−1 )ϕ ej0 ϕn ∂ei1 ⊗A · · · ⊗A ∂ein−1

π(a ⊗ ∗ ek0 ∗ ej0 (ek0 ei0 ))π(b0 ⊗ ∗ ei0 ) · · · π(bn−1 ⊗ ∗ ein−1 )ϕ ej0 ϕn ∂ei1 ⊗A · · · ⊗A ∂ein−1

π(a ⊗ ∗ ek0 )π(b0 ⊗ ∗ ei0 ) · · · π(bn−1 ⊗ ∗ ein−1 ) ∗ ej0 (ek0 ei0 )ϕ ej0 ϕn ∂ei1 ⊗A · · · ⊗A ∂ein−1 π(a ⊗ ∗ ek0 )π(b0 ⊗ ∗ ei0 ) · · · π(bn−1 ⊗ ∗ ein−1 ) ϕ

k0 , j0 , i0 , i1 ,··· , in−1

=

X

∗

ej0 (ek0 ei0 )ej0 ϕn ∂ei1 ⊗A · · · ⊗A ∂ein−1

π(a ⊗ ek0 )π(b0 ⊗ ei0 ) · · · π(bn−1 ⊗ ein−1 ) ϕ ek0 ei0 ϕn ∂ei1 ⊗A · · · ⊗A ∂ein−1 ∗

k0 , i0 , i1 ,··· , in−1

∗

∗

where in the second equality we have used the definition of the multiplication in L and in the fifth one the left Re -action of L . On the other hand, we have canB a∂un ⊗B (ϕ ⋆ ∂ϕn ) X π(a ⊗ ∗ ek0 )π(b0 ⊗ ∗ ei0 ) · · · π(bn−1 ⊗ ∗ ein−1 )(ϕ ⋆ ϕn )(ek0 ∂ei0 ⊗A ∂ei1 ⊗A · · · ⊗A ∂ein−1 ) = k0 , i0 , i1 ,··· , in−1

X

=

k0 , i0 , i1 ,··· , in−1

−

X

π(a ⊗ ek0 )π(b0 ⊗ ∗ ei0 ) · · · π(bn−1 ⊗ ∗ ein−1 )ϕ ek0 ϕn (ei0 ∂ei1 ⊗A · · · ⊗A ∂ein−1 )

k0 , i0 , i1 ,··· , in−1

π(a ⊗ ek0 )π(b0 ⊗ ∗ ei0 ) · · · π(bn−1 ⊗ ∗ ein−1 )ϕ ek0 ei0 ϕn (∂ei1 ⊗A · · · ⊗A ∂ein−1 ) .

Therefore, the sum of these two terms leads to


canB

41

(a ⊗B ϕ).(un ⊗B ϕn ) = canB aun ⊗B (ϕ ⋆ ∂ϕn ) + canB a∂un ⊗B (ϕ ⋆ ∂ϕn ) X π(a ⊗ ek0 )π(b0 ⊗ ∗ ei0 ) · · · π(bn−1 ⊗ ∗ ein−1 )ϕ ek0 ϕn (ei0 ∂ei1 ⊗A · · · ⊗A ∂ein−1 ) = k0 , i0 , i1 ,··· , in−1

Set rn,α = ϕn (ei0 ∂ei1 ⊗A · · ·⊗A ∂ein−1 ) ∈ R, where α = (i0 , · · · , in−1 ). Henceforth, we compute the following term canB (a ⊗B ϕ).(un ⊗B ϕn ) = X

=

π(a ⊗∗ ek0 )π(b0 ⊗ ∗ ei0 ) · · · π(bn−1 ⊗ ∗ ein−1 )ϕ(ek0 rn,α )

k0 , i0 , i1 ,··· , in−1

X

=

π(a ⊗ ϕ(ek0 rn,α )∗ ek0 )π(b0 ⊗ ∗ ei0 ) · · · π(bn−1 ⊗ ∗ ein−1 )

k0 , i0 , i1 ,··· , in−1

X

=

π(a ⊗ rn,α ϕ)π(b0 ⊗ ∗ ei0 ) · · · π(bn−1 ⊗ ∗ ein−1 )

i0 , i1 ,··· , in−1

X

=

i0 , i1 ,··· , in−1

h i o π(a ⊗ ϕ) (1 ⊗ rn,α ) π(b0 ⊗ ∗ ei0 ) · · · π(bn−1 ⊗ ∗ ein−1 )



= π(a ⊗ ϕ) 



= π(a ⊗ ϕ) 



= π(a ⊗ ϕ) 

X

i0 , i1 ,··· , in−1

X

i0 , i1 ,··· , in−1

X

i0 , i1 ,··· , in−1

 o (1 ⊗ rn,α ) π(b0 ⊗ ∗ ei0 ) · · · π(bn−1 ⊗ ∗ ein−1 )  

π(b0 ⊗ ∗ ei0 ) · · · π(bn−1 ⊗ ∗ ein−1 )rn,α 



π(b0 ⊗ ∗ ei0 ) · · · π(bn−1 ⊗ ∗ ein−1 ) ϕn (ei0 ∂ei1 ⊗A · · · ⊗A ∂ein−1 )

= canB (a ⊗B ϕ).canB (un ⊗B ϕn ).

The equality (62), is now derived by linearity from the last one we proved. Since canB preserves the unit, we deduce that canB is a morphism of Re -rings. The inverse of canB is constructed as follows. It is clear that the map ζ : (A ⊗ ∗ A)† → D sending a ⊗ ϕ 7→ a ⊗B ϕ is an Re -bilinear map. Therefore, it is canonically extended to the tensor algebra ζ : TRe ((A ⊗ ∗ A)† ) → D, as D is an Re -ring. Now, for every a, b ∈ A and ϕ ∈ ∗ A, we have ! X X ζ = (a ⊗B ei ϕ).(b ⊗B ∗ ei ) (a ⊗ ei ϕ) ⊗Re (b ⊗ ∗ ei ) i

i

=

X

ab ⊗B (ei ϕ ⋆ ∂ ∗ ei ) +

i

= ab ⊗B

X

(ei ϕ ⋆ ∂ ∗ ei )

i

= ab ⊗B

X i

= ab ⊗B ϕ,

X

!

!

i

+ a∂b ⊗B

(ei ϕ ⋆ ∂ ∗ ei ) , X i

∗

a∂b ⊗B (ei ϕ ⋆ ∗ ei ) X

(ei ϕ ⋆ ∗ ei )

i

X i

(ei ϕ ⋆ ∂ ei ) = ϕ

(ei ϕ ⋆ ∗ ei ) = 0

!

42


= ζ(ab ⊗ ϕ).

This means that ζ factors throughout the canonical projection π : TRe (A ⊗ ∗ A)† → L , and so we have an algebra map ζ : L → D. Given a ∈ A and ϕ ∈ ∗ A, we have X canB ◦ ζ(π(a ⊗ ϕ)) = canB (a ⊗B ϕ) = π(a ⊗B ∗ ei )ϕ(ei ) i

= π

X

a ⊗ ∗ ei ϕ(ei )

i

!

= π(a ⊗ ϕ).

This implies that canB ◦ ζ = idL . Now, take un ∈ Qn , n ≥ 1, of the form un = a0 ∂a1 ⊗A · · · ⊗A ∂an−1 and ϕn ∈ ∗ Qn . Then we have X ζ π(a0 ⊗ ∗ ei0 ) · · · π(an−1 ⊗ ∗ ein−1 )ϕn (ei0 ∂ei1 ⊗A · · · ⊗A ∂ein−1 ) ζ ◦ canB (un ⊗B ϕn ) = i0 , i1 ,··· , in−1

=

X

(a0 ⊗B ∗ ei0 ) · · · (an−1 ⊗B ∗ ein−1 )ϕn (ei0 ∂ei1 ⊗A · · · ⊗A ∂ein−1 )

i0 , i1 ,··· , in−1

= un ⊗B ϕn ,

by Proposition 4.3.13.

This shows that ζ ◦ canB = idD .

Corollary 4.3.15. Let (L l )∗ be the right convolution ring of the R-coring L l . Then there is an isomorphism of rings (L l )∗ ∼ = End(QB ). Proof. We know that each hn ∨ Q = ∗ Qn is finitely generated and projective right R-module. The same property holds true for each right R-module of the form ei1 ,in ∨ Q, where ei1 ,in = hi1 + · · · + hin . This means that the unital bimodule B ∨ QR satisfies the second condition of [13, Proposition 5.1] for each idempotent which belong to the set of local units of B. Therefore we have, as in the proof of [13, Proposition 5.1], that the functor − ⊗B ∨ Q is left adjoint to − ⊗R Q. Hence Hom−R D, R = Hom−R Q ⊗B ∨ Q, R ∼ = Hom−B Q, Q . Now, we conclude by Theorem 4.3.14.

4.4. Monoidal equivalence between chain complexes of k-modules and left L -comodules. In this subsection we will use the isomorphism of bialgebroids stated in Theorem 4.3.14 to show that the following are equivalent: 1) QB is faithfully flat, 2) the underlying module R⊗1o L of L is flat and the functor Q ⊗B − : B Mod → L Comod is a monoidal equivalence of categories. This is our first main result, and stated below as Theorem 4.4.18. Remark 4.4.16. Let B = k(N) ⊕ k(N) be the ring with enough orthogonal idempotents associated to the small k-linear category k(N) considered in Subsection 4.2. We have already observed in 4.2 that the category of unital left B-modules B Mod is in a canonical way isomorphic to the category Ch + (k) of chain complexes of k-modules. Therefore, B Mod inherits a structure of monoidal category. Recall that B is generated as a free k-module by the set of elements {hn , vn }n∈N with {hn }n∈N as a set of orthogonal idempotents. The multiplication of two object X, Y ∈ B Mod, is then given by M X ⊖Y = ⊕ hi X ⊗ hj X , n∈N

i+j=n

That is, hn (X ⊖ Y ) = ⊕i+j=n hi X ⊗ hj Y , for every n ∈ N, and for every k ≥ 1, l ≥ 1 with k + l = m, we have vm−1 (hk x ⊗ hl y) = vk−1 x ⊗ hl y + (−1)k hk x ⊗ vl−1 y,


43

(i.e. the Leibniz rule), and vm−1 (h0 x ⊗ hm y) = h0 x ⊗ vm−1 y,

vn (hn x ⊗ h0 y) = vn−1 x ⊗ h0 y

for every x ∈ X, y ∈ Y , and m, n ≥ 1. The multiplication of B-linear maps is obvious. The unit object is the left unital B-module k[0] whose underlying k-module is k, and whose B-action is given by ( 0, if n 6= 0 hn k[0] = k, if n = 0. We know that the cochain complex Q• of Subsection 4.1 induces an L -comodule Q = ⊕n∈N Qn whose coaction is easily seen to be right B-linear. Thus, Q ⊗B − : B Mod → L Comod, acting in the obvious way, is a well defined functor. This functor is in fact monoidal Lemma 4.4.17. Consider the monoidal categories B Mod and L Comod, with structure given in Remark 4.4.16 and Lemma 3.4.7 respectively. Then Q ⊗B − : B Mod → L Comod is a monoidal functor, with structure Γ2X,Y : (Q ⊗B X) ⊗R (Q ⊗B Y ) −→ Q ⊗B (X ⊖ Y ), given by Γ2X,Y (un ⊗B hn x) ⊗R (um ⊗B hm y)   (un ⊗A um ) ⊗B (hn x ⊗ vm−1 y) + (un ⊗A ∂um ) ⊗B (hn x ⊗ hm y), n, m ≥ 1 =  u u ⊗ (h x ⊗ h y), n = 0 or m = 0, n m B n m

for every un ∈ Qn , um ∈ Qm , x ∈ X and y ∈ Y , and Γ0 : R → Q ⊗B k[0] sending r 7→ r ⊗B h0 1.

Proof. The fact that Γ2X,Y is a well defined map comes from the observation that the right R-action of Q ⊗B X as left L -comodule is given by the right R-action of Q viewed as left L -comodule. That is, the one given by the rule (47). Now, it is easily seen that the right R-action of Q given by (47) is exactly the right R-action of Q we started with (i.e. that which comes from the inclusion R KR ⊂ A ⊗R A). A direct computation, using Lemma 4.1.6, shows that Γ2X,Y is left L -colinear, for each X, Y . We leave to the reader the proof of the associativity and unitary properties of (Γ2−,− , Γ0 ). Our first main result is the following. Theorem 4.4.18. Let R be an algebra over a commutative ground ring k, and A an R-ring which is finitely generated and projective as left R-module. Consider the associated left R-bialgebroid constructed in Proposition 3.3.5 and let B = k(N) ⊕ k(N) be the ring with enough orthogonal idempotents of (58). Consider the cochain complex Q• of Subsection 4.1 with its canonical right unital B-action and left L -coaction. Then the following statements are equivalent (1) The right module LRl is flat and the functor Q ⊗B − : monoidal categories; (2) QB is a faithfully flat module.

B Mod

−→

L Comod

is an equivalence of

Proof. The monoidal condition is, by Lemma 4.4.17, always satisfied, so it can be omitted in the proof of item (1). Henceforth, we only need to show that LRl is flat and Q ⊗B − is an equivalence if and only if QB is a faithfully flat module. By the left version of the Theorem of generalized faithfully flat descent [15, Theorem 5.9], we know that QB is faithfully flat if and only if DRl = 1⊗Ro (Q ⊗B ∨ Q) is flat and Q ⊗B − : B Mod → D l Comod is an equivalence of category. We then conclude by Theorem 4.3.14.

44


Notice that, when QB is faithfully flat, the inverse of the functor of Q ⊗B − : B Mod → L Comod is given by the cotensor product ∨ QL − : L Comod → B Mod. The structure of bicomodule on ∨ Q is given as follows. Recall that Q is in fact an (L , B)-bicomodule, that is, the left L -coaction of Q is right B-linear. So, since each of the Qn , n ≥ 0, is finitely generated and projective left R-module, each of the left duals ∗ Qn admits, using dual bases a right L -coaction, for which ∨ Q becomes a (B, L )-bicomodule. The condition LRl is flat in item (1) of Theorem 4.4.18, seems to be redundant. But, although we can deduce form the equivalence of categories that the category of left L -comodule is abelian, we can not affirm that the forgetful functor L Comod → R Mod is left exact. Thus, LRl is not necessarily a flat module, see [16, Proposition 2.1]. Consider the category Ch + (k) of chain complexes of k-modules and denote by O : Ch + (k) → B Mod the canonical isomorphism of categories, see Subsection 4.2. In the case when R = k is a field, it is known that QB is always faithfully flat wherever dimk (A) < ∞. A complete proof for a non commutative field, that is, a division ring is given in Theorem 4.6.25 below. We thus obtain the following corollary Corollary 4.4.19. [37, Theorem 4.4] Let k be a field and A an k-algebra such that 1 < dimk (A) < ∞. Consider the associated coendomorphism k-bialgebra L constructed in Proposition 3.3.5. Then the category Ch + (k) of chain complexes of k-modules is monoidally equivalent, via the functor (Q ⊗B −) ◦ O : Ch + (k) → L Comod, to the category of left L -comodules. Proof. By the foregoing observations, this is a direct consequence of Theorem 4.4.18. The composition of the functor given in Corollary 4.4.19 with the forgetful functor gives, for any chain complex V• in Ch + (k), ⊕ Qn ⊗ Vn n≥0 Q ⊗B O(V• ) = , h∂un ⊗ xn+1 − un ⊗ ∂xn+1 i n≥0

L Comod

→ k Mod

which is the functor used by D. Tambara in [37] to establish his equivalence of categories i.e. Corollary 4.4.19. 4.5. Equivalence between chain complexes of left R-modules and left L -comodules. Our main aim here is to extend the result of Theorem 4.4.18 to the category Ch + (R) of chain complexes over left R-modules. In other words, we are interested in relating the category of chain complexes of left R-modules and the category of left L (A)-comodules over the left R-bialgebroid of Proposition 3.3.5. Precisely, we show an analogue of Theorem 4.4.18 where L is replaced by its quotient R-coring L := L (A)/ h1L (r ⊗ 1o − 1 ⊗ r o )ir ∈ R and the ring B by its extension C = R(N) ⊕ R(N) . This is our second main result i.e. Theorem 4.5.24. Of course, in this case, the monoidal equivalence of categories is reduced to an equivalence, unless the base ring R is commutative and the extension A is an R-algebra. The later case will be analyzed separately in Subsection 4.7 below. Let A be an R-ring and assume that R A is a finitely generated and projective module. Fix a dual basis {(ei , ∗ ei }i for R A, and consider the left R-bialgebroid of Proposition 3.3.5: TRe (A ⊗ ∗ A)† E . L := D P † ⊗ e (a′ ⊗ ∗ e )† − (aa′ ⊗ ϕ)† , (1 ⊗ ϕ)† − 1 ⊗ ϕ(1)o (a ⊗ e ϕ) i i R i a,a′ ∈A, ϕ∈∗ A We denote by π : TRe (A ⊗ ∗ A)† → L the canonical projection. We will remove the dag up-script when writing elements of L whenever there is no matter of confusion. Lemma 4.5.20. Let J be the left ideal of L generated by the following set of elements n o π(ar ⊗ ϕ) − π(a ⊗ rϕ) . a∈A, ϕ∈∗ A, r∈R


45

Then J is a coideal of the underlying R-coring L l . Proof. An easy computation shows that

π(ar ⊗ ϕ) − π(a ⊗ rϕ) = π(a ⊗ ϕ) r ⊗ 1o − 1 ⊗ r o ,

for every elements a ∈ A, ϕ ∈ ∗ A and r ∈ R. Thus, J as left Re -bimodule is generated by the set {gr := 1L .(r ⊗ 1o − 1 ⊗ r o )}r∈R . For x ∈ L an arbitrary element and r ∈ R, we get ε(xgr ) = ε x.(1 ⊗ ε(gr )o ) = 0,

as ε(gr ) = 0. Hence, ε(J ) = 0. On the other hand, for every r ∈ R , we have

∆(gr ) = (1L ⊗R 1L )(r ⊗ 1o ) − (1L ⊗R 1L )(1 ⊗ r o ). Using these equalities we can show that, for every x ∈ L and r ∈ R, we have X X ∆(xgr ) = x(1) ⊗R x(2) (r ⊗ 1o ) − x(1) ⊗R x(2) (1 ⊗ r o ) (x)

=

X

(x)

x(1) ⊗R x(2) (r ⊗ 1o − 1 ⊗ r o ),

(x)

P

where ∆(x) = (x) x(1) ⊗R x(2) . Therefore, (π ⊗R π) ◦ ∆(xgr ) = 0, for every x ∈ L and r ∈ R, where π : L → L /J is the canonical projection. Thus J is a coideal of L . Denote by L := L /J the quotient R-coring and by π : L → L the canonical projection. Notice that π is also left L -colinear. Consider the cochain complex Q• of Subsection 4.1. We know, by Proposition 4.1.6, that each Qn is a left L -comodule whence a left L -comodule with coaction λn : Qn → L ⊗R Qn → L ⊗R Qn ,

n ≥ 0.

Lemma 4.5.21. Let n ∈ N. The L -coaction λn is right R-linear that is Qn is an (L , R)-bicomodule (here R is the trivial R-coring). Proof. For n = 0 the statement is trivial since λ0 (r) = (r ⊗ 1o )π(1L ) = π(1L )(1 ⊗ r o ), for every r ∈ R. Take n ≥ 1 and an element un ∈ Qn of the form un = a0 ∂a1 ⊗A · · · ⊗A ∂an−1 . Then, for every r ∈ R, we have λn (un r) =

X π π(a0 ⊗ ∗ ei0 ) · · · π(an−1 r ⊗ ∗ ein−1 ) ⊗R ωn,α , α, in

where α = (i0 , · · · , in−1 ), and ωn,α = ei0 ∂ei1 ⊗A · · · ⊗A ∂ein−1 X = π π(a0 ⊗ ∗ ei0 ) · · · π(an−1 ⊗ r ∗ ein−1 ) ⊗R ωn,α α

=

X α

=

X α

h i (π ⊗R Qn ) π(a0 ⊗ ∗ ei0 ) · · · π(an−1 ⊗ ∗ ein−1 ) (1 ⊗ r o ) ⊗R ωn,α h i ∗ ∗ (π ⊗R Qn ) π(a0 ⊗ ei0 ) · · · π(an−1 ⊗ ein−1 ) ⊗R ωn,αr

X = π π(a0 ⊗ ∗ ei0 ) · · · π(an−1 ⊗ ∗ ein−1 ) ⊗R ωn,α r α

= λn (un )r,

where in the fourth equality we have used that each Qn is in fact a left ×R -L -comodule, see the proof of Lemma 4.1.6. We then conclude by linearity.

46


Remark 4.5.22. The quotient R-coring L does not admit, in general, a structure of left R-bialgebroid. However, if we assume that R is commutative (i.e. a commutative k-algebra) and that A is an R-algebra, then the left ideal J is in fact a two-sided ideal, since in this case we have the following equalities gr π(a ⊗ ϕ) = π(a ⊗ ϕ)gr , for every r ∈ R, a ∈ A, and ϕ ∈ A∗ . In view of this, a direct verification shows that L is an R-bialgebroid such that the canonical surjection π : L → L is a morphism of R-bialgebroids. Notice, that here the prefix ”left” was removed before bialgebroid. This is due to the fact that L is actually an R ⊗ R-algebra, that is, there is only one structure of R ⊗ R-module. Let us consider the k-linear category R(N) whose objects are the natural numbers N and homomorphisms sets are defined by   / {n, n + 1} 0, if m ∈ (63) HomR(N) n, m = R.1n = 1n .R, if n = m  R.n+1 = n+1.R, if m = n + 1. n n

The last two terms are copies of R RR viewed as an R-bimodule which is free as left and right R-module of rank one generated by an invariant element. The composition is defined using the regular R-biactions of R RR . The induced ring with enough orthogonal idempotents is the free left R-module C = R(N) ⊕ R(N) generated by elements {hn , un }n∈N subject to the following relations: hn hm = δn, m hn , ∀n, m ∈ N (Kronecker delta) un um = um un = 0, ∀n, m ∈ N un hn+1 = un = hn un , ∀n, m ∈ N. In other words C is the ring of (N × N)-matrices  R R 0 R  0 0  (64) C =    

over R of the form  0 0  R 0   R R  .. .. ..  . . .   0 R R  .. .. .. . . .

i.e. with possibly non-zero entries in each row: (i, i) and (i, i + 1). C is also free as right R-module, since the generators are invariant. One can easily check that the category of chain complexes of left R-modules Ch + (R) is equivalent to the category of unital left C-modules. Let B be the ring with enough orthogonal idempotents of (58). There is a morphism of rings B → C with the same set of orthogonal idempotents. In this way, we have by [13, page 733] the usual adjunction between left unital B-modules and C-modules using restriction of scalars and the tensor product functor C ⊗B −. By Lemma 4.5.21, we have a morphism of rings R → EndL (Qn ), for every n ≥ 0. This leads to a faithful functor from the category R(N) to the category of (L , R)-bicomodules (here R is considered as a trivial R-coring) χ′ : R(N) → L ComodR . The composition of χ′ with the forgetful functor gives rise then to a fiber functor ω : R(N) → R ModR whose image is in add(R R). Therefore, we can apply the constructions performed in Subsection 4.2. Thus, we have an infinite comatrix R-coring Q⊗C ∨ Q together with a canonical map canC : Q ⊗C ∨ Q −→ L sending P canC ∗ ∗ / π π(a ⊗ e ) · · · π(a ⊗ e ) ϕ e ∂e ⊗ · · · ⊗ ∂e (65) un ⊗C ϕn 0 i0 n−1 in−1 i0 i1 A A in−1 . i0 , ··· , in−1


47

Clearly we have a surjective map ϑ : Q ⊗B ∨ Q → Q ⊗C ∨ Q. Moreover, we have a commutative diagram with exact rows relating the two R-corings morphisms canB and canC (see equations (61) and (65)) (66)

/ Ker(ϑ) /J

0

0

/ Q ⊗B ∨ Q

ϑ

/ Q ⊗C ∨ Q

canC

canB

/L

/0

/L

π

/0

Proposition 4.5.23. In diagram (66), we have the following equality canB (Ker(ϑ)) = J . In particular, the map canC of equation (65) is an isomorphism of R-corings. Proof. The inclusion canB (Ker(ϑ)) ⊆ J is clear from the commutative diagram (66). Conversely, let y ∈ L and r ∈ R be arbitrary elements. We need to show that ygr ∈ canB (Ker(ϑ)). There is no loss of generality if we assume that y = xπ(a ⊗ ϕ), for some x ∈ L and a ∈ A, ϕ ∈ ∗ A. Since canB is, by Theorem 4.3.14, bijective, there exists u ∈ Q ⊗B ∨ Q such that x = canB (u). In view of this, ygr = canB u(ar ⊗B ϕ − a ⊗B rϕ) , as canB is multiplicative. We will show that ϑ u (ar ⊗B ϕ − a ⊗B rϕ) = 0. This is directly obtained from the following claim: ϑ (un ⊗B ϕn ) (ar ⊗B −a ⊗B rϕ) = 0, for every un ∈ Qn , ϕn ∈ ∗ Qn . This is true for n = 0. Let us check the claim for n ≥ 1. So we have ϑ (un ⊗B ϕn ) (ar ⊗B ϕ − a ⊗B rϕ) = = = = =

(un ar ⊗C (ϕn ⋆ ∂ϕ)) + (un ⊗A ∂ar) ⊗C (ϕn ⋆ ϕ) − (un a ⊗C (ϕn ⋆ r∂ϕ)) − (un ⊗A ∂a) ⊗C (ϕn ⋆ rϕ) (un ar ⊗C (ϕn ⋆ ∂ϕ)) + (un ⊗A ∂ar) ⊗C (ϕn ⋆ ϕ) − (un a ⊗C r (ϕn ⋆ ∂ϕ)) − (un ⊗A ∂a) ⊗C r (ϕn ⋆ ϕ) (un ar ⊗C (ϕn ⋆ ∂ϕ)) + (un ⊗A ∂ar) ⊗C (ϕn ⋆ ϕ) − (un ar ⊗C (ϕn ⋆ ∂ϕ)) − (un ⊗A ∂ar) ⊗C (ϕn ⋆ ϕ) 0,

where in the second equality we have used the fact that s(ϕn ⋆ ψ) = ϕn ⋆ (sψ), for all s ∈ R and ψ ∈ ∗ A. The last statement to prove is a consequence of the first one, since the diagram (66) has exact rows. Our second main result is the following Theorem 4.5.24. Let R be an algebra over a commutative ground ring k, and A an R-ring which finitely generated and projective as left R-module. Consider the associated left R-bialgebroid L constructed in Proposition 3.3.5 and J the coideal of L generated by the set of elements {1L (r ⊗ 1o − 1 ⊗ r o )}r∈R ; denote by L = L /J the corresponding quotient R-coring. Let C = R(N) ⊕ R(N) be the ring with enough orthogonal idempotents induced from the small k-linear category R(N) defined by relations (63). Consider the cochain complex Q• given in Subsection 4.1 with its canonical right unital C-action and left L -coaction as in Lemma 4.5.21. Then the following statements are equivalent l

(1) The right module L R is flat and the functor Q ⊗C − : categories; (2) QC is a faithfully flat module.

C Mod

−→

L Comod

is an equivalence of

Proof. By the left version of the generalized faithfully flat descent Theorem [15, Theorem 5.9], we know that (Q ⊗C ∨ Q)R is flat and Q ⊗C − : C Mod → Q⊗C ∨ Q Comod is an equivalence of categories, if and only if QC is faithfully flat. We then deduced the stated equivalence by using the isomorphism of R-corings canC : Q ⊗C ∨ Q ∼ = L established in Proposition 4.5.23.

48


Notice that, if QC is faithfully flat, then the inverse functor of Q ⊗C − : C Mod → L Comod is given by the cotensor product ∨ QL − : L Comod → C Mod. Here the structure of (C, L )-bicomodule of ∨ Q is deduced, as was observed in Subsection 4.4, from that of Q using the fact that each of the Qn ’s is finitely generated and projective left R-module. 4.6. Conditions under which QC is faithfully flat. As was seen in Theorems 4.4.18 and 4.5.24, a necessary condition for establishing an equivalence of categories of left comodules and chain complexes, is the faithfully flatness of the unital right module Q. The proof of this fact is actually the most difficult task in this theory. In this subsection we will analyze assumptions under which QC is faithfully flat. The following is our third main result. Theorem 4.6.25. The notations and assumptions are that of Theorem 4.5.24. Assume further that AR is finitely generated and projective, and the cochain complex Q• is exact and splits, in the sense that, for every m ≥ 1, Qm = ∂Qm−1 ⊕ Qm = Ker(∂) ⊕ Qm as right R-modules, for some right R-module Qm . Then QC is a flat module. Furthermore, if k is a field and R is a division k-algebra, then QC is faithfully flat. Proof. We first consider the following family of right R-modules   ∂Qm ⊕ Qm , for m ≥ 1 (m) Q =  ∂Q ⊕ Q , for m = 0 0 0

which we claim to be a family of right unital flat C-modules. Using this claim we can easily deduce that (m) QC is a flat module since we know that QC = ⊕m≥0 QC . The structure of unital right C-module of each Q(m) is given as follows: Denote by im : ∂Qm → Q(m) , im : Qm → Q(m) the canonical injections and by jm , jm their canonical projections. For every element u(m) ∈ Q(m) , we set  (  / {m, m + 1} 0, if n ∈ 0, if n 6= m u(m) hn = im jm (u(m) ), if n = m u(m) un =  im γm jm (u(m) ) , if n = m i j (u(m) ), if n = m + 1 m m where γm : Qm → Qm → ∂Qm . That is, the obtained cochain complexes have the following form (m)

Q•

:0

/

0······0 /

γm Qm _ _ _ _ _ _ _ _/ ∂Q < m

BB BB BB BB !

Qm

/

0

/

0······

z zz zz z z zz

Put en, n+1 = hn + hn+1 , for every n ≥ 0. These are idempotents elements in C, and the induced rings, i.e. en, n+1 Cen, n+1 are all isomorphic to the upper-triangular matrices over R. That is, we have R R Cn, n+1 : = en, n+1 Cen, n+1 = , for every n ∈ N. 0 R

It is clear that, for every m ≥ 0, we have Q(m) em, m+1 = Q(m) . Therefore, there is an isomorphism of right unital C-modules (67) Q(m) em, m+1 ⊗C em, m+1 C ∼ = Q(m) . m, m+1

Next we will show that each of the right Cm, m+1 -modules Q(m) em, m+1 = Q(m) is finitely generated and projective. This fact, combined with the isomorphisms (67), establish the above claim. For m = 0, it is clear that the right C0, 1 -module R R 1R 0 (0) Q = R ⊕ R = 0 0 = 0 0 C0, 1


49

is finitely generated and projective. Now take m ≥ 1, under the hypothesis AR is finitely generated and projective, we can show, as in Lemma 4.1.4, that each right R-module Qm is also finitely generated and projective. Thus, we can consider a dual basis {(qm,k , q∗m,k )}k for each right R-module Qm . In this way, we have a right Cm, m+1 -linear map    ∗ qm,k (jm (u(m) ) q ∗m,k (xm ) ∗ (m) u(m) 7−→   , θm, −→ Cm, m+1 , k : Q 0 0

where xm ∈ Qm is the projection of xm ∈ Qm = ∂Qm−1 ⊕ Qm , defined by jm (u(m) ) = ∂xm ∈ ∂Qm . We ∗ should mention that, under our assumptions, the maps θm,k are well defined. Effectively, if there is some (m) other element ym ∈ Qm such that jm (u ) = ∂xm = ∂ym , then xm −ym ∈ Ker(∂m ) = ∂Qm−1 which means ∗ that they have equal image xm = y m in Qm ∼ = Qm /∂Qm−1 . It is convenient to check that θm, k are right Cm, m+1 -linear. But first we will identify the right module Qm with the quotient of Qm , Qm = Qm /∂Qm−1 . The right Cm, m+1 -action of Q(m) is given as follows: Take an element u(m) ∈ Q(m) and write it in the form u(m) = (qm , ∂pm ) for some elements qm , pm ∈ Qm . Here jm (u(m) ) = ∂pm and jm (u(m) ) = qm . So r11 r12 = qm r11 , ∂qm r12 + ∂pm r22 , (qm , ∂pm ) 0 r22 r11 r12 for every element in Cm, m+1 . Therefore, 0 r22    r11 r12 ∗ ∗     θm, , ∂p ) q r , ∂q r + ∂p r (q = θ m m m 11 m 12 m 22 k m, k 0 r22   q ∗m, k (qm r11 ) q ∗m,k qm r12 + pm r22   =   0 0  ∗  q m, k (qm r11 ) q ∗m,k (qm r12 ) + q∗m,k (pm r22 )  =  0 0  ∗  q m, k (qm ) r11 q ∗m,k (qm ) r12 + q∗m,k (pm ) r22  =  0 0  ∗    q m,k (qm ) q ∗m,k (pm ) r11 r12   =  0 r22 0 0   r11 r12 ∗  . = θm, (q , ∂p ) m k m 0 r22 Take an arbitrary element (qm , ∂pm ) ∈ Q(m) , we have (qm , ∂pm ) = (qm , 0) + (0, ∂pm ) 0 1 = (qm , 0) + (pm , 0) 0 0

50


=

X k

=

X

(q m, k ,

k

=

X

(q m, k ,

k

=

X

(q m, k ,

k

= which shows that

n

X

0 1 0) + 0) 0 0 k ∗ X q m, k (qm ) 0 q ∗m, k (pm ) 0 0 1 + (qm, k , 0) 0) 0 0 0 0 0 0 k ∗ X q m, k (qm ) 0 0 q ∗m, k (pm ) (qm, k , 0) + 0) 0 0 0 0 k ∗ q (q ) q ∗m, k (pm ) 0) m, k m 0 0 ∗ 0) θm, k qm , ∂pm ,

(q m, k q ∗m, k (qm ),

(q m, k ,

k

(qm, k , 0),

∗ θm, k

o

k

X

(qm, k q ∗m, k (pm ),

is a dual basis for the right Cm, m+1 -module Q(m) , and this finishes

the proof of the main statement. If we assume now that k is a field and R is a division k-algebra, then one can show as follows that each Q(m) em, m+1 is a progenerator in the category of right Cm, m+1 -modules. This will imply that Q(m) em, m+1 ⊗Cm, m+1 − is afaithful functor. Thus, by identifying each ring Cm, m+1 with the upper triangular matrix ring R R , we know that T = eT ⊕ (1 − e)T , where e is the obvious idempotent element. The T := 0 R (m) structure of right T -module of Q(m) is given by the decomposition QT = ∂Qm ⊕ Qm with a surjective canonical map γm : Qm → ∂Qm . Since R is a division ring and each component of Q(m) is by assumption finite dimensional with d = dimR (Qm ) ≤ dimR (∂Qm ) = d′ , we can split Q(m) as d′ −d Q(m) ∼ , = (eT )d ⊕ (1 − e)T (m)

and this shows that QT is a progenerator. Let f : X → Y be a morphism of right unital C-modules such that Q ⊗C f = 0. Hence Q(m) ⊗C f = 0, for every m ≥ 0, as QC = ⊕m≥0 Q(m) . Therefore, we have 0 = Q(m) ⊗C f ∼ em, m+1 C ⊗C f, ∀m ≥ 0 =⇒ em, m+1 C ⊗C f = 0, ∀m ≥ 0. = Q(m) em, m+1 ⊗C m, m+1

This means that hm C ⊗C f = 0, for every m ≥ 0, and so f = 0. This shows that Q ⊗C − is a faithful functor, which completes the proof. Remark 4.6.26. As one can see, the hypothesis on the complex Q• in Theorem 4.6.25, is not easy to check. However, under further conditions on the ring extension R → A, this hypothesis is satisfied. For instance, it is clear from Lemma 4.1.1 and Remark 4.1.2 that it is satisfied by assuming that the ring extension R → A splits either in the category of right or left R-modules. Obviously this includes the case when A is free as right (or left) R-module with 1A as an element of the canonical basis. In particular, this is the case when R is a division ring. Corollary 4.6.27. Let D be a division k-algebra over a field k, and A a D-ring which is finite dimensional as left and right D-vector space with dimension ≥ 2. Consider the associated left D-bialgebroid L constructed in Proposition 3.3.5 and its coideal J of Lemma 4.5.20. Then the category Ch + (D) of chain complexes of left D-vector spaces is equivalent to the category of left (L /J )-comodules. Proof. It follows from Theorems 4.5.24 and 4.6.25.

The following diagram displays the relationship between chain complexes and left comodules treated in this section. There, the second and third horizontal arrows mean a canonical adjunction, while the first one is not always an adjunction unless some flatness or purity conditions are provided. Of course, the dashed


51

arrows are by Theorems 4.4.18 and 4.5.24 an equivalence of categories, whenever QC and QB are faithfully flat. Obviously, the last condition can be obtained form the first one, providing the extension B → C is right faithfully flat. /

L Comod I ]