DG-INJECTIVE COVERS, #- INJECTIVE COVERS ALINA IACOB
Abstract. Let R be a left noetherian ring. We show that every complex of left R-modules has a # -injective cover. We use this result to prove that over a commutative noetherian ring of finite Krull dimension every complex has a DG-injective cover if and only if the ring has finite global dimension.
1. Introduction The class of DG-injective complexes (see Section 2 for definitions) was introduced by Avramov and Halperin ([2]) in the more general context of DG-modules over DG-rings. Avramov and Foxby defined the injective dimension for unbounded complexes by means of DG-injective resolutions([1]). The DG-injective complexes were also a key ingredient in Enochs, Jenda and Xu’s work on orthogonality in the category of complexes ([6]). They proved the existence of various precovers and preenvelopes with respect to the classes of DG-injective, DG-projective and exact complexes. In particular they showed that every complex has a DGinjective envelope (see section 2 for definitions). The use of covers and envelopes in their work is fundamental and affords a new description of the derived category of a ring ([6], Theorem 4.5). We consider here the question of the existence of DG-injective covers. The question was raised by Garcia-Rozas in his book, ”Covers and envelopes in the category of complexes”. He proved in [8] (Theorem 3.2.18) that over a Gorenstein ring the following are equivalent: a) every complex has a DG-injective cover; b) l.gl.dimR < ∞. We prove (Proposition 3) that over a left noetherian ring R the following are equivalent: a) every complex of left R-modules has a DG-injective cover; Key words and phrases. cover, DG-injective complex, #-injective complex. 1
2
ALINA IACOB
b) every complex of injective left R-modules is a DG-injective complex. We use this result to show (Theorem 3) that if R is a commutative noetherian ring of finite Krull dimension then the following are equivalent: a) every complex has a DG-injective cover; b) gl.dimR < ∞. We also consider the question of the existence of #-injective covers. The #-injective complexes were introduced by Avramov and Foxby; these are the complexes of injective R-modules. We prove (Proposition 1) that over a left noetherian ring every complex of left R-modules has a #-injective precover. Since the class of #-injective complexes over a left noetherian ring is closed under direct limits it follows that over such a ring every complex has a #-injective cover (Theorem 2).
2. Preliminaries ∂
2 A (chain) complex C of R-modules is a sequence C = . . . → C2 − →
∂
∂
∂−1
1 0 C1 − → C0 − → C−1 −−→ C−2 → . . . of R-modules and R-homomorphisms such that ∂n−1 ◦ ∂n = 0 for all n ∈ Z.
∂ −2
∂ −1
∂0
A chain complex of the form C = . . . → C −2 −−→ C −1 −−→ C 0 −→ ∂1
C 1 −→ C 2 → . . . is called a cochain complex. In this case ∂ n+1 ◦ ∂ n = 0 for all n ∈ Z. We note that a cochain complex is simply a chain complex with C i replaced by C−i and ∂ i by ∂−i . Throughout the paper we use both the subscript notation for complexes and the superscript notation. If X and Y are both complexes of R-modules then Hom(X, Y ) denotes the complex with Hom(X, Y )n = Πq=p+n HomR (Xp , Yq ) and with differential given by ∂(f ) = ∂ ◦ f − (−1)n f ◦ ∂, for f ∈ Hom(X, Y )n . Definition 1. ([1]) A complex I is DG-injective if each In is injective and if Hom(E, I) is exact for any exact complex E. It is known that a complex I = . . . → 0 → 0 → I n0 → I n0 +1 → I n0 +2 → . . . with each I n an injective R-module is DG-injective i.e. every bounded below complex of injective modules is DG-injective ([1], Remark 1.1.I).
DG-INJECTIVE COVERS, #- INJECTIVE COVERS
3
It is also known ([1], Proposition 3.4 I) that if l.gl.dimR < ∞ then any complex I with all In injective left R-modules is a DG-injective complex. It is not known whether or not the converse is true. We proved ([7]) that the converse is true in the case when R is a commutative noetherian ring of finite Krull dimension. More precisely we have: Theorem 1. For a commutative noetherian ring R of finite Krull dimension the following are equivalent: (1)Any complex of injective R-modules is DG-injective; (2) gl.dim.R < ∞. (the result follows from [7], theorems 18 and 24). Throughout this paper, for two objects X and Y of an abelian category A with enough injectives and enough projectives, Hom(X, Y ) denotes the set of morphisms from X to Y and Exti (X, Y ) are the right derived functors of Hom. So these extension functors are computed using the classical projective resolutions of X or the classical injective resolutions of Y . Definition 2. If F is a class of objects in an abelian category A, ⊥ F will denote the class of objects M of A such that Ext1 (M, F ) = 0 for all F ∈ F and F⊥ will denote the class of objects N such that Ext1 (F, N ) = 0 for all F ∈ F. Definition 3. ([8]) Let F be a class of objects of an abelian category A. Let X be an object of A. We say that F ∈ F is an F-precover if there is a homomorphism Φ : F → X such that the diagram FÂ 0 Ä
F
Â
φ
ÂÂÂ
ÂÂ
² / X
can be completed for each homomorphism F 0 → X with F 0 ∈ F. If, moreover, any f : F → F such that Φ = Φ ◦ f is an automorphism of F , then Φ : F → X is called an F-cover of X. Dually we have the concepts of F-preenvelope and F-envelope.
4
ALINA IACOB
When A is the category of complexes and F is the class of DG-injective complexes we obtain the definition of a DG-injective envelope. Enochs, Jenda and Xu proved ([6], Theorem 3.12) that every complex has a DG-injective envelope. Definition 4. An F-precover φ : F → X is special if Kerφ ∈ F⊥ . An F-preenvelope φ : X → F is said to be special if Cokerφ ∈⊥ F. It is easy to find examples of preenvelopes and precovers that are not special, but under suitable conditions, all envelopes and covers are special. Lemma 1. (Wakamatsu’s Lemma) If F is a class of objects of A closed under extensions, then all F-covers and all F-envelopes are special. 3. Over a noetherian ring every complex has a #-injective cover Let R be an associative ring with identity. By R-module we mean a left R- module. Definition 5. ([1]) A complex of injective R-modules is called a #injective complex. By Definition 3, a morphism of complexes φ : E → X is a #-injective precover of X if E is #-injective and if for any #-injective complex J the sequence Hom(J, E) → Hom(J, X) → 0 is exact. If moreover, any f : E → E such that φ ◦ f = φ is an automorphism of E then φ : E → X is a #-injective cover of X. We show first (Lemma 2) that over a left noetherian ring every bounded complex has a # -injective precover . We use this to show (Lemma 3) that if R is a left noetherian ring then every bounded above complex of R-modules X = . . . → X2 → X1 → X0 → 0 has a #- injective precover. Using Lemma 3 we prove that over a left noetherian ring every complex has a #- injective precover. Since every direct limit of # -injective complexes over a noetherian ring is still # -injective , it follows that over such a ring every complex has a #-injective cover (Theorem 2). We recall first the following definition: Definition 6. ([4]) If N is a left R-module then a complex . . . → E1 → E0 → N → 0 is called an injective resolvent of N if each Ei is an injective left R-module, if E0 → N , E1 → Ker(E0 → N ) and
DG-INJECTIVE COVERS, #- INJECTIVE COVERS
5
Ei → Ker(Ei−1 → Ei−2 ) for i ≥ 2 are injective precovers (in the sense of Definition 3 with A the category of left R-modules and F the class of injective left R-modules). If all these maps are injective covers then we say that the complex is a minimal injective resolvent of N . We note that a complex E = . . . → E1 → E0 → N → 0 is an injective resolvent of N if and only if each Ei is injective and Hom(S, E) is an exact complex for any injective module S. By [4], Theorem 1.2, over a noetherian ring R, every module has an injective cover. Thus over such a ring every module has a minimal injective resolvent. Our first result is the following: Lemma 2. If R is a left noetherian ring then every bounded complex of R-modules has a #-injective precover. Proof. For each n ≥ 0 let X(n) = . . . → 0 → Xn → . . . → X1 → X0 → 0 with X0 in the zeroth place. We show by induction that for each n ≥ 0 there is a #-injective precover D(n) → X(n) with kernel Ln such that D(n) ⊂ D(n + 1), D(n)k = D(n + 1)k for 1 ≤ k ≤ n, H k Hom(S, D(n)) = 0, for any k ≥ n + 1 and for any injective module S, such that Ln ⊂ Ln+1 , Lkn+1 = Lnk for 1 ≤ k ≤ n, and such that Hom(S, Ln ) is an exact complex for any injective R-module S. I. n = 0 f1 f0 Let . . . → E1 − → E0 − → X0 → 0 be a minimal injective resolvent of f2 f1 0 X0 and let E = . . . → E2 − → E1 − → E0 → 0 be the deleted minimal injective resolvent. Let A be a complex of injective R-modules and let β ∈ Hom(A, X(0)). ...
...
/ A1 Â ÂÂÂ Â γ1 ÂÂÂ ² / E1
0
a1
f1
/ A0 Â ÂÂÂ Â γ0 ÂÂÂ ² / E0 ÂÂ ÂÂÂ f0 ÂÂÂ ²
/ X0
/ A−1
/ 0
/ 0
/ ...
6
ALINA IACOB
f0
Since β0 ∈ Hom(A0 , X0 ), A0 is injective and E0 − → X0 is an injective cover there is γ0 ∈ Hom(A0 , E0 ) such that β0 = f0 ◦ γ0 . Then f0 γ0 a1 = β0 a1 = 0. Since A1 is injective, γ0 a1 ∈ Hom(A1 , Kerf0 ) and E1 → Kerf0 is an injective cover there is γ1 ∈ Hom(A1 , E1 ) such that γ0 a1 = f1 γ1 . Similarly, there is γi ∈ Hom(Ai , Ei ) such that fi γi = γi−1 ai , for any i ≥ 1. So E 0 → X(0) is a #-injective precover. Since E 0 is a deleted minimal injective resolvent of X0 , any map E 0 → E 0 that makes the diagram E0
 ÄÄ ÂÂ Ä Â φ ÄÄ ÄÄ ÂÂÂ Ä ÄÄ Â² ÄÄ Ä / X(0) E0
commutative is an isomorphism ([5], pp. 169). So E 0 → X(0) is a # -injective cover. Let L0 = Ker(E 0 → X(0)). Since L0 = . . . → E2 → E1 → Kerf0 → 0 is a minimal injective resolvent of Kerf0 , the complex Hom(S, L0 ) is exact for any injective R-module S. II. n → n + 1 ln+1
l
n X(n + 1) = . . . → 0 → Xn+1 −−→ Xn − → . . . → X0 → 0.
t
t
t
2 1 0 Let E = . . . → E2 − → E1 − → E0 − → Xn+1 → 0 be a minimal injective t1 resolvent of Xn+1 . We denote by E the complex . . . → E2 → E1 − → E0 → 0, with E0 in the nth place. By the above E → Xn+1 is a #-injective cover (where Xn+1 = · · · → 0 → Xn+1 → 0 → . . ., with Xn+1 in the nth place).
.
.
By induction hypothesis, there is a #-injective precover D(n) → X(n). D(n) = . . .
X(n) = . . .
/ Dn+1
/ 0
The map of complexes
dn+1
/ Dn ²
dn
/ ...
φn
/ Xn
/ ...
/ D1 ²
φ1
/ X1
/ D0 ²
/ 0
φ0
/ X0
/ 0
DG-INJECTIVE COVERS, #- INJECTIVE COVERS
Xn+1 = . . .
/ 0
X(n) = . . .
/ 0
/ ...
/ 0
/ 0
/ Xn−1
/ ...
/ X 0
/ 0
/ Xn+1
² ²
/ 0
7
ln+1 ln
/ X n
.
v
induces a map of complexes E − → D(n) such that the diagram
. Â
v
²
/ D(n)  Â  φ  ²
Xn+1
/ X(n)
EÂ Â
ÂÂÂ
 Â Â
is commutative. In particular, the diagram v0
E0 t0
ÂÂÂ
ÂÂÂ
ÂÂ Â
²
ln+1
Xn+1
/ Dn  Â Â φn ² / Xn
is commutative (so φn v0 = ln+1 t0 ).
.
/ E2
E = ... D(n) = . . .
t2
t1
/ E1
² v2
² v1
/ Dn+2
/ Dn+1
(1) / E0 ² v0
dn+1
/ Dn
dn
/ 0
/ ...
/ 0
/ 0
/ Dn−1
/ ...
/ D1
/ D
.
Let C be the mapping cone of v : E → D(n). γn+3 γn+2 γn+1 dn So C = . . . → Dn+3 ⊕ E2 −−−→ Dn+2 ⊕ E1 −−−→ Dn+1 ⊕ E0 −−−→ Dn −→ Dn−1 → . . . → D1 → D0 → 0 with γn+1 (x, y) = dn+1 (x)+v0 (y), and γn+k (x, y) = (dn+k (x)+vk−1 (y), −tk−1 (y)) for k ≥ 2. We show that C → X(n + 1) is a #-injective precover. C = ...
γn+3
X(n + 1) = . . .
/ Dn+2 ⊕ E1 ²
/ 0
γn+2
/ Dn+1 ⊕ E0 ²
γn+1
(0,t0 )
/ Xn+1
ln+1
/ Dn ²
dn
φn
/ Xn
ln
/ Dn−1 ²
/ ...
φn−1
/ Xn−1
Let A be a complex of injective R-modules and let β ∈ Hom(A, X(n + 1)).
/ ...
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ALINA IACOB
X(n + 1) = . . . This gives a map A = ... ²
/ An+1
/ An+2
A = ...
/ An
² βn+1 ² βn ln+1 / Xn+1 / X n
² / 0
/ 0
an
/ A n
β
X(n) = . . .
an+1
²
/ 0
βn ln
/ X n
/ ...
/ A0
/ A−1
ln
/ ...
² / X 0
/ 0
a1
/ ...
/ An−1 ²
an
/ A 0
βn−1
²
/ ...
/ Xn−1
/ A−1
/
/ ...
β0
/ X 0
/ 0
Since D(n) → X(n) is a #-injective precover, there is θ : A → D(n) such that β = φθ (βi = φi θi , 0 ≤ i ≤ n, and di θi = θi−1 ai for 1 ≤ i ≤ n) ...
/ An+2 ² θn+2
...
/ Dn+2 ⊕ E1
...
/ 0
an+2
/ An+1 ² θn+1
/ Dn+1 ⊕ E0
an+1
γn+1
² (0,t0 ) ln+1 / Xn+1
/ An ² θn / Dn
an
/ ...
dn
/ ...
² φn / X n
/ ...
/ A0 ² θ0 / D0 ² φ0 / X
t
0 Since βn+1 : An+1 → Xn+1 , An+1 is injective and E0 − → Xn+1 is an injective cover, there is r ∈ Hom(An+1 , E0 ) such that βn+1 = t0 r. Then (0, t0 )(0, r) = t0 r = βn+1 . We have γn+1 (0, r) = v0 r and dn (θn an+1 −v0 r) = dn θn an+1 −dn v0 r = θn−1 an an+1 − 0 = 0. So θn an+1 − γn+1 (0, r) : An+1 → Ker dn . Also φn (θn an+1 − γn+1 (0, r)) = βn an+1 − φn γn+1 (0, r) = βn an+1 − φn v0 r = βn an+1 − ln+1 t0 r (by (1)). So φn (θn an+1 − γn+1 (0, r)) = βn an+1 − ln+1 βn+1 = 0. Hence θn an+1 − γn+1 r : An+1 → Ker φn = Lnn Thus θn an+1 − γn+1 r : An+1 → Lnn ∩ Ker dn . By inductive hypothesis, Hom(S, Ln ) is exact, for any injective R-module S.
dn+1
d
n Hom(An+1 , Ln−1 ) → So . . . → Hom(An+1 , Ln+1 ) −−−→ Hom(An+1 , Ln ) −→ . . . is exact (with dn : Hom(An+1 , Ln ) → Hom(An+1 , Ln−1 ), dn (g) = dn ◦ g). Since θn an+1 − γn+1 (0, r) ∈ Ker dn = Im dn+1 it follows that θn an+1 − γn+1 (0, r) = dn+1 ◦ s for some s : An+1 → Ln+1 ⊂ Dn+1 . So θn an+1 = γn+1 (0, r) + dn+1 s = γn+1 (0, r) + γn+1 (s, 0) = γn+1 (s, r). Let θn+1 : An+1 → Dn+1 ⊕ E1 be given by θn+1 (x) = (s(x), r(x)). Then by the above γn+1 θn+1 = θn an+1 . Also, (0, t0 )θn+1 = 0 + t0 r = βn+1 . We have t0 ran+2 = βn+1 an+2 = 0, so ran+2 ∈ Hom(An+2 , Ker t0 ).
0
/ A−1 / 0 / 0
DG-INJECTIVE COVERS, #- INJECTIVE COVERS
9
t
1 Since E1 − → Ker t0 is an injective cover there is ω ∈ Hom(An+2 , E1 ) such that ran+2 = t1 ω. Also, dn+1 s = θn an+1 − γn+1 (0, r). Then dn+1 san+2 = 0 − γn+1 (0, ran+2 ) = −v0 ran+2 = −v0 t1 ω = −dn+1 v1 ω. So dn+1 (san+2 + v1 ω) = 0. By induction hypothesis
dn+2
dn+1
. . . → Hom(An+2 , Dn+2 ) −−−→ Hom(An+2 , Dn+1 ) −−−→ Hom(An+2 , Dn ) is exact. Since san+2 + v1 ω ∈ Ker dn+1 = Im dn+2 we have that san+2 = dn+2 e − v1 ω for some e ∈ Hom(An+2 , Dn+2 ). Then θn+1 an+2 = (san+2 , ran+2 ) = (dn+2 e − v1 ω, t1 ω) = γn+2 (e, −ω) = γn+2 θn+2 with θn+2 : An+2 → Dn+2 ⊕ E1 , θn+2 = (e, −ω). We show that for each k ≥ n + 3 there is θk : Ak → Dk ⊕ Ek−n−1 such that θk ak+1 = γk+1 θk+1 . Since v : E → D(n) is a map of complexes and C is its mapping cone we have an exact sequence of complexes 0 → D(n) → M (v) → E [1] → 0. Each Dj is injective, so the sequence is split exact in each degree. Thus for any R S we have an exact sequence 0 → Hom(S, D(n)) → Hom(S, M (v)) → Hom(S, E [1]) → 0 and therefore an associated long exact sequence: . . . → H k Hom(S, D(n)) → H k Hom(S, M (v)) → H k Hom(S, E [[1]) → H k−1 Hom(S, D(n)) → . . .
. .
.
.
.
n+1
Since E [1] = . . . → E2 → E1 → E 0 → 0 → . . . is a deleted injective resolvent, we have H k Hom(S, E [1]) = 0 for any ≥ n + 2, for any injective R-module S. By induction hypothesis H k Hom(S, D(n)) = 0 for any k ≥ n + 1, for any injective R-module S. So H k Hom(S, C) = 0, for any k ≥ n + 2, for any injective module S. We have γn+2 θn+2 an+3 = θn+1 an+2 an+3 = 0. Since Hom(An+3 , Dn+3 ⊕ E2 ) → Hom(An+3 , Dn+2 ⊕ E1 ) → Hom(An+3 , Dn+1 ⊕ E0 ) is exact and θn+2 an+3 ∈ Ker Hom(An+3 , γn+2 ) it follows that there is θn+3 ∈ Hom(An+3 , Dn+3 ⊕ E2 ) such that θn+2 an+3 = γn+3 θn+3 . Similarly there is θk ∈ Hom(Ak , Dk ⊕ Ek−n−1 ) such that θk ak+1 = γk+1 θk+1 . So C → X(n + 1) is a #-injective precover.
.
We have D(n) ⊂ C = D(n + 1). D(n) = . . .
/ Dn+2 ²
D(n + 1) = . . .
dn+2
(1,0)
/ Dn+2 ⊕ E1
γn+2
/ Dn+1 ²
dn+1
/ Dn
dn
/ ...
(1,0)
/ Dn+1 ⊕ E0
γn+1
/ Dn
/ ...
10
ALINA IACOB
and D(n)k = D(n + 1)k for 1 ≤ k ≤ n. Let Ln+1 = Ker(D(n + 1) → X(n + 1)). γn+2 Then Ln+1 = . . . → Dn+2 ⊕ E1 −−−→ Ker (0, t0 ) → Ker φn → Ker φn−1 → . . . → Ker φ1 → Ker φ0 → 0. γn+2 γn+1 So Ln+1 = . . . → Dn+2 ⊕ E1 −−−→ Dn+1 ⊕ Ker t0 −−−→ Ker φn → Ker φn−1 → . . . → Ker φ1 → Ker φ0 → 0 We have Ln ⊂ Ln+1 , and Ln+1 = Lnk for 1 ≤ k ≤ n. k Ln = . . .
/ Dn+2
dn+2
/ Dn+1
(1,0)
²
dn+1
/ Ker φn
dn
/ ...
(1,0) γn+2
²
γn+1
/ Dn+1 ⊕ Ker t0 / Dn+2 ⊕ E1 / Ker φn Ln+1 = . . . We have an exact sequence 0 → Ln → Ln+1 → Ln+1 /Ln → 0 −t2 −t1 where Ln+1 /Ln ' . . . → E2 −−→ E1 −−→ Ker t0 → 0 is Hom(Inj, −) t2 t1 exact (since . . . → E2 − → E1 − → E0 → Xn+1 → 0 is a minimal injective resolvent). The sequence 0 → Ln → Ln+1 → Ln+1 /Ln → 0 is split exact in each degree, so for any R-module S we have an exact sequence 0 → Hom(S, Ln ) → Hom(S, Ln+1 ) → Hom(S, Ln+1 /Ln ) → 0. If S is an injective module then Hom(S, Ln+1 /Ln ) is an exact complex. Also, by induction hypothesis, if S is injective then Hom(S, Ln ) is exact. It follows that Hom(S, Ln+1 ) is exact for any injective R-module S. So for each n ≥ 0 we have Ln ⊂ Ln+1 , Ln+1 = Lnk = Lkk , for any k n+1 0 ≤ k ≤ n and the complex Hom(S, L ) is exact for any injective R-module S.
So every bounded complex has a # - injective precover. ¤ We use Lemma 2 to show that over a noetherian ring every bounded above complex has a #-injective precover. Lemma 3. Over a left noetherian ring every bounded above complex has a #-injective precover. Proof. Let X = . . . → X2 → X1 → X0 → 0. Then X = lim X(n) with →
X(n) = . . . → 0 → Xn → Xn−1 → . . . → X1 → X0 → 0 (with X0 in the zeroth place).
/ ...
DG-INJECTIVE COVERS, #- INJECTIVE COVERS
11
By the proof of Lemma 1, for each n ≥ 0 there is a #-injective precover D(n) → X(n) with kernel Ln such that D(n) ⊂ D(n + 1), D(n)k = D(n + 1)k for 1 ≤ k ≤ n, H k Hom(S, D(n)) = 0 for any k ≥ n + 1 for any injective module S, Ln ⊂ Ln+1 , Lnk = Ln+1 for k 1 ≤ k ≤ n, and such that Hom(S, L(n)) is an exact complex for any injective R-module S. Let L = lim Ln . Then Lk = lim Lnk = Lkk for any k ≥ 0 and Lk = 0 for → → k ≤ −1. So Hom(S, Lk+1 ) → Hom(S, Lk ) → Hom(S, Lk−1 ) is exact for any injective R-module S, for any k ≥ 0 (with L−1 = 0) Thus Hom(S, L) is exact for any injective R-module S. (2) We show that D → X is a #-injective precover, where D = lim D(n) →
(so Dk = D(k)k for any k ≥ 0). Let A be a complex of injective R-modules and let β ∈ Hom(A, X) ...
...
/ A2 Â ÂÂÂ β2 ÂÂ ÂÂ ²
/ X 2
a2
l2
/ A 1 ÂÂ ÂÂ β1 ÂÂ ÂÂ ²
/ X 1
a1
l1
/ A 0 ÂÂÂ Â β0 ÂÂ ÂÂ ²
/ X 0
a0
/ A−1  Â  ² / 0
/ ...
/ ...
β gives a map of complexes β : A → X(n), where A = . . . → 0 → An → An−1 → . . . → A0 → A−1 → . . .. 0
0
/ An   Â βn ²
/ Xn
/ An−1  Â  βn−1 ² / Xn−1
/ ...
/ ...
/ A0 Â ÂÂÂ ÂÂ ÂÂÂ ²
/ X 0
/ ···
/ 0
Since D(n) → X(n) is a # -injective precover there is a map of complexes α that makes the diagram A
 ÄÄ ÂÂ Ä Ä Â α ÄÄÄ Ä Â Â ÄÄ ÄÄ Â Ä ² ÄÄ / X(n) D(n)
commutative. So there is α0 ∈ Hom(A0 , D0 ) such that φ0 α0 = β0 , and
12
ALINA IACOB
for each i ≥ 1 there is ri ∈ Hom(Ai , Di ) such that φi ri = βi (since φn
Dn = D(n)n and Dn −→ Xn is in fact D(n)n → Xn ). ...
/ A2
a2
/ A1
a1
...
/ D 2   Â φ2 ²
d2
/ D1 Â ÂÂÂ Â φ1 ÂÂÂ ²
d1
...
/ X2
l2
/ X1
l1
/ A0  Â  α0 ²Â
/ D0 ÂÂÂ ÂÂ φ0 ÂÂÂ ² / X0
a0
/ A−1
d0
/ 0
l0
/ 0
/ ...
We have φ0 (α0 a1 − d1 r1 ) = φ0 α0 a1 − φ0 d1 r1 = β0 a1 − l1 φ1 r1 = β0 a1 − l1 β1 = 0. So α0 a1 − d1 r1 : A1 → Ker φ0 = L0 A1
L = ...
/ L2
d2
Ä ÂÂ ÄÄ ÂÂ Ä s1 ÄÄ ÂÂÂ α0 a1 −d1 r1 ÄÄ Ä Ä ÂÂÂ ÄÄ ² ÄÄ Ä d1 / L / L 1 0
/ 0
Since Hom(A1 , L) is exact (by (2)), there is s1 ∈ Hom(A1 , L1 ) such that α0 a1 − d1 r1 = d1 s1 , that is α0 a1 = d1 (s1 + r1 ). Let α1 = s1 + r1 . Then φ1 α1 = φ1 (s1 + r1 ) = φ1 s1 + φ1 r1 . But φ1 s1 = 0 (since Im s1 ⊆ L1 = Ker φ1 ) and φ1 r1 = β1 . So φ1 α1 = β1 and α0 a1 = d1 α1 . Similarly there is αi ∈ Hom(Ai , Di ) such that di αi = αi−1 ai , and φi αi = βi , for any i ≥ 1. So Hom(A, D) → Hom(A, X) → 0 is exact for any #-injective complex A. Thus D → X is a #-injective precover. So every bounded above complex . . . → X2 → X1 → X0 → 0 has a #-injective precover. ¤ We can prove now that over a left noetherian ring every complex has a #-injective precover. The proof uses the fact that the class of injective modules is closed under inverse transfinite extensions.
DG-INJECTIVE COVERS, #- INJECTIVE COVERS
13
We recall that an inverse system (Xα )α≤λ is said to be continuous if X0 = 0 and if for each limit ordinal β ≤ λ we have Xβ = lim Xα with ← the limit over the α < β. The inverse limit is said to be a system of epimorphisms if all the morphisms in the system are epimorphisms. Definition 7. ([3], definition 2.1) Let L be a class of R-modules. A module X is said to be an inverse transfinite extension of objects of L if X = lim Xα for a continuous inverse system of epimorphisms (Xα )α≤λ ←
such that Ker(Xα+1 → Xα ) is in L whenever α + 1 ≤ λ. L is said to be closed under inverse extensions if each inverse transfinite extension of objects in L is also in L. Remark 1. ([3], Corollary 1.7) The class of injective modules is closed under inverse transfinite extensions. Proposition 1. If R is a left noetherian ring then every complex of R-modules has a #-injective precover. Proof. Let X = . . . → X2 → X1 → X0 → X 1 → X 2 → . . . Then X = lim Y n with Y n = . . . → X1 → X0 → X 1 → . . . → X n → 0 ←
Y n+1 = . . . ²
/ X1
/ X0
/ ...
/ Xn
/
X n+1 ²
/ X / X / ... / Xn / 0 Y n = ... 1 0 Y n+1 → Y n is surjective for each n ≥ 0. We show that for each n there is a # -injective precover D(n) → Y (n) with D(n) = . . . → D1 → D0 → D1 → . . . → Dn → 0 such that D(n + 1) → D(n) → 0 is exact, L(n + 1) → L(n) → 0 is exact (where L(n) = Ker(D(n) → X(n)) and such that T (n) = Ker(L(n + 1) → L(n)) ∈ # − inj ⊥ , for any n ≥ 0 (i.e. Ext1 (M, T (n)) = 0 for any #-injective complex M ). I. n = 0 By lemma 3, Y 0 = . . . → X2 → X1 → X0 → 0 has a # -injective precover D(0) = . . . → D2 → D1 → D0 → 0. II. n → n + 1 ln → X n+1 → 0 Y n+1 = . . . → X1 → X0 → X 1 → X 2 → . . . → X n − n+1 Let . . . → E2 → E1 → E0 → X → 0 be a minimal injective resolvent of X n+1 and let E = . . . → E2 → E1 → E0 → 0 be the deleted minimum injective resolvent of X n+1 . Let D(n) = . . . → D1 → D0 → . . . → Dn−1 → Dn → 0 be a #-injective precover of Y n .
/ 0
14
ALINA IACOB
D(n) : . . .
/ D1 ²
²
r1
/ X Y n : ... 1 The map of complexes
Y n : . . . ²
ÂÂÂ
ÂÂ
/ ...
/ D0 ²
/ ...
/ X0
/ 0
/ 0
/
/
dn−1
Dn−1
r0
/ X 0
/ X1
/
²
rn−1
X n−1
X1
/ ...
/ 0
/ ...
/ Dn ²
/
ln−1
/ 0
rn
/ Xn
X n−1
/ 0
ln−1
Â
Xn+1 : . . .
/ 0
/
/ Xn  Â Â ln ²
X n+1
induces a map of complexes α : D(n) → E. ... /
/
Dn−2 Â ÂÂ
... /
²Â
Â
Dn−1 Â ÂÂ
 Â α2
E2
dn−1
ÂÂ ÂÂÂ α1
f2
/
²Â
E1
f1
/ Dn ÂÂ ÂÂÂ Â Â α0 ÂÂÂ ² / 0
E
/ 0
/ 0
such that the diagram
Â
/ Yn ÂÂ ÂÂÂ Â ÂÂÂ ²
E
/ Xn+1
D(n) ÂÂÂ
ÂÂÂ
ÂÂ
²
is commutative. In particular D n ²
ÂÂÂ
ÂÂ
rn
ÂÂ
α0
E0
f0
/ Xn   Â ln ² / Xn+1
is a commutative diagram (so f0 α0 = ln rn ). δn−2
Let C be the mapping cone of α, C = . . . → E 3 ⊕ Dn−2 −−→ E 2 ⊕ δn−1 δn Dn−1 −−→ E 1 ⊕ Dn −→ E0 → 0 with
/ 0
/ 0
DG-INJECTIVE COVERS, #- INJECTIVE COVERS
15
δn (x, y) = f1 (x) + α0 (y) δn−1 (x, y) = (f2 (x) + α1 (y), −dn−1 (y)) δn−2 (x, y) = (f3 (x) + α2 (y), −dn−2 (y)) ....................................... δn−k (x, y) = (fk+1 (x) + αk (y), −dn−k (y)) / E 3 ⊕ D n−2
C = ... ²
²
δn−2
/ E 2 ⊕ D n−1
(0,+rn−2 )
²
ln−2
δn−1
δn
/ E 1 ⊕ Dn
(0,−rn−1 )
E0
(0,+rn )
²
ln−1
/
²
ln
f0
/ / / Xn / Y n+1 = . . . X n−2 X n−1 X n+1 We show that C → Y n+1 is a # -injective precover. Let A be a complex of injective R-modules and let β ∈ Hom(A, Y n+1 ) β
β gives a map of complexes A − →Yn
... /
An−1 Â ÂÂ
ÂÂ
 ÂÂ
ÂÂ
...
²
...
/
ÂÂ ÂÂÂ ÂÂÂ ÂÂÂ ÂÂÂ ÂÂ ÂÂ
/ Xn ÂÂ ÂÂÂ ÂÂÂ ÂÂ ÂÂÂ ÂÂÂ ÂÂ ÂÂ
X n−1
/ Xn
X n−1 ÂÂ
/
/ An   Â ²Â
/
An+1 Â ÂÂ
ÂÂ
/ ...
ÂÂ
 ÂÂ
ÂÂ
ÂÂ
²
X n+1 Â ÂÂ
An+2 Â ÂÂ
 ÂÂ
/
/
ÂÂÂ
² / 0
ÂÂ
²Â
/ 0
Since D(n) → Y n is a #-injective precover, there is γ : A → D(n) such that the diagram
A
Ä Â ÄÄ ÂÂ Ä γ Ä Ä Â β ÄÄ Ä Â Ä Â² ÄÄ Ä / D(n) Yn
is commutative.
/ 0
/ 0
16
ALINA IACOB
... /
... /
²
/
ÂÂ
 γn−2
 γn−1
² /
DÂ ÂÂ
dn−1
n−1
DÂ Â
 Â rn−2
X n−2
 ÂÂ
 ÂÂ
n−2
²Â
an−1
An−1 Â
Â
ÂÂÂ
ÂÂÂ
Â
...
/
An−2 Â
² /
Â
ÂÂ ÂÂÂ rn−1 ln−1
X n−1
/ An  Â  Â γn ²Â
/
/
An+1
/ ...
An+2
/ 0
/ Dn  Â  Â rn ²Â
/ Xn
/ 0
Since An+1 is injective and f0 : E 0 → X n+1 is an injective cover there is η0 ∈ Hom(An+1 , E 0 ) such that βn+1 = f0 η0 . ... ... ...
/
² η3 / E 3 ⊕ D n−2 /
/
An−2
² (0,rn−2 )
X
n−2
An−1
an−1
an
/ An
² η2 ² η1 δ / E 2 ⊕ D n−1 n−1/ E 1 ⊕ D n /
² (0,−rn−1 )
X
/ δn
² (0,rn ) l n / Xn
n−1
An+1
/
/
² η0
E0 ² f0
X
n+1
/
An+2 / 0 / 0
Let γ n : An → E 1 ⊕ Dn , γ n = (0, γn ). Then (0, rn )(0, γn ) = rn γn = βn , and f0 (δn γ n − η0 an ) = f0 α0 γn − f0 η0 an = ln rn γn − f0 η0 an = ln βn − βn+1 an = 0 f1 → Ker f0 is an injective So δn γ n − η0 an : An → Ker f0 . Since E1 − precover there is χn : An → E1 such that δn γ n − η0 an = f1 χn AÂ n ÂÂÂ
ÂÂ δn γ n −η0 an
χn
Ä
E1
f1
 Â ²
/ Ker f0
Thus η0 an = δn γ n − f1 χn = δn γ n − δn (χn , 0) = δn (0, γn ) − δn (χn , 0). Let η1 : An → E 1 ⊕ Dn , η1 (x) = (−χn (x), γn (x)) Then by the above η0 an = δn η1 . Also, (0, rn ) ◦ η1 = rn γn = βn . Let γ n−1 : An−1 → E 2 ⊕Dn−1 , γ n−1 (x) = (0, −γn−1 ). Then (0, −rn−1 )γ n−1 = βn−1 .
/ ...
DG-INJECTIVE COVERS, #- INJECTIVE COVERS
17
We have η1 an−1 = (−χn an−1 , γn an−1 ) = (−χn an−1 , dn−1 γn−1 ) and δn−1 γ n−1 = δn−1 (0, −γn−1 ) = (−α1 γn−1 , dn−1 γn−1 ) So −η1 an−1 + δn−1 γ n−1 = (χn an−1 − α1 γn−1 , 0) Since f1 ◦(χn an−1 −α1 γn−1 ) = (δn γ n −η0 an )an−1 −f1 α1 γn−1 = δn γ n an−1 − α0 dn−1 γn−1 = δn (0, γn an−1 ) − α0 γn an−1 = α0 γn an−1 − α0 γn an−1 = 0 it follows that χn an−1 − α1 γn−1 : An−1 → Ker f1 . f2 Since An−1 is injective and E2 − → Ker f1 is an injective precover, there is χn−1 : An−1 → E2 such that χn an−1 − α1 γn−1 = f2 χn−1 An−1 Â ÂÂ
χn−1
ÂÂÂ
Ä
 χn an−1 −α1 γn−1
² / Ker f1
f2
E2
 ÂÂ
So −η1 an−1 + δn−1 γ n−1 = (f2 χn−1 , 0) = δn−1 (χn−1 , 0). Thus η1 an−1 = δn−1 ((−χn−1 , 0) + γ n−1 ) = δn−1 (−χn−1 , −γn−1 ) Also (0, −rn−1 )(−χn−1 , −γn−1 ) = +rn−1 γn−1 = βn−1 Let η2 = (−χn−1 , −γn−1 ). We have (0, −rn−1 )η2 = βn−1 and δn−1 η2 = η1 an−1 . Similarly, for each i there is ηn+1−i : Ai → E n+1−i ⊕ Di such that ηn−i αi = δi ηn+1−i So C → Y n+1 is a # -injective precover. There is a surjective map D(n + 1) = C → D(n)
² /
D(n) . . . ... ...
/ E 3 ⊕ D n−2
/
²
δ0
/ E n+1 ⊕ D 0
D(n + 1) . . .
²
δn−2
/ E 2 ⊕ D n−1
(0,1)
Dn−2
d0
D0
/
²
/
²
/ ...
D1
δn−1
(0,−1)
Dn−1
/ ...
/ E n ⊕ D1
dn−1
/ E 1 ⊕ Dn ²
δn
/
E0
/ 0
/ 0
/ 0
(0,1)
/ Dn
There is also a surjective map L(n + 1) → L(n) where L(n) = Ker(D(n) → X(n)). Ker(0, rk ) = {(x, y) ∈ E n+1−k ⊕Dk / (0, rk )(x, y) = 0} = {(x, y) / rk (y) = 0} = E n+1−k ⊕ Ker rk
18
ALINA IACOB
L(n + 1) = . . .
/ E n+1 ⊕ Ker r 0
L(n) = . . .
² / Ker r0
...
/ E 3 ⊕ Ker r n−2 ²
...
δ0
/ Ker rr−2
² / Ker r1
d0
δn−2
/ E 2 ⊕ Ker r n−1
(0,id)
²
dn−2
δ1
/ E n ⊕ Ker r 1
d1
δn−1
/ ...
/ E1 ⊕ r n
(0,−id)
²
dn−1
/ Ker rn−1
/ ...
/ Ker f0
(0,id)
/ Ker rn
/ 0
We have fn+1 fn Ker(L(n + 1) → L(n)) = T (n + 1) ∼ = . . . → E n+1 −−→ E n −→ . . . → f2 f1 E3 → E2 − → E1 − → Ker f0 → 0 But ... /
E n+1
/ En
/ ... /
E3
/
E2
f2
/
E1
f1
/ E0
/ 0
f0
²
0
/ 0
² / Xn+1
is a #-injective cover with kernel . . . → E 2 → E 1 → Ker f0 → 0. So by Wakamatsu’s lemma, Ext1 (S, T (n + 1)) = 0, for any #-injective complex S (i.e. T (n + 1) ∈ # − inj ⊥ ). Let S be a complex of injective R-modules. The exact sequence 0 → T (n + 1) → L(n + 1) → L(n) → 0 gives an exact sequence 0 → Hom(S, T (n + 1)) → Hom(S, L(n + 1)) → Hom(S, L(n)) → Ext1 (S, T (n + 1)) = 0 So Hom(S, L(n + 1)) → Hom(S, L(n)) → 0 is exact, for any n ≥ 0, for any # -injective S. Since for each n ≥ 0 D(n) → X(n) is a # -injective precover, the sequence 0 → Hom(S, L(n)) → Hom(S, D(n)) → Hom(S, X(n)) → 0 is exact for any # -injective S. Since the map Hom(S, L(n + 1)) → Hom(S, L(n)) is surjective for any n ≥ 0, it follows ([5], Theorem 1.5.13) that the sequence 0 → lim Hom(S, L(n)) → lim Hom(S, D(n)) → lim Hom(S, X(n)) → ← ← ← 0 is exact, so 0 → Hom(S, lim L(n)) → Hom(S, lim D(n)) → Hom(S, lim X(n)) → ← ← ← 0
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DG-INJECTIVE COVERS, #- INJECTIVE COVERS
19
is exact for any #-injective complex S. By Remark 1, lim D(n) is a complex of injective R-modules. So lim D(n) → ← ← X is a # -injective precover. ¤ A similar argument to that in the proof of Corollary 5.2.7 in [5] gives the following useful result: Proposition 2. Let F be a class of complexes of R-modules that is closed under well ordered inductive limits, and let M be a complex of R-modules. If M has an F precover then it has an F-cover. Theorem 2. If R is a left noetherian ring then every complex of left R-modules has a #-injective cover. Proof. Since the class of #-injective complexes over a noetherian ring is closed under direct limits ([5], Theorem 3.1.17), the result follows from Propositions 1 and 2. ¤ 4. DG-injective covers We show in this section (Theorem 3) that for a commutative noetherian ring R of finite Krull dimension the following are equivalent: a) every complex has a DG-injective cover; b) gl.dim.R < ∞. We start by showing (Lemma 4) that if R is a left noetherian ring such that every complex has a DG-injective cover then every complex of injective left R-modules is DG-injective (i.e. every #-injective complex is DG-injective). We prove then (Proposition 3) that over a left noetherian ring R the following are equivalent: a) every complex has a DG-injective cover; b) every complex of injective R-modules is a DG-injective complex. Using this result and Theorem 1 we obtain Theorem 3: If R is a commutative noetherian ring of finite Krull dimension then every complex of R-modules has a DG-injective cover if and only if gl.dim.R < ∞. Lemma 4. Let R be a left noetherian ring such that every complex has a DG-injective cover. Then every complex of injective left R-modules is a DG-injective complex. Proof. Let . . . → E2 → E1 → E0 → E−1 → . . . be a complex of injective R -modules.
20
ALINA IACOB
For each n ≥ 0 let Xn = . . . → 0 → 0 → En → En−1 → . . . with En in the nth place. Then Xn ⊂ Xn+1 for each n ≥ 0 and E = lim Xn . →
φ
Let D − → E be a DG-injective cover and let K = Kerφ. Then 0 → Hom(A, K) → Hom(A, D) → Hom(A, E) → 0 is exact for any DGinjective complex A. In particular, the sequence 0 → Hom(Xn , K) → Hom(Xn , D) → Hom(Xn , E) → 0 is exact for any n ≥ 0. For each n ≥ 0 we have an exact sequence of DG-injective complexes 0 → Xn → Xn+1 → XXn+1 → 0 and therefore an exact sen Xn+1 quence 0 → Hom( Xn , K) → Hom(Xn+1 , K) → Hom(Xn , K) → Ext1 ( XXn+1 , K) = 0 ( by Wakamatsu’s Lemma, Ext1 (A, K) = 0 for any n DG-injective complex A). So Hom(Xn+1 , K) → Hom(Xn , K) → 0 is exact for any n ≥ 0. It follows that the sequence 0 → lim Hom(Xn , K) → ←
lim Hom(Xn , D) → lim Hom(Xn , E) → 0 is exact. But lim Hom(Xn , K) ∼ = ← ← ← Hom(E, K), lim Hom(Xn , D) ∼ = Hom(Xn , D) and lim Hom(Xn , E) ∼ = ←
←
Hom(E, E). Since the sequence 0 → Hom(E, K) → Hom(E, D) → Hom(E, E) → 0 is exact it follows that there is r ∈ Hom(E, D) such that φ ◦ r = IdE . So φ is surjective and the sequence 0 → K → D → E → 0 is split exact. Since E is isomorphic to a direct summand of a DG-injective complex, E is DG-injective. ¤ Proposition 3. Let R be a left noetherian ring. The following are equivalent: a) Every complex has a DG-injective cover; b) Every complex of injective left R-module is DG-injective. Proof. a) ⇒ b) is the lemma; b) ⇒ a) in this case the class of DG-injectives is the class of #-injective complexes and we showed (Theorem 2) that every complex has a #injective cover. ¤ Theorem 3. Let R be a commutative noetherian ring of finite Krull dimension. The following are equivalent: a) Every complex has a DG-injective cover; b) gl.dim.R < ∞. Proof. a) ⇒ b) By the previous result every complex of injective R modules is a DG-injective complex. By Theorem 1 it follows that R has finite global dimension. b) ⇒ a) follows from the previous proposition, since gl.dim.R < ∞ implies that every complex of injective R modules is a DG-injective complex. ¤
DG-INJECTIVE COVERS, #- INJECTIVE COVERS
21
References [1] L. Avramov and H.-B. Foxby. Homological dimensions of unbounded complexes. J. Pure Appl. Algebra, (71):129–155, 1991. [2] L.L. Avramov and S. Halperin. Through the looking glass: A dictionary between rational homotopy theory and local algebra. Lecture Notes in Math., 1183:1–27, 1986. [3] E.E. Enochs, A. Iacob, and O.M.G. Jenda. Closure under transfinite extensions. Illinois Journal of Mathematics, to appear. [4] E.E. Enochs and O.M.G. Jenda. Gorenstein injective and projective modules. Mathematische Zeitschrift, (220):611–633, 1995. [5] E.E. Enochs and O.M.G. Jenda. Relative Homological Algebra. Walter de Gruyter, 2000. De Gruyter Exposition in Math; 30. [6] E.E. Enochs, O.M.G. Jenda, and J. Xu. Orthogonality in the category of complexes. Math. J. Okayama Univ., 38:25–46, 1996. [7] A. Iacob. Direct sums of DG-injective complexes. submitted. [8] J.R. Garc´ıa Rozas. Covers and evelopes in the category of complexes of modules. CRC Press LLC, 1999. Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30458 USA, Email:
[email protected]