Embeddings of canonical modules and resolutions of connected sums

EMBEDDINGS OF CANONICAL MODULES AND RESOLUTIONS OF CONNECTED SUMS

arXiv:1704.03072v1 [math.AC] 10 Apr 2017

ELA CELIKBAS, JAI LAXMI, AND JERZY WEYMAN Abstract. For an ideal Im,n generated by all square-free monomials of degree m in a polynomial ring R with n variables, we obtain a specific embedding of a canonical module of R/Im,n to R/Im,n itself. The construction of this explicit embedding depends on a minimal free R-resolution of an ideal generated by Im,n . Using this embedding, we give a resolution of connected sums of several copies of certain Artinian k-algebras where k is a field.

1. Introduction For a Cohen-Macaulay ring S with a canonical module ωS , it is well-known that, if S is generically Gorenstein (e.g. S is reduced), then ωS can be identified with an ideal of S, that is, ωS embeds into S; see, for example [2, 3.3.18]. In this paper we give an explicit construction of such an embedding for a certain ring. More precisely, if R is a polynomial ring in n variables over a field k, Im,n is the ideal of R generated by all square-free monomials of degree m and ωR/Im,n is the canonical module of R/Im,n , then, in Theorem 5.5, we establish an explicit standard graded embedding of ωR/Im,n into R/Im,n . Our motivation for this study comes from obtaining minimal free resolutions of connected sums of Gorenstein rings. As given in [1], a connected sum of several Gorenstein rings Si is a Gorenstein ring S that is a special quotient of the fiber product (pullback) of Si ’s. Indeed, as a consequence of our argument, we give a construction of a resolution of a connected sum of several copies of Si := k[x]/(xei +1 ) over a field k; see Corollary 6.3. In order to construct a specific embedding from ωR/Im,n to R/Im,n , we use generators of the R/Im,n -module HomR (ωR/Im,n , R/Im,n ). In section 3, we give a set of generators of HomR (R/Im,n , R/Im,n ) in Theorem 3.2. Moreover, as an immediate result of Theorem 3.2, we get a presentation of HomR (ωR/Im,n , R/Im,n ). Section 4 deals with the computation of Hilbert-Poincar´e functions of R/Im,n and ωR/Im,n . The main result of this paper is Theorem 5.5 which gives a specific standard graded embedding of a canonical module ψ := ψm,n : ωR/Im,n −→ R/Im,n . In Corollary 5.6, the image of ψm,n is identified with an ideal of R/Im,n generated by maximal minors of a certain Vandermonde-like matrix D. We use Theorem 5.5 and Corollary 5.6 to Date: April 12, 2017. 2010 Mathematics Subject Classification. Primary 13D02, 13D40, 13H10, 20C30; Secondary 13F55. Key words and phrases. squarefree monomial ideal, canonical module, Gorenstein ring, connected sum. Jerzy Weyman was partially supported by NSF grant DMS-1400740. 1

2

ELA CELIKBAS, JAI LAXMI, AND JERZY WEYMAN

get a resolution of a Gorenstein ring obtained from an embedding of a canonical module of R/Im,n in Corollary 5.7. In section 6 we specialize to m = 2. In this case, in Theorem 6.1, we give another, Nn graded embedding of a canonical module of R/I2,n into the ring R/I2,n . As a corollary of this theorem, a canonical module of R/I2,n is identified with an Nn -graded ideal of R/I2,n . The mapping cone of the map of free resolutions over R covering the embedding ωR/Im,n −→ R/Im,n gives a minimal free resolution of the connected sum of algebras Si := k[x]/(xei +1 ). Section 2 contains known results regarding the main tools used in the rest of the paper including the definition of connected sums, resolutions of the ideals generated by square-free monomials of a given degree and of corresponding Stanley-Reisner rings. 2. Preliminaries 2.1. Notation. a) For a positive integer n, let [n] = {1, . . . , n}. If σ ⊂ [n], then |σ| denotes the number of elements contained in σ. b) Let R = k[x1 , . . . , xn ] be a polynomial ring in n variables over a field k with x1 > . . . > xn . We order the monomials in R with graded lexicographic order. c) Let m and n be positive integers with m ≤ n, then Im,n denotes an ideal generated by all square-free monomials of degree m in n variables. Furthermore, ωR/Im,n denotes a canonical module of R/Im,n . d) For an R-module M, ℓ(M) and µ(M) denote the length and the minimal number of generators of M, respectively. e) For a commutative Noetherian ring T , dim(T ) denotes the Krull dimension of T . f) Let M = ⊕i≥0 Mi be a graded The Hilbert-Poincar´ e function of M is the P R-module. i formal power series HM (t) = i≥0 ℓ(Mi )t . g) Let (T, m, k) be an Artinian local ring. Then the socle of T is soc(T ) = (0 :T m). h) For a Noetherian local ring T and a T -module M, a finite presentation of M is an exact sequence T ⊕m → T ⊕n → M → 0 with m, n positive integers. 2.2. Connected Sums. Definition 2.1. Let Si = k[xi ]/(xei i +1 ), soc(Si ) = (xei i ), and J = hxi xj , xei i − xe11 |1 ≤ i ≤ ni where ei ≥ 1 be the ideal in R := k[x1 , . . . , xn ] defining the connected sum S1 #k . . . #k Sn of the algebras Si (compare [1] for the definition of connected sums). Therefore we have S1 #k . . . #k Sn = R/J. Remark 2.2. With notation in Definition 2.1, S1 #k . . . #k Sn is Gorenstein by [1]. 2.3. Specht Modules and Free Resolution of the Ring R/Im,n. We recall the definition q of Specht module S (p,1 ) associated to a hook partition (p, 1q ) of n where p, q are nonnegative integers. We follow [3, Section 7.4]. Let n = p + q and let Sn be a symmetric group on [n]. Let (p, 1q ) be a hook partition of n. An oriented column tabloid of shape (p, 1q ) is filling of Young diagram of (p, 1q ) with positive integers 1, 2, . . . , n, with each number appearing once, which is skew-symmetric in the first column and symmetric in the remaining rows. q The Specht module S (p,1 ) is the k-vector space generated by the equivalence classes [T ] of oriented column tabloids of shape (p, 1q ) with entries in [n] modulo the following relations:

EMBEDDINGS OF CANONICAL MODULES AND RESOLUTIONS OF CONNECTED SUMS 3

a) Alternating columns: σ[T ] = sign(σ)[T ] for all σ ∈ Sn fixing the columns of T (so in the case of hook, just permuting P the numbers in the first column). b) Shuffling relations: [T ] = [T ′ ], where sum is over all T ′ acquired from T by exchanging the element of the second column of [T ] with one of the element of the first column of T . We recall some facts about Specht modules associated to a hook partition (p, 1q ). q a) The Specht module S (p,1 ) is a k-vector space. By using hook length formula from [5, Theorem 20.1], we have p+q−1 (p,1q ) . dim(S )= q q

b) The symmetric group Sn acts on S (p,1 ) by permuting the numbers in oriented column tabloids. c) An oriented column tabloid of shape (p, 1q ) is called standard tableau of shape (p, 1q ) if the entries in each row and column are increasing (in the case of hooks it means the entries in the first column are increasing and the first entry in the first column is 1). q The equivalence classes of standard tableaux of shape (p, 1q ) form a k-basis of S (p,1 ) . Let SY T ([p + q], (p, 1q )) denote the set of standard tableaux of shape (p, 1q ) with entries 1, 2, . . . , p + q (each number appearing once). In the remaining part of this subsection, we state the results from [4]. Definition 2.3. (a) Let n, m and i be integers. For 1 ≤ m ≤ n and 0 ≤ k ≤ n − m, (1)

k

Ukm,n := IndSSnm+k ×Sn−m−k (S (m,1 ) ⊗k S (n−m−k) ). q

Here S (n−m−k) is the Specht module S (p,1 ) with p := n−m−k, q := 0. The right hand side of k Equation 1 is the k[Sn ]-module induced by the k[Sm+k × Sn−m−k ]-module S (m,1 ) ⊗k S (n−m−k) . If any of the inequalities involving n, m, and i are violated, then we set Uim,n := 0. (b) A k[Sn ]-module Fkm,n is defined as Fkm,n := Ukm,n ⊗k R(−m − k).

(2)

Remark 2.4. Let n, m, and k be positive integers with 1 ≤ m ≤ n and 0 ≤ k ≤ n − m. (a) The module Ukm,n is generated by the equivalence classes of oriented column tabloids of shape (m, 1k ), filled with numbers 1, 2, . . . , n without repetitions. Moreover, the equivalence classes of standard tableaux of shape (m, 1k ) form a k-basis of Ukm,n . (b) The module Fkm,n is a free R-module generated by the equivalence classes of oriented column tabloids of shape (m, 1k ), filled with numbers 1, 2, . . . , n without repetitions. (c) The equivalence classes of standard tableaux of the shape (m, 1k ) with entries in [n] (without repetitions) form an R-basis of Fkm,n and the rank of Fkm,n is βk := rank(Fkm,n ) = n m+k−1 . m+k k We define an R-linear map

m,n ∂km,n : Fkm,n −→ Fk−1

by setting ∂km,n ([T ]) :=

k X p=0

(−1)k−p xip [T \ ip ]

4


where [T ] is an oriented column tabloid of shape (m, 1k ), and T \ ip is an oriented column tabloid of shape (m, 1k−1 ) obtained from T by omitting the number ip in position p − 1 in the first column of T . Proposition 2.5. Let n, m and k be positive integers with m ≤ n and 0 ≤ k ≤ n − m. m,n Then (Fm,n • , ∂• ) is a complex of free R-modules which is a minimal free resolution of the R-module R/Im,n . This complex is Sn -equivariant, where Sn acts on Fkm,n diagonally (the action on R just permutes the variables xi ). 2.4. Simplicial Complex and Stanley-Reisner Rings. Definition 2.6. [2, Definition 5.1.1] Let V = {v1 , . . . , vn } be a finite set. (1) A non-empty set ∆ of subsets of V with the property that τ ∈ ∆ whenever τ ⊂ σ for some σ ∈ ∆ is called a simplicial complex on the vertex set V . The elements of ∆ are called faces, and the dimension, dim σ, of a face σ is the number |σ| − 1. The dimension of the simplicial complex ∆ is dim(∆) = max{dim σ : σ ∈ ∆}. (2) Let k be a field. The Stanley-Reisner ring of the complex ∆ is the homogeneous k-algebra k[∆] = k[x1 , . . . , xn ]/I∆ , where I∆ is the ideal generated by all monomials xi1 . . . xis such that {vi1 , . . . , vis } 6∈ ∆. The Krull dimension of the Stanley-Reisner ring k[∆] is dim(∆) + 1. Lemma 2.7. Let I∆ = Im,n , then k[∆] is Cohen-Macaulay. Proof. If I∆ = Im,n , then all monomials xi1 . . . xim−1 ∈ I∆ , so {vi1 , . . . , vim−1 } 6∈ ∆. Hence, dim(∆) = m − 2. Then dim(k[∆]) = m − 1. By Proposition 2.5, projective dimension of k[∆] is n − m + 1. By graded AuslanderBuchsbaum formula, depth(k[∆]) = m − 1. Thus, k[x1 , . . . , xn ]/Im,n is Cohen Macaulay. m,n Remark 2.8. By Proposition 2.5, (Fm,n ) is a minimal free resolution of R/Im,n . Let • , ∂• m,n m,n G• = HomR (F• , R) be the dual complex. Then Gm,n is a minimal free R-resolution of • ωR/Im,n .

3. Generators of Hom The goal of this section is to find the generators (Theorem 3.2) and a presentation (Corollary 3.3) of the R-module HomR (ωR/Im,n , R/Im,n ). We start with the example n = 4, m = 2. Example 3.1. Let R = k[x "1 , x2 ,#x3 , x4 ] and I2,4 " = hx # 1 x2 , x1 x3 , x1 x4 , x"2 x3 , x#2 x4 , x3 x4 i be an ideal of R. Let [T[4]\{2} ] =

1 2 3 4

, [T[4]\{3} ] =

1 3 2 4

and [T[4]\{4} ] =

1 4 2 3

. The formulas


for differentials in the complex F4,2 • are 2,4 3 2 ∂2 ([T[4]\{2} ]) = x1 4 − x3 14 2 3 − x1 23 = x1 4 2,4 2 3 ∂2 ([T[4]\{3} ]) = x1 4 − x2 14 2,4 2 4 − x2 13 ∂2 ([T[4]\{4} ]) = x1 3

2

4

3

4

+ x4

− x3

+ x4

+ x3

1 2 3

1 2 4

+ x4

. 4 .

1 2 3

.

1 3 2

1 2

Let P be the matrix of ∂22,4 with respect to the bases of standard tableaux in modules F24,2 and F14,2 .   0 0 x4  0 x4 0     0 0 −x3      x3 0 0  P =   0 −x2 0   −x2 0 0     0 x1 x1  x1 0 −x1 Columns are listed in order T[4]\{4} = are listed in order

1 2 3

,

1 3 2

,

1 2 4

,

1 4 2

1 4 2 3

,

, T[4]\{3} =

1 3 4

,

1 4 3

,

1 3 2 4

2 3 4

, and T[4]\{2} = 2 4 3

, and

1 2 3 4

, and rows

.

Then the transpose of P , denoted by P T , gives a matrix presentation of ωR/I2,4 . For 2 ≤ i ≤ 4, let f{i} : ωR/I2,4 → R/I2,4 be defined as ( xi , if i = j (3) f{i} ([T[4]\{j} ]) = 0, if i 6= j. In order to show that f{i} is well defined, it is enough to prove that f{i} satisfies the relations of P T . Note that the entry xi is missing in the column corresponding to ∂22,4 ([T[4]\{i} ]) in P , T hence f{i} satisfies the relations " #of P . The tableau [T[4]\{1} ] =

2 1 3 4

"

2 1 3 4

is expressed in terms of standard tableaux such as

#

=

"

1 2 3 4

#

−

"

1 3 2 4

#

+

"

1 4 2 3

#

.

Let (4)

f{1} ([T[4]\{2} ]) = −f{1} ([T[4]\{3} ]) = f{1} ([T[4]\{4} ]) = x1 .

Since ∂22,3 ([T[4]\{1} ]) = ∂22,4 ([T[4]\{2} ])−∂22,4 ([T[4]\{3} ])+∂22,4 ([T[4]\{4} ]), there is no term involving x1 in ∂22,3 ([T[4]\{1} ]). Hence, f{1} is well defined.

6


Now suppose ψ : ωR/I2,4 → R/I2,4 satisfies ψ([T[4]\{i} ]) = c + u for some c ∈ k, and u ∈ hx1 , . . . , xn i. Then ψ satisfies the relations of P T , which implies, cxi = 0, hence c = 0. Then, taking into account relations given by the first six columns of P T , we can write X (e) (e) X (e) (e) ψ([T[4]\{4} ]) = a1 x1 + a4 x4 , e≥1

ψ([T[4]\{3} ]) =

X

e≥1

(e) (e)

b1 x1 +

e≥1

ψ([T[4]\{2} ]) =

X

(e) bi ,

(e) ci

(e) (e)

b3 x3 ,

e≥1

(e) (e) c1 x1

+

e≥1

(e) ai ,

X

X

(e) (e)

c2 x2 ,

e≥1

and are all in k. where (e) (e) (e) Using the relations from last two columns of P T , we get a1 = c1 = −b1 for each e. This means X (e) (e−1) X (e) (e−1) X (e) (e−1) X (e) (e−1) c2 x2 )f2 + ( b3 x3 )f3 + ( a4 x4 )f4 . c1 x1 )f1 + ( ψ=( e≥1

e≥1

e≥1

e≥1

This shows that {f{1} , f{2} , f{3} , f{4} } is a minimal generating set of HomR (ωR/I2,4 , R/I2,4 ).

In the light of the example above, the following theorem gives a general description of a minimal generating set of HomR (ωR/Im,n , R/Im,n ). Theorem 3.2. For 1 < j1 < . . . < jm−1 ≤ n, let Θ = {j1 , . . . , jm−1 }. Suppose fj1 ,...,jm−1 and f1,j2 ,...,jm−1 are maps from ( ωR/Im,n to R/Im,n defined as xj1 xj2 . . . xjm−1 , if Γ = Θ and fj1 ,...,jm−1 ([T[n]\Γ]) = 0 , otherwise. ( x1 xj2 . . . xjm−1 , if Γ = Θ or Γ = (Θ \ {j1 }) ∪ {l} for 1 6= l ∈ [n] \ Θ, f1,j2 ,...,jm−1 ([T[n]\Γ ]) = 0 , otherwise. Then {fj1 ,...,jm−1 , f1,j2 ,...,jm−1 : 1 < j1 < . . . < jm−1 ≤ n} is a minimal generating set of HomR (ωR/Im,n , R/Im,n ). Proof. First note that by Remark 2.4, Bk := {[T ] : T ∈ SY T ((m, 1k ), [n])} is a basis of Fkm,n . For 1 = i0 < j1 < . . . < jm−1 ≤ n and 1 = i0 < i1 < i2 < . . . < in−m ≤ n, we set standard tableau of shape (m, 1n−m ) as i0

j1

j2

. . . jm−1

i1 (5)

[T[n]\{j1 ,...,jm−1 } ] = [T1,i1 ,...,in−m ] =

i2 .. . in−m

m,n m,n m,n By Proposition 2.5, we get the differential ∂n−m : Fn−m → Fn−m−1 as


(6)

m,n ∂n−m ([T1,i1 ,...,in−m ]) =

n−m X

(−1)n−m−k xik [T1,i1 ,...,in−m \ ik ].

k=0

m,n ∂n−m

Let P be the matrix of with respect to the bases Bn−m and Bn−m−1 . Then P T , the transpose of P , is a presentation of ωR/Im,n by Remark 2.8. In order to show that fj1 ,...,jm−1 and f1,j2 ,...,jm−1 are well defined, it is enough to see that fj1 ,...,jm−1 and f1,j2 ,...,jm−1 satisfy the relations of P T . Since the column with respect to m,n ∂n−m ([T1,i1 ,...,in−m ]) does not involve xjk , the corrensponding row in P T has no xjk as well. Thus fj1 ,...,jm−1 satisfies the relations of P T . A non-standard tableau [T[n]\{1,j2,...,jm−1 } ] can be expressed in terms of standard tableaux as [T[n]\{1,j2 ,...,jm−1 } ] = [T[n]\{j1 ,...,jm−1 } ] +

n−m X

(−1)k [T[n]\{ik ,j2 ,...,jm−1 } ].

k=0

m,n By direct computation one sees that column with respect to ∂n−m ([T[n]\{1,j2 ,...,jm−1 } ]) does not involve x1 and xjk for k = 2, . . . , m − 1. Therefore, f{1,j2 ,...,jm−1 } satisfies the relations of P T , hence f{1,j2 ,...,jm−1 } is well defined. We now claim that {fτ : τ ⊂ [n], |τ | = m−1} is a generating set of HomR (ωR/Im,n , R/Im,n ). Let ϕ ∈ HomR (ωR/Im,n , R/Im,n ). For 1 < l1 < . . . < lm−1 ≤ n, let σ = {l1 , . . . , lm−1 } ⊂ [n]. Since ϕ([T[n]\σ ]) ∈ R/Im,n , we can write X X npm−1 mp mp np + bτ xp1 1 . . . xpk k ϕ([T[n]\σ ]) = aτ xp1 1 . . . xpm−1 τ ={p1 ,...,pk }⊂[n],k