An Upper Bound for the First Hilbert Coefficient of Gorenstein Algebras and Modules
aa r X i v : . [ m a t h . A C ] D ec AN UPPER BOUND FOR THE FIRST HILBERT COEFFICIENT OFGORENSTEIN ALGEBRAS AND MODULES
SABINE EL KHOURY, MANOJ KUMMINI, AND HEMA SRINIVASAN
Abstract.
Let R be a polynomial ring over a field and M = L n M n a finitely generatedgraded R -module, minimally generated by homogeneous elements of degree zero with agraded R -minimal free resolution F . A Cohen-Macaulay module M is Gorenstein when thegraded resolution is symmetric. We give an upper bound for the first Hilbert coefficient, e in terms of the shifts in the graded resolution of M . When M = R/I , a Gorenstein algebra,this bound agrees with the bound obtained in [ES09] in Gorenstein algebras with quasi-pureresolution. We conjecture a similar bound for the higher coefficients. Introduction
Let k be any field and R = L R i be a polynomial ring over k , finitely generated in degreeone. Let M = L i M i be a finitely generated graded R -module. We write H ( M, i ) := dim k M i for the Hilbert function of M . For i ≫ H ( M, i ) = P M ( i ) where P M ( x ) is a polynomialcalled the the Hilbert polynomial of M . The degree of P M ( x ) is dim M −
1. We can write P M ( x ) = d − X i =0 ( − i e i (cid:18) x + d − − ix (cid:19) = e ( d − x d − + . . . + ( − d − e d − where the coefficients e i are non-negative integers (depending on M ), called the Hilbertcoefficients of M . The first one, e , also denoted by e , is the multiplicity of M .We now introduce the notion of a Gorenstein R -module, generalizing the notion of aGorenstein quotient ring of R . Assume that M is generated minimally by elements of degree0. Let F = F ( M ) : 0 → T s M j = t s R ( − j ) β sj δ s → . . . → T i M j = t i R ( − j ) β ij δ i → . . . → T M j = t R ( − j ) β j δ → R β → M be a minimal graded free resolution of M as an R -module. We assume that β i,t i = 0 and β i,T i = 0 for every i . The codimension of M , denoted codim M , is the height of the annihilatorof M . Note that s ≥ codim M , with equality if and only if M is Cohen-Macaulay. We saythat M is Gorenstein if it is Cohen-Macaulay and if F is self-dual, i.e., Hom R ( F , R ) ≃ R after appropriate shifts.The numbers β ij appearing in F are called the graded Betti numbers of M ; note that β i,j is the number of copies of R ( − j ) that appear at homological degree i in any minimal gradedfree resolution of M , and that β i,j = dim k Tor Ri ( k , M ) j . If M is Gorenstein, then t s = T s and β ij = β s − i,N − j for N = T s .There has been a lot of work on bounding the multiplicity and the Hilbert coefficients offinitely generated modules, in terms of the minimal and maximal shifts of their minimal free resolutions. C. Huneke and Srinivasan (see [[HS98], Conjecture 1]) and of J. Herzog andSrinivasan (see [[HS98], Conjecture 2]) proposed upper and lower bounds for the multiplicityin terms of t i and T i . These conjectures were proven by using the Boij and S¨oderberg in full.In [HZ09], Herzog and Zheng extend the result on the multiplicity to all coefficients in theCohen Macaulay case, and found upper and lower bounds for the e i in the sense of [HS98]. Inthe Gorenstein case, when the resolution is quasi-pure, i.e., t i ≥ T i − for all i , the duality ofthe resolution is used to find sharper bounds for the multiplicity and all Hilbert coefficients,see [Sri98] and [ElS12] respectively. When the resolution is not quasi-pure, the duality of theresolution can be picked up in the Betti table from the Boij and S¨oderberg decomposition.This gives a generalization for the upper bound of the multiplicity to all Gorenstein algebras,see [EKS] for instance. In this paper, we extend the upper bound found in [EKS] to the firstcoefficient e , and prove the following theorem: Theorem 1.1.
Let M be a finitely generated graded Gorenstein R -module, minimally gen-erated by homogeneous elements of degree zero. Let s = codim M and k = ⌊ s ⌋ . Let β ( M ) denote the minimal number of generators of M and F ( M ) be the graded resolution of M asabove. Set ˜ t i = ( min (cid:8) T i , (cid:4) t s (cid:5)(cid:9) , i = 1 , . . . , k ;max (cid:8) t i , (cid:6) t s (cid:7)(cid:9) , i = k + 1 , . . . , s. Then e ( M ) ≤ β ( M )( s + 1)! s Y i =1 ˜ t i s X i =1 (˜ t i − i ) . When the resolution is quasi-pure, t i ≥ T i − and this coincides with the bounds in [ES09].Other authors gave also bounds for the first Hilbert coefficient e , see [HH], [RV10], [RV05],and [E05]. We compare our bounds to that of Rossi-Valla in [RV10] [RV05], and Elias in[E05], then conjecture that our result can be extended to all coefficients. Acknowledgements.
The authors also thank the University of Michigan for its hospitalitywhere part of this work was done when the authors were visiting for the conference in honorof Craig Huneke. The second author is partially supported by a grant from the InfosysFoundation and a travel grant from the National Board of Higher Mathematics, India. Wethank the referee for a thorough reading of the manuscript and helpful comments.2.
Notation
Let k be a field and R = k [ x , . . . , x n ] be dimensional polynomial ring over k with deg x i =1 for all 1 ≤ i ≤ n . Let M be a finitely generated graded R -module. Suppose M is Gorensteinthen the duality of the resolution can be recorded as followIf codim M = 2 k + 1 then0 → R ( − d s ) → ⊕ β j =1 R ( − ( d s − d j ) → . . . → ⊕ β k j =1 R ( − ( d s − d kj )) → ⊕ β k j =1 R ( − d kj ) → . . . → ⊕ β j =1 R ( − d j ) → R β ILBERT COEFFICIENTS OF GORENSTEIN MODULES 3 and if codim M = 2 k then0 → R ( − d s ) → ⊕ β j =1 R ( − ( d s − d j ) → . . . → ⊕ β k / r k j =1 R ( − ( d s − d kj )) ⊕ ⊕ β k / r k j =1 R ( − d kj ) → . . . → ⊕ β j =1 R ( − d j ) → R β The minimal shifts in the resolution are: t i = min j d ij ≤ i ≤ kd s − max j d s − i,j k + 1 ≤ i < sd s i = s and the maximal shifts in the resolution are: T i = max j d ij ≤ i ≤ kd s − min j d s − i,j k + 1 ≤ i < sd s i = s The graded Betti numbers of M denoted by β i,j ( M ) are the number of copies of R ( − d ij )that appear at homological degree i , in a minimal R -free resolution of M . We have X j β i,j = β i i ≤ kβ s − i k + 1 ≤ i < s i = s We think of the collection { β i,j ( M ) : 0 ≤ i ≤ n, j ∈ Z } as an element β ( M ) = ( β i,j ( M )) ≤ i ≤ n,j ∈ N ∈ B := n M i =0 M j ∈ Z Q , and call it the Betti table of M . In general, a rational Betti table β is an element β =( β i,j ) ≤ i ≤ n,j ∈ N ∈ B such that:(i) for all 0 ≤ i ≤ n , β i,j = 0 for finitely many j ,(ii) for all i > j , if β i,j = 0 then there exists j ′ < j such that β i − ,j ′ = 0.Let β = ( β i,j ) ≤ i ≤ n,j ∈ N be a rational Betti table. Its length is max { i : β i,j = 0 for some j } . Vandermonde matrices.
Given ( α , α , . . . , α k ) a sequence of real numbers, we denote thefollowing Vandermonde determinants by V t = V t ( α , α , . . . , α k ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) .... α α ... α k α α ... α k ... ... ... ... α k − α k − ... α k − k α k − t α k − t ... α k − tk (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Y ≤ j
Remark 2.1. V t ( α , α , . . . , α k ) ≥ degree sequence of length s is an increasing sequence d = ( d < d < · · · < d s ) ofintegers. An R -module M is said to have a pure resolution of type d if its Betti table β ij = 0if and only if j = d i , ≤ i ≤ s . By the Herzog-K¨uhl equations [HK84], β ( M ) is a positiverational multiple of the pure Betti table, which we denote by β ( d ), given by:(2.2) β ( d ) i,j = ( Q l = i | d l − d i | , ≤ i ≤ s and j = d i , otherwise . We call the Betti table β ( d ) defined in (2.2), the pure Betti table associated to d . For0 ≤ i ≤ s , write β i ( d ) = β ( d ) i,d i .Eisenbud, Fløystad and Weyman [EFW11, Theorem 0.1] (in characteristic zero) and Eisen-bud and Schreyer [ES09, Theorem 0.1] showed that for all degree sequences d , there is aCohen-Macaulay R -module M such that β ( M ) is a rational multiple of β ( d ). Moreover,for all R -modules M , β ( M ) can be written as a non-negative rational combination of the β ( d ) [BS08]; if we take a saturated chain of degree sequences, (in the set of all degreesequences, a saturated chain with respect to the partially order given by by point-wise com-parison) then the non-negative rational coefficients in the decomposition are unique [ES09,Theorem 0.2].2.1. Self-dual resolutions and symmetrized Betti tables.
Let β be a Betti table. Let s and N be integers. We say that β is ( s, N ) -self-dual if β i,j = β s − i,N − j for all i, j . We saythat β is self-dual if there exist s and N such that β is ( s, N )-self-dual. If β is self-dual, then s is the length of β and N = max { j : β s,j = 0 } + min { j : β ,j = 0 } . Definition 2.3.
Let d = ( d < · · · < d s ) be a degree sequence and N ≥ d + d s . Let d ∨ ,N = ( N − d s < · · · < N − d ). Denote the pure Betti table associated to d ∨ ,N by β ∨ ,N ( d ).Similarly, set β ∨ ,Ni ( d ) = β ∨ ,N ( d ) i,N − d s − i . Let β sym ( d, N ) = β ( d )+ β ∨ ,N ( d ). We call β sym ( d, N )the symmetrized pure Betti table , given by symmetrizing d with respect to N .Peskine and Szpiro [PS74] showed that for a finitely generated graded R -module M with s = codim M , pd M X i =0 ( − i X j d tij = ( ≤ t < s ( − s s ! e ( M ) if t = s. Moreover a similar expression to all coefficients was given in [E09, Lemma 3.2] and [ElS12,Theorem 3.2], for every finitely generated graded R -module M of codimension s . We have, l X r =0 ( − l − r ν l − r pd M X i =0 ( − i b i X j =1 d t + rij = ( − s ( s + l )! e l ( M ) if t = s. with ν l − r = X ≤ ξ <ξ < ··· <ξ l − r ≤ s + l − ξ ξ · · · ξ l − r and ν = 1 . Let d be a degree sequence of length s . Since the pure Betti table β ( d ) is, up to multipli-cation by a rational number, the Betti table of a Cohen-Macaulay R -module of codimension ILBERT COEFFICIENTS OF GORENSTEIN MODULES 5 s , we see that P si =0 ( − i β i ( d ) d li = 0 for all 0 ≤ l < s . Further, by a direct computation, wecan see that P si =0 ( − i β i ( d ) d s + ri = ( − s V r ( d ,...d s ) V ( d ,...d s ) = ( − s X α + ...α s = r d α d α . . . d α s s . Definition 2.4.
Let d = ( d , . . . , d s ) be a degree sequence and β ( d ) and β ( d ∨ ,N ) be as inDefinition 2.3. Then we define, e l as follows:( s + l )! e l ( β ( d )) = l X r =0 ( − l − r ν l − r X α + ...α s = r d α d α . . . d α s s and( s + l )! e l ( β ( d ∨ ,N )) = l X r =0 ( − l − r ν l − r X α + ...α s = r ( N − d s − ) α ( N − d s − ) α . . . ( N − d ) α s Note that the above sums can be factored as follows
Lemma 2.5.
For any sequence of integers y < y < . . . < y sl X r =0 ( − l − r ν l − r X α + ...α s = r y α y α . . . y α s s = X ≤ i ≤··· i l ≤ s l Y t =1 ( y i t − ( i t + t − where ν l − r = Y ≤ β < ··· <β l − r ≤ s + l − β · · · β l − r . This has been proved in [ElS12, Lemma 4.8] and [E09, Theorem 4.2]. We repeat it for thesake of completeness.
Proof.
For any given tuples 1 ≤ γ ≤ · · · ≤ γ r ≤ s with 0 ≤ r ≤ l , we have 1 ≤ β ≤ · · · ≤ β l − r ≤ s such that { γ , · · · γ r } ∪ { β , · · · β l − r } = { i , · · · i l } . In the product X ≤ i ≤··· i l ≤ s l Y t =1 ( y i t − ( i t + t − Y ≤ γ ≤···≤ γ l ≤ s y γ t is Y ≤ β < ··· <β l − r ≤ s + l − β · · · β l − r = ν l − r since i t + t − i l + l − (cid:4) Since β sym ( d, N ) = β ( d ) + β ∨ ,N ( d ), we see that(2.6)( s + l )! e l ( β sym ( d, N )) = X ≤ i ≤··· i l ≤ s l Y t =1 ( d i t − ( i t + t − X ≤ i ≤··· i l ≤ s l Y t =1 ( N − d s − i t − ( i t + t − Proposition 2.7. [EKS] [Proposition 2.4] Let M be finitely generated graded Cohen-Macaulay R -module with codim M = s , generated minimally by homogeneous elements of degree zero. S. EL KHOURY, MANOJ KUMMINI, AND H. SRINIVASAN
Suppose that β ( M ) is self-dual. Let N = T s = t s . Then there exist degree sequences d α , ≤ α ≤ a for some a ∈ N and positive rational numbers r α , ≤ α ≤ a such that β ( M ) = a X α =0 r α β sym ( d α , N ) . Moreover,(i) the d α are degree sequences of length s and they are not ( s, N ) -dual to each other.(ii) d α +1 > d α for all ≤ α ≤ a − .(iii) N ≥ d αi + d αs − i for all α and i , or equivalently, d α ≤ ( d α ) ∨ ,N for all α . As a consequence, there is a formula for all the Hilbert coefficients, e ℓ . Let M, N, d α be asin the theorem above. e ℓ ( M ) = a X α =0 r α β sym ( d α , N ) . Using this, one can get a stronger upper bound for the multiplicity e of Gorensteinmodules [EKS][Theorem 3.1]. That is, e ( M ) ≤ β ( M ) s ! k Y i =1 min (cid:26) T i , (cid:22) t s (cid:23)(cid:27) s Y i = k +1 max (cid:26) t i , (cid:24) t s (cid:25)(cid:27) . Again, when the resolution is quasi-pure this generalizes the bound in [ ? ]. This paper isan attempt to generalize this to higher Hilbert coefficients. We prove an analogous boundfor the first coefficient, e and conjecture a bound for the higher coefficients. e i , i ≥ Upper bound for e Notation 3.1.
We define some notation used in the statements and proofs of our results.Let d = (0 , d , . . . , d s ) be a non-decreasing sequence of integers.(i) For a sequence d ′ = (0 , d ′ , . . . , d ′ s ) of integers, we write d < d ′ if d = d ′ and d i ≤ d ′ i for every 1 ≤ i ≤ s .(ii) ˜ d = ( ˜ d , . . . , ˜ d s ) be the (non-decreasing) sequence˜ d i = ( min (cid:8) d s − d s − i , (cid:4) d s (cid:5)(cid:9) , i = 1 , . . . , k ;max (cid:8) d i , (cid:6) d s (cid:7)(cid:9) , i = k + 1 , . . . s. (iii) Suppose that d s ≥ d i + d s − i for every 0 ≤ i ≤ s . Write(a) b d = β ( d ) + β ( d ∨ ,d s ).(b) Ψ d = Q si =1 ˜ d i .(iv) For 1 ≤ l ≤ s , define f l ( ˜ d ) = f l ( ˜ d , . . . , ˜ d s ) = X ≤ i ≤···≤ i l ≤ s l Y t =1 ( ˜ d i t − ( i t + t − . Set f ( ˜ d , . . . , ˜ d s ) = 1. (cid:4) ILBERT COEFFICIENTS OF GORENSTEIN MODULES 7
This section is devoted to the proof of Theorem 3.2, which is about finding an upper boundfor the first Hilbert coefficient of Gorenstein algebras.
Theorem 3.2.
Let M be a finitely generated graded Gorenstein R -module, minimally gen-erated by homogeneous elements of degree zero. Let s = codim M and k = ⌊ s ⌋ . Let β ( M ) denote the minimal number of generators of M . For ≤ i ≤ s , write t i = t i ( M ) = min { j :Tor Ri ( k , M ) j = 0 } and m = ( t i ) i =0 ,...,s . Then e ( M ) ≤ β ( M )( s + 1)! Ψ m f (˜ t ) . This upper bound coincides with the upper bound of Gorenstein algebras with quasi-pureresolutions found in [ElS12]. In order to prove this theorem, we find the upper bound for e ( β sym ( d, N )) of a symmetrized pure Betti table then use the Boij and S¨oderberg decom-position in order to generalize our result to e ( M ). To proceed, we first need the followinglemma, Lemma 3.3.
Let d and d ′ be degree sequences such that d = 0 and d < d ′ ≤ ( d ′ ) ∨ ,d s < d ∨ ,d s .Then Ψ d f ( ˜ d ) ≥ Ψ d ′ f ( ˜ d ′ ) .Proof. By induction on P i d ′ i − d i , we may assume, without loss of generality, that thereexists j such that d ′ j = d j + 1 and d ′ i = d i for all i = j . Moreover, if 1 ≤ i ≤ s − k −
1, then d i does not figure in the expression for Ψ d f l ( ˜ d ), so we may assume that j ≥ s − k . Additionally, j ≤ s −
1. We rewrite Ψ d as(3.4) Ψ d = s − Y i = s − k min (cid:26) d s − d i , (cid:22) d s (cid:23)(cid:27) s Y i = k +1 max (cid:26) d i , (cid:24) d s (cid:25)(cid:27) . We note that f ( ˜ d ) = X ≤ i ≤ s ( ˜ d i − i ) = X ≤ i ≤ s ˜ d i − (cid:18) s + 12 (cid:19) Two cases arise: j < k + 1 and j ≥ k + 1. The first case is possible if and only if s = 2 k and j = k . In this case, d k appears only once in (3.4), and since d k ≤ d s − d s − k = d s − d k ,we get d s − d k ≥ (cid:4) d s (cid:5) . By the hypothesis that d ′ ≤ ( d ′ ) ∨ ,d s , d s − d k − ≥ d k + 1, so d s − d k − ≥ (cid:4) d s (cid:5) . We get min (cid:8) d s − d k , (cid:4) d s (cid:5)(cid:9) = min (cid:8) d s − d k − , (cid:4) d s (cid:5)(cid:9) = (cid:4) d s (cid:5) . In theexpression f ( ˜ d ′ ), only d s − d k − f ( ˜ d ′ )Ψ d ′ = f ( ˜ d )(Ψ d ).In the second case (i.e., j ≥ k + 1), d j appears twice in in (3.4). We need to show thatΨ d ′ f l ( ˜ d ′ )Ψ d f l ( ˜ d ) ≤ d j < (cid:6) d s (cid:7) , then d s − d j > (cid:4) d s (cid:5) . So we get that d j + 1 ≤ (cid:6) d s (cid:7) and d s − d j − ≥ (cid:4) d s (cid:5) . Hence, min (cid:8) d s − d j , (cid:4) d s (cid:5)(cid:9) = min (cid:8) d s − d j − , (cid:4) d s (cid:5)(cid:9) = (cid:4) d s (cid:5) and max (cid:8) d j , (cid:6) d s (cid:7)(cid:9) =max (cid:8) d j , (cid:6) d s (cid:7)(cid:9) = (cid:6) d s (cid:7) . As a result we obtain f ( ˜ d ′ )Ψ d ′ = f ( ˜ d )(Ψ d ). S. EL KHOURY, MANOJ KUMMINI, AND H. SRINIVASAN
Let us now consider the case where d j ≥ (cid:6) d s (cid:7) so d j + 1 ≥ (cid:6) d s (cid:7) . This implies that d s − d j − < d s − d j ≤ (cid:4) d s (cid:5) . It suffices to show thatΨ d ′ f l ( ˜ d ′ )Ψ d f l ( ˜ d ) = ( d s − d j − d j + 1) f ( ˜ d ′ )( d s − d j ) d j f ( ˜ d ) ≤ . We compute the difference between the numerator and the denominator. We write(3.5)( d s − d j − d j +1) f ( ˜ d ′ ) − ( d s − d j ) d j f ( ˜ d ) = − (2 d j − d s +1) f ( ˜ d )+( d s − d j − d j +1)( f ( ˜ d ′ ) − f ( ˜ d ))where f ( ˜ d ) = X ≤ i ≤ s ˜ d i − (cid:18) s + 12 (cid:19) = s (cid:22) d s (cid:23) − (cid:18) s + 12 (cid:19) and f ( ˜ d ′ ) = X ≤ i ≤ s ˜ d ′ i − (cid:18) s + 12 (cid:19) = s (cid:22) d s (cid:23) − − (cid:18) s + 12 (cid:19) So Equation 3.5 is equal to: − (2 d j − d s + 1)( s (cid:22) d s (cid:23) − (cid:18) s + 12 (cid:19) ) + ( d s − d j − d j + 1)( − d j ≥ (cid:6) d s (cid:7) and s (cid:4) d s (cid:5) − s ( s +1)2 = s ( (cid:4) d s (cid:5) − s +12 ) ≥ d s ≥ s + 2. Hence( d s − d j − d j + 1) f ( ˜ d ′ ) − ( d s − d j ) d j f ( ˜ d ) ≤
0, and the proof is done. (cid:4)
The next proposition is needed for the proof of Theorem 3.2. Let d = (0 , d , . . . , d s ) be adegree sequence such that d s ≥ d i + d s − i for all 0 ≤ i ≤ s . We show that the first coefficient e ( β sym ( d, d s ) of symmetrized pure sequences satisfy the upper bound in our main theorem.When l = 1, we write from equation 2.6:(3.6) ( s + l )! e ( β sym ( d, d s )) = X ≤ i ≤ s ( d i − i ) + X ≤ i ≤ s ( d s − d s − i − i )= f ( d ) + f ( d ∨ ,d s )We note that f ( d ) = X ≤ i ≤ s d i − (cid:18) s + 12 (cid:19) ; f ( d ∨ ,d s ) = sd s − X ≤ i ≤ s d i − (cid:18) s + 12 (cid:19) . Proposition 3.7.
Let d = (0 , d , . . . , d s ) be a degree sequence such that d ≤ d ∨ ,d s . Then f ( d ) + f ( d ∨ ,d s ) ≤ b d Ψ d f ( ˜ d ) . ILBERT COEFFICIENTS OF GORENSTEIN MODULES 9
Proof.
We again prove this by induction on P i (cid:0) ( d ∨ ,d s ) i − d i (cid:1) = P i ( d s − d s − i − d i ), whichis non-negative by our hypothesis. If P i ( d s − d s − i − d i ) = 0 (equivalently, d = d ∨ ,d s ), then˜ d = d , so the assertion follows from noting that b d Ψ d ≥ d < d ∨ ,d s , then there exists j ≥ k such that d j < d s − d s − j . Pick j to be maximal withthis property. Note that j < s , so d ′ s = d s . Since d j < d s − d s − j , d j +1 = d s − d s − j − and d s − j > d s − j − , we see that d ′ := (0 , d , · · · , d j − , d j + 1 , d j +1 , · · · , d s ) is a degree sequence,that d ′ ≤ ( d ′ ) ∨ ,d s and that P i ( d s − d s − i − d i ) > P i ( d s − d ′ s − i − d ′ i ). Hence by induction f ( d ′ ) + f (( d ′ ) ∨ ,d s ) ≤ b d ′ Ψ d ′ f ( ˜ d ′ ) . Therefore f ( d ′ ) + f (( d ′ ) ∨ ,d s ) = f ( d ) + f ( d ∨ ,d s ). We now show that f ( ˜ d ′ ) = f ( ˜ d ). Tothis end, we consider various cases:(i) j < s − k : Then ˜ d ′ = ˜ d , so f ( ˜ d ′ ) = f ( ˜ d ).(ii) j = s − k : We consider two sub-cases.(a) s = 2 k : Then( ˜ d ′ ) i = ( ˜ d i , ≤ i ≤ s, i = k min (cid:8) d s − d k − , (cid:4) d s (cid:5)(cid:9) , i = k Note that since d k < d s − d k , d s − d k − ≥ (cid:4) d s (cid:5) , so ( ˜ d ′ ) k = ˜ d k = (cid:4) d s (cid:5) . Therefore d ′ = d and hence f ( ˜ d ′ ) = f ( ˜ d ).(b) s = 2 k + 1: Then( ˜ d ′ ) i = ˜ d i , ≤ i ≤ s, i = k and i = k + 1min (cid:8) d s − d k +1 − , (cid:4) d s (cid:5)(cid:9) , i = k max (cid:8) d k +1 + 1 , (cid:6) d s (cid:7)(cid:9) , i = k + 1Therefore f ( ˜ d ′ ) = f ( ˜ d ).(iii) j > s − k : Then( ˜ d ′ ) i = ˜ d i , ≤ i ≤ s, i = j and i = s − j min (cid:8) d s − d j − , (cid:4) d s (cid:5)(cid:9) , i = s − j max (cid:8) d j + 1 , (cid:6) d s (cid:7)(cid:9) , i = j Therefore f ( ˜ d ′ ) = f ( ˜ d ).Now note that b d Ψ d ≥ b d ′ Ψ d ′ [EKS, Proof of Proposition 3.5, p.126]. This completes theproof of the Proposition. (cid:4) Proof of Theorem 3.2.
Pick degree sequences d α and non-negative rational numbers r α asin Proposition 2.7. We need to show that ( s + 1)! e ( M ) ≤ β ( M )Ψ t f (˜ t ). We get this asfollows: ( s + 1)! e ( M ) = ( s + 1)! X α r α e ( β sym ( d α ))= X α r α (cid:0) f ( d α ) + f (( d α ) ∨ ,d s ) (cid:1) (by (3.6) ≤ X α r α b d α Ψ d α f ( ˜ d α ) (by Proposition 3 . ≤ X α r α b d α ! Ψ t f (˜ t ) (by Lemma 3 . β ( M )Ψ t f (˜ t ) . (3.8) (cid:4) Next we give examples for the bound that we found and we compare our result to thebound found by Rossi-Valla in [RV05, Theorem 3.2] and Elias in [E05, Theorem 2.3]. Weobtain a much sharper bound.
Example 3.9.
Let R = k [ x, y, z, w, t, u, v ] and I = ( y , z , w , x − zwt, x y − wt , x z, xyz − x t, yzt − xt , yt , zt , t ) a homogeneous Gorenstein ideal of R . Note that e ( R/I ) = 90,and the Betti diagram is given as follow:
We have T = 3 , T = 5 , t = 5 , t = 7, and d = T = t = 10 and d = 5. By theorem3.2, e ( R/I ) ≤ f ( T , T , t , t , t ) T T t t t , which implies that e ( R/I ) ≤
16! (2 + 3 + 2 + 3 + 5)3 . . . .
10 = 109 . (cid:0) e (cid:1) − (cid:0) µ ( I ) − d (cid:1) where µ ( I ) = 11is the minimal number of generator of I and d = 7 the dimension of R . Since e = 90, thenthe bound in [E05] gives e ( R/I ) ≤ (cid:4) In the above example, the minimal free resolution of
R/I is quasi-pure. This case wasdone in [ElS12]. In the next example,
R/I has a non quasi-pure minimal free resolution.
Example 3.10.
Let R = k [ x, y, z, w, s, t, u, v ] and I = ( wt, zt, xt, zw, yw, xw, yz, xz, x y + yt , x − y + yt − t , w + t , z − t ) a homogeneous Gorenstein ideal of R . Note that e ( R/I ) = 65, and the Betti diagram is given as follow:
ILBERT COEFFICIENTS OF GORENSTEIN MODULES 11
3: . . 2 4 2 .4: . 2 9 14 8 .5: . . . . . 1
We have T = 5 , T = 6 , t = 4 , t = 5, and d = T = t = 10 and d = 5. By theorem3.2, e ( R/I ) ≤ f ( T , d , d , t , t ) T d d t t , which implies that e ( R/I ) ≤
16! (4 + 3 + 2 + 1 + 5)5 . . . . ≃ . (cid:0) e (cid:1) − (cid:0) µ ( I ) − d (cid:1) − λ ( R/I ) + 1 where µ ( I ) = 12 is the minimal number of generator of I , d = 8 the dimensionof R , and λ ( R/I ) = e − e + e . We note that e ( R/I ) = 26 and e ( R/I ) = 68. Hence thebound in [RV05] gives e ( R/I ) ≤ (cid:4) Huneke and Hanumanthu [HH] find a bound for e in a slightly different setting. Fora CM local ring ( R, m ) of dim d and an ideal I contained in m k , for some k >
2, let λ ( R/I n +1 ) = P i ( − i e i ( I ) (cid:0) ( n + d − id ) (cid:1) . In [HH] corollary 3.7, they show that e ≤ (cid:0) (( e − k )2) (cid:1) .Finally, we conjecture that the above upper bound can be extended to all Hilbert coeffi-cients e i ’s. We believe the following is true. Conjecture 3.11.
Let M be a finitely generated graded Gorenstein R -module, minimallygenerated by homogeneous elements of degree zero. Let s = codim M and k = ⌊ s ⌋ . Let β ( M ) denote the minimal number of generators of M and write m = ( t i ) i =0 ,...,s . Then e j ( M ) ≤ β ( M )( s + 1)! Ψ t f j (˜ t ) . for all ≤ j ≤ d − . Example 3.12.
In Example 3.10, we considered the ideal I = ( wt, zt, xt, zw, yw, xw, yz, xz, x y + yt , x − y + yt − t , w + t , z − t ) where T = 5 , T = 6 , t = 4 , t = 5, d = T = t = 10and d = 5. We have e ( R/I ) = 68, and we note that e ( R/I ) ≤
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Department of Mathematics, American University of Beirut, Beirut, Lebanon.
Email address : [email protected] Chennai Mathematical Institute, Siruseri, Tamilnadu 603103 India.
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