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BETTI NUMBERS OF GRADED MODULES AND THE MULTIPLICITY CONJECTURE IN THE NON-COHEN-MACAULAY CASE

¨ MATS BOIJ AND JONAS SODERBERG Abstract. We use the results by Eisenbud and Schreyer [3] to prove that any Betti diagram of a graded module over a standard graded polynomial ring is a positive linear combination Betti diagrams of modules with a pure resolution. This implies the Multiplicity Conjecture of Herzog, Huneke and Srinivasan [5] for modules that are not necessarily Cohen-Macaulay. We give a combinatorial proof of the convexity of the simplicial fan spanned by the pure diagrams.

arXiv:0803.1645v1 [math.AC] 11 Mar 2008

1. Introduction The formula for the multiplicity of a standard graded algebra with a pure resolution found by C. Huneke and M. Miller [7], led to the formulation of the Multiplicity Conjecture by C. Huneke and H. Srinivasan, which was later generalized and published by J. Herzog and H. Srinivasan [5]. In a series of papers1 these versions of the Multiplicity conjecture have been proven in many cases. Recently D. Eisenbud and F.-O. Schreyer proved the conjecture in the CohenMacaulay case by proving a set of conjectures formulated by the authors [1] on the set of possible Betti diagrams up to multiplication by positive rational numbers. In the work of D. Eisenbud and F.-O. Schreyer, they introduce a set of linear functionals de?ned on the space of possible Betti diagrams. The linear functionals are given by certain cohomology tables of vector bundles on Pn?1 and they show that the supporting hyperplanes of the exterior facets of the simplicial fan given by the pure Betti diagrams are given by the vanishing of these linear functionals, while the functionals are non-negative on the Betti diagram of any minimal free resolution. In this paper we generalize the contruction given in our previous paper in order to include Cohen-Macaulay pure Betti diagrams of various codimensions. We show that the linear functionals, similar to the ones introduced by D. Eisenbud and F.-O. Schreyer de?ne the supporting hyperplanes of the simplicial fan. Furthermore, these new linear functionals are limits of the functionals given by D. Eisenbud and F.-O. Schreyer which allows us to conclude that all Betti diagrams of graded modules can be uniquely written as a positive linear combination of pure diagrams in a totally ordered chain. Together with the existence of modules with pure resolutions, proved by D. Eisenbud, G. Fl?ystad and J. Weyman [2]

Date : March 11, 2008 (Preliminary version). 1 Citations will be added.

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¨ MATS BOIJ AND JONAS SODERBERG

in characteristic zero, and by D. Eisenbud and F.-O. Schreyer [3] in general, this gives a complete classi?cationo of the possible Betti diagrams up to multiplication by scalars. As a consequence, we get the Multiplicity Conjecture for algebras and modules that are not necessarily Cohen-Macaulay. In fact, we get a stronger version of the inequalities of the Multiplicity Conjecture in terms of the Hilbert series of the module which is bounded from below by the Hilbert series corresponding to the lowest shifts in a minimal free resolution while it is bounded from above by the Hilbert series corresponding to the highest shifts in the ?rst s + 1 terms of the resolution, where s is the codimension of the module. We also give a combinatorial proof of the convexity of the simplicial fan spanned by the pure diagrams, even though this convexity is an implicit consequence of the results involving the linear functionals. Furthermore, we show that if we choose the basis of pure diagrams in a certain way, all the coe?cients in the expansion of an actual Betti diagram into a chain of pure diagrams are non-negative integers. 2. The partially ordered set of pure Betti diagrams Let R = k [x1 , x2 , . . . , xn ] be the polynomial ring with the standard grading. For any ?nitely generated graded module M , we have a minimal free resolution 0 ?→ Fn ?→ Fn?1 ?→ · · · ?→ F1 ?→ F0 ?→ M ?→ 0 where Fi =

j ∈Z

R(?j )βi,j ,

i = 0, 1, . . . , n,

and we get that the Hilbert series of M can be recovered from the Betti numbers, βi,j , by 1 H (M, t) = (1 ? t)n

n

(?1)i βi,j tj .

i=0 j ∈Z

It was noted by J. Herzog and M. K¨ uhl [4] that this gives us s linearly independent equations

n

(?1)i βi,j j m = 0,

i=0 j ∈Z

m = 0, 1 , . . . , s ? 1

for i = 0, 1, . . . , s.

where s is the codimension of M . Furthermore, they proved that in the case when M is Cohen-Macaulay and the resolution is pure, i.e., if Fi = R(?di ) for some integers d0 , d1 , . . . , ds , we get a unique solution to these equations given by ? s ? 1 ? i ? (?1) , j = di d ? di βi,j = , j =0 j ? j =i ? ? 0, j = di

BETTI NUMBERS OF GRADED MODULES

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De?nition 1. For an increasing sequence of integers d = (d0 , d1 , . . . , ds ), where 0 ≤ s ≤ n, we denote by π (d) the matrix in Qn?1 × QZ given by π (d)i,j = (?1)

i

1 , d ? di j =0 j

j =i

s

for j = di

and zero elsewhere. We will call this the pure diagram given by the degree sequence d = (d0 , d2 , . . . , ds ). We will use the notation di (π ) to denote the degree, di , when π = π (d) = π (d0 , d1, . . . , ds ). For degree sequences with d0 = 0, we will also use the normalized pure diagram π ? (d) = d1 d2 · · · ds π (d) so that normalized pure diagrams have π ?0,0 = 1. We de?ne a partial ordering on the set of pure diagrams, extending the ordering used for Cohen-Macaulay diagrams of a ?xed codimension. De?nition 2. We say that π (d0 , d1, . . . , ds ) ≤ π (d′0 , d′1 , . . . , d′t ) if the s ≥ t and di ≤ d′i for i = 0, 1, . . . , t. As in the case of Cohen-Macaulay pure diagrams, we get a simplicial structure given by the maximal chains of pure diagrams, which spans simplicial cones. In order to have maximal chains in this setting, we have to ?x a bound on the region we are considering. We can do this by restricting the degrees to be in a given region M + i ≤ di ≤ N + i. We will denote the subspace generated by Betti diagrams with these restrictions by BM,M = s Qn+1 × QN ?M +1 . Furthermore, we denote by BM,N the subspace of BM,N of diagrams satisfying the s ?rst Herzog-K¨ uhl equations. Proposition 1. For s = 0, 1, . . . , n, we have that any maximal chain of pure diagrams of s codimension at least s in BM,N form a basis for BM,N . Proof. For any interval of length one π < π ′ there is a unique non-zero entry in π which is zero in all pure diagrams above π ′ . Thus the pure diagrams in any maximal chain are linearly independent. The number of elements in any maximal chain of pure diagrams of codimension at least s is (n + 1)(N ? M ) + n ? s + 1, since we have n + 1 positions that has to be raised N ? M steps and then n ? s + 1 times when the codimension is lowered by s one. On the other hand, we have that the dimension of BM,N is (n + 1)(N ? M + 1) ? s, since we have s independent equations on BM,N . By looking at the order in which the di?erent positions of a Betti diagram in BM,N disappear when going along a maximal chain of pure diagrams we get the following observation: Proposition 2. The maximal chains of pure diagrams in BM,N are in one to one correspondence with numberings of the entries of an (N ? M + 1) × (n + 1)-matrix which are increasing to the left and downwards.

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¨ MATS BOIJ AND JONAS SODERBERG

Example 1. For n = 2, M = 0 and N = 1, the numbering 4 3 1 6 5 2 corresponds to the maximal chain ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? < < < < < ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? and there are four other maximal chains corresponding to 3 2 1 , 6 5 4 4 2 1 , 6 5 3 5 2 1 , 6 4 3 and 5 3 1 . 6 4 2

3. Description of the boundary facets We know that the partially ordered set of pure diagrams in BM,N give rise to a simplicial fan ?, where the faces are the totally ordered subsets. The facets of this simplicial complex correspond to maximal chains in the partially ordered set. According to Proposition 1 each such set is a basis for the space BM,N . If we look at normalized Betti diagrams, we get a simplicial complex which is the hyperplane section of the simplicial fan. This is the complex described in our previous paper, in the case of Cohen-Macaulay diagrams. Proposition 3. The simplicial cones spanned by the totally ordered sets of pure diagrams in BM,N form a simplicial fan. Proof. We need to show that the cones meet only along faces. This is the same as to say that any element which can be written as a positive linear combination of pure diagrams in a chain can be written so in a unique way. If such a sum has only one term, the uniqueness is trivial. Thus, suppose that a diagram in BM,N can be written as a positive linear combination of totally ordered pure diagrams in two di?erent ways and that this is the minimal number of terms in such an example. We then have

m k

β=

i=1

λi πi =

j =1

′ ?j πj

where all the coe?cients are positive. Look at the lowest degree in which β is non-zero for each column. These degrees have to be given by the degrees in π1 and by the degrees ′ ′ in π1 . Thus we must have π1 = π1 . If λ > ?, we can subtract ?π1 from β and from both sums to get

m k

β ? λ1 π1 =

i=2

λi πi = (?1 ? λ1 )π1

j =2

′ ?j πj

Since all the pure diagrams in the left expression are greater than π1 , the degrees in which β ? λ1 π1 is non-zero have to be given by π2 , but then the coe?cient of π1 in the right hand expression has to be zero, i.e., λ1 = ?1 . Since this was assumed to be a minimal example, we get that the expressions for β ? λ1 π1 are term wise equal and so were the original expressions for the diagram β .

BETTI NUMBERS OF GRADED MODULES

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We will now show that the coe?cients of the pure diagrams when a Betti diagram is expanded in the basis given by a maximal chain have nice expressions. In particular, we see that the coe?cients are integers. Moreover, it will give us expressions for the inequalities that de?ne the simplicial fan similar to the inequalities used by D. Eisenbud and F.-O. Schreyer to prove our conjectures in the Cohen-Macaulay case.

s Proposition 4. The coe?cient of π1 = π (d0 , d1 , . . . , dm ) when a Betti diagram β in BM,N is expanded in a basis containing π0 < π1 < π2 is given by n di (π0 ) m i j =0

j =k

(?1)

i=0 d=M

(dj ? d)βi,d ,

when π1 di?ers from π0 in codimension and from π2 in column k , by

n di (π0 ) m?1

(?1) (dk ? dm )

i=0 d=M j =0

j =k

i

(dj ? d)βi,d ,

when π1 di?ers from π2 in codimension and from π0 in column k , by

n di (π0 ) m

(?1) (d? ? dk )

i=0 d=M j =0

i

(dj ? d)βi,d ,

j ∈{ / k,?}

when π1 di?ers from π2 in column k and from π0 in column ? = k , and by

n di (π0 ) m?1

(?1)i

i=0 d=M j =0

(dj ? d)βi,d ,

when π1 di?ers from π0 and π2 in codimension. In particular, all the coe?cients are integers. Remark 1. For simplicity of notation, we use the convention that di(π ) = N if i is greater than the codimension of π . Proof. In all four cases, the coe?cients of βi,d are zero for pure diagrams π ≥ π2 by construction since the sums are taken only up to degree di (π0 ) and the coe?cients are zero in all positions where di (π0 ) = di (π2 ). Furthermore, the expressions are zero on all pure diagrams π , where π ≤ π0 by the Herzog-K¨ uhl equations. Indeed, for such pure diagrams, we can extend the summation over d be over all degrees from M to N . When we do this we get and expand the products we get polynomials in d and therefore, the expressions are linear combinations of the Herzog-K¨ uhl equations

n N

(?1)i βi,d dj = 0,

i=0 d=M

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¨ MATS BOIJ AND JONAS SODERBERG

for j = 0, 1, . . . , m ? 1 in the second and third case, and for j = 0, 1, . . . , m in the ?rst and fourth case. Observe that all pure diagrams π with π ≤ π0 , have codimension at least the codimension of π0 and hence satisfy these Herzog-K¨ uhl equations. It remains to show that the value of the expressions are one on the pure diagram π1 . In the ?rst expression, the only non-zero term is in column k and equals

m

(?1)k

j =0

j =k

(dj ? dk ) · (?1)k

1 = 1, d ? d j k j =0

j =k

m

in the second expression, the only non-zero term is in column m and equals

m m?1

(?1)m (dk ? dm )

j =0

j =k

(dj ? dm ) · (?1)m

j =0

1 = 1, dj ? dm

in the third expression, the only non-zero term is in column k and equals

m

(?1) (d? ? dk )

j =0

j ∈{ / k,?}

k

(dj ? dk ) · (?1)

k

1 =1 d ? dk j =0 j

j =?

m

and in the fourth expression, the non-zero term is in column m and equals

m?1 m?1

(?1)

m j =0

(dj ? dm ) · (?1)

m j =0

1 = 1. dj ? dm

Example 2. When n = 3, M = 0 and N = 2 we can choose the basis of B0,2 given by the numbering 10 4 3 1 11 6 5 2 12 9 8 7 When expanding a Betti diagram β0 β1 β2 β3 0 : β0,0 β1,1 β2,2 β3,3 1 : β0,1 β1,2 β2,3 β3,4 2 : β0,2 β1,3 β2,4 β3,5

BETTI NUMBERS OF GRADED MODULES

7

into this chain we get the coe?cients by applying the linear functionals corresponding to the following matrices: 0 0 0 6 0 0 0 0 0 0 0 0 1 0 ?4 6 ?6 0 0 6 4 0 0 0 0 5 0 1 ?2 3 0 2 ?3 4 0 3 ?4 5 9 0 0 0 2 0 0 0 6 1 0 0 10 0 0 0 0 0 0 0 24 0 0 0 0 0 0 0 3 0 ?6 0 ?8 0 0 7 0 0 1 0 0 0 11 6 ?18 0 ?36 0 0 8 6 0 0 0 0 ?6 0 10 0 0 0 0 0 0 4 0 0 0 8 0 0 1 12 8 ?12 0 0 0 0 2 2 0 0 0 0 ?2 0 4 0 0 0 12 8 0 0 ?4 ?10 0 0 0

8 ?12 12 12 ?12 8 0 0 0 ?1 ?1 ?1 1 1 1 ?1 ?1 ?1

Matrices 1, 2, 11 and 12 correspond to individual Betti numbers as we will see in the next proposition. Matrices 3, 4, 5 and 6 are of the third kind described in Proposition 4 since they correspond to two consecutive degree changes in distinct columns. Matrix 7 is of the second kind, since it corresponds to a degree change followed by a change in codimension. Matrices 8 and 9 are of the fourth kind, since they correspond to two consecutive changes of codimension. Matrix 10 is of the ?rst kind since it corresponds to a change in codimension followed by a degree change. In order to prove that the simplicial fan is convex and in order to prove that any Betti diagram of a module of codimension s is a positive linear combination of pure diagrams, we need to know what is the boundary of the simplical fan. The description is very similar to the description given in our previous paper and we only have to add one more kind of boundary facet. Proposition 5. A facet of the boundary of the simplical fan given by the pure diagrams in s BM,N is given by removing one element from a maximal chain such that there is a unique way to complete it to a maximal chain. There are four di?erent cases: i) The removed element is maximal or minimal, or ii) the removed element is in the middle of a chain of three degree sequences which di?er in one single column, i.e., π (d) < π (d′ ) < π (d′′ ), di = d′i ? 1 = d′′ i ? 2, di+1 = d′i+1 ? 1, or

iii) the removed element di?er from the adjacent vertices in two adjacent degrees, i.e., π (d) < π (d′ ) < π (d′′ ), d′i = d′′ i ? 1, or

iv) the removed element di?er from the adjacent vertices in codimension, i.e., π (d) < π (d′ ) < π (d′′ ), codim(π (d)) = codim(π (d′′ )) + 2.

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¨ MATS BOIJ AND JONAS SODERBERG

Proof. A boundary facet is a codimension one face of a simplicial cone and hence given by the removing of one vertex. Since it is on the boundary, it is contained in a unique maximal dimensional cone, so there has to be a unique extension of the chain into a maximal chain. This clearly happens if we remove the maximal or the minimal element, since there are unique such elements in the partially ordered set. Suppose therefore that we remove π (d′ ) and that the adjacent vertices in the chain are π (d) < π (d′ ) and π (d′′ ) > π (d′ ). If the di?erence between π (d) and π (d′′ ) are in two columns that are not adjacent, we can ?nd another element between π (d) and π (d′′ ) by exchanging the order in which the two degrees are increased. This can also be done if the columns are adjacent, but the degrees di?er by more than one. If one of the two di?er from π (d′ ) in codimension and the other by an increase of the degree in one column, we can alter the order and obtain another element between π (d) and π (d′′ ). The remaining cases are those described by ii), iii) and iv). Remark 2. In Example 2 matrices 1 and 12 correspond to the ?rst kind of facets. Matrices 2 and 11 correspond to the second kind of boundary facets. Matrices 4 and 6 correspond to boundary facets of the third kind and matrices 8 and 9 correspond to boundary facets of the fourth kind. The remaining matrices correspond to inner faces of the fan.

s Theorem 1. The simplicial fan of pure diagrams in BM,N is convex.

Proof. We will use the following observation which allows us to go from local convexity to global convexity: If the simplicial fan is not convex, there will be one boundary facet which supporting hyperplane pass through the interior of a neighboring simplicial cone. We can see this by looking a line segment between two vertices which contains points outside the simplicial fan. If we take a two-dimensional plane through these two points and through generic inner point of the fan, we get a two-dimensional picture where we can ?nd two edges meeting at an inwards angle. The supporting hyperplane of the boundary facet meeting the two-dimensional plane in one of these edges meets the interior of the simplicial cone corresponding to the other edge. Thus we will prove that any two boundary facets meet in a convex manner. If any of the two facets are of the ?rst or second kind, described in Proposition 5, it is clear that any Betti diagram will lie on the correct side of the supporting hyperplane, since this hyperplane is given by the vanishing of a single Betti number. Thus we will assume that the facets are of the third or fourth kind. s Let K denote the number of pure diagrams in a maximal chain in BM,N . Any two boundary facets meeting along a codimension one face gives us a chain with K ? 2 pure diagrams. If the two missing vertices are on levels di?ering by more than one, there is a unique way of completing the chain into a maximal chain and the two facets are faces of the same simplicial cone. Hence they meet in a convex way. Thus we can assume that the two vertices missing are on adjacent levels and that there a least element π3 above these and a greatest element π0 below them. Now we can see from the di?erence in codimension between π0 and π3 that facets of the third kind described in Proposition 5 cannot meet facets of the fourth kind in this way.

BETTI NUMBERS OF GRADED MODULES

9

′ ′′ We must have that the two facets are given by removing π1 or π2 from the chains ′ ′ ′′ ′′ π0 < π1 < π2 < π3 and π0 < π1 < π2 < π3 int the following lattice

eπ3 ?d ′ ′′ deπ2 ? eπ2 ?d d ′ ′′ d? deπ1 eπ1 d ? d? eπ0

′ ′′ We need to show that the coe?cient of π1 is positive when π1 is expanded into the chain ′ ′ π0 < π1 < π2 < π3 . Note that the codimension cannot di?er by more than three between π0 and π3 and if it di?ers by three, there is only one chain between π0 and π3 . Thus we can assume that the di?erence in codimension is zero, one or two. In order to do this we use Proposition 4 which gives us an expression of this coe?cient involving only two terms. Suppose that π0 and π3 di?er in codimension by two. Then the coe?cient is given ′ by the fourth expression of Proposition 4 and the coe?cient for π1 = π (d 0 , d 1 , . . . , d m ) ′′ when π1 = π (d0 , d1 , . . . , dk?1, dk + 1, dk+1, . . . , dm ) is expanded in the basis containing ′ ′ π0 < π1 < π2 is given by n m?1

(?1)

i=0 d≤di (π0 )

i j =0

′′ (dj ? d)βi,d (π1 )

which has only two non-zero terms in columns m and m + 1 and equals dk ? dm 1 (?1) (dj ? dm )(?1) · dk + 1 ? dm dm+1 ? dm j =0

m m m?1 m+1 m+1 m?1 m?1

j =0

1 dj ? dm

m?1

1 dk ? dm+1 1 (dj ? dm+1 )(?1) +(?1) dk + 1 ? dm+1 dm ? dm+1 j =0 dj ? dm+1 j =0 1 dk ? dm dk ? dm+1 1 = ? = > 0, dm+1 ? dm dk + 1 ? dm dk + 1 ? dm+1 (dk + 1 ? dm+1 )(dk + 1 ? dm ) since dk + 1 < dm < dm+1 . ′ ′′ If π0 and π3 di?er in codimension by one we get that the coe?cient of π1 when π1 = ′ ′ π (d0 , d1 , . . . , dm?1 is expanded in the basis π0 < π1 < π2 < π3 is given by

n m

(?1) (dk + 2 ? dk )

i=0 d≤di (π0 ) j =0

j ∈{ / k,k+1}

i

′′ (dj ? d)βi,d (π1 )

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¨ MATS BOIJ AND JONAS SODERBERG

which has only two non-zero terms from columns k and k + 1 and equals

m m?1

2(?1)

k j =0

(dj ? dk )(?1)

j ∈{ / k,k+1}

k j =0

j =k

1 dj ? dk

m?1

m

+2(?1)k+1

j =0

(dj ? dk ? 1)(?1)k+1

j =0

j =k+1 j ∈{ / k,k+1}

1 dj ? dk ? 1

dm ? dk ? 1 dm ? dk ?2 = 2 > 0. dk + 1 ? dk dk ? dk ? 1 The last possibility is the codimension of π0 equals the codimension of π3 and in this ′ ′ case π0 < π1 < π2 corresponds to a facet of the third kind and we get that the coe?cient of ′ ′′ π1 = π (d0 , d1 , . . . , dk , dk +2, dk+2, . . . , dm ), when π1 = π (d0 , d1 , . . . , d??1 , d? +1, d?+1, . . . , dm ) ′ ′ is expanded in the basis π0 < π1 < π2 < π3 is given by =2

n m

(?1) (dk + 2 ? dk )

i=0 d≤di (π0 ) j =0

j ∈{ / k,k+1}

i

′′ (dj ? d)βi,d (π1 ).

Again there are only two non-zero terms and the coe?cient is equal to

m

2(?1)

k j =0

d? ? dk (dj ? dk )(?1) d? + 1 ? dk

k m

1 d ? dk j =0 j

j =k

m

j ∈{ / k,k+1}

+2(?1) =2

k +1 j =0

(dj ? dk ? 1)(?1)

j ∈{ / k,k+1}

k +1

d? ? dk ? 1 d? + 1 ? dk ? 1

m

j =0

j =k+1

1 dj ? dk ? 1

d? ? dk d? ? dk ? 1 2 ?2 = > 0, d? + 1 ? dk d? ? dk (d? ? dk )(d? ? dk + 1)

since d? ? dk and d? ? dk + 1 are both negative or both positive. 4. The expansion of any Betti diagram into sums of pure diagrams We know from the work of D. Eisenbud and F.-O. Schreyer that the inequalities given by the exterior facets of the cone are valid on all minimal free resolutions, not only on the resolutions of Cohen-Macaulay modules. The inequalities that we have to add because we now look at chains of pure diagrams of di?erent codimensions can be seen to be limits of the inequalities already known. As in the previous section, we look at Betti diagrams of graded modules under the restriction that βi,j = 0 unless M + i ≤ j ≤ N + i, i.e., diagrams in BM,N . Theorem 2. Any Betti diagram of a ?nitely generated graded module P of codimension s s can be uniquely written as a positive linear combination of pure diagrams in BM,N , where M is the least degree of any generator of M and N is the regularity of P .

BETTI NUMBERS OF GRADED MODULES

11

Proof. By Proposition 5 we know what are the boundary facets of the simplicial fan given by the pure diagrams in BM,N and by Proposition 4 we know how to obtain the inequalities given by the boundary facets. We need to show that any Betti diagram of a graded module satis?es these inequalities. We have two di?erent kinds of inequalities. The ?rst comes from the boundary facets of the third kind and can be written as

n di (π0 ) m

(?1)

i=0 d=M

i j =0

j ∈{ / k,k+1}

(dj ? d)βi,d (P ) ≥ 0

and the other comes from boundary facets of the fourth kind and can be written as

n di (π0 ) m?1 i j =0

(?1)

i=0 d=M

(dj ? d)βi,d (P ) ≥ 0.

As we can see from these expressions, they are very similar and the work of D. Eisenbud and F.-O. Schreyer shows that the ?rst kind of inequality holds for m = n. We can see that the second kind of inequality is equal to the ?rst kind of inequality if we increase N and m by one and choose k = m ? 1. Thus it is su?cient to prove that the ?rst kind of inequality always holds for any m ≤ n. In order to do this we look at the inequality we get by exchanging π1 = π (d0 , d1 , . . . , dm ) t by π1 = π (d0 , d1 , . . . , dm , dm + 1 + t, dm + 2 + t, . . . , dm + n ? m + t), and similarly exchanging t t π0 and π2 by π0 and π2 . lim 1 tn?m

n n

t) di (π0

m

n

t→∞

(?1)i

i=0 d=M di (π0 ) m j =0

(d j ? d )

j ∈{ / k,k+1}

(dm + j ? m + t ? d)βi,d

j =m+1

=

i=0 d=M

(?1)i

j =0

j ∈{ / k,k+1}

(dj ? d)βi,d (P )

and the limit is non-negative since for each integer t ≥ 0, the expression under the limit in the left hand side is non-negative. Remark 3. One of the questions raised by D. Eisenbud and F.-O. Schreyer was what was the description of the convex cone cut out by all of their inequalities. The answer to this question is that this convex cone in BM,N equals the convex cone of all Betti diagrams in BM,N of graded modules up to multiplication by non-negative rational numbers. This follows from the theorem and the fact that all the inequalities we need when bounding the regularity are given by limits of their inequalities. We can also see that the unique way of writing any Betti diagram of a module into a chain of pure diagrams leads to a way of writing the diagram as a linear combination of diagrams of Cohen-Macaulay modules, one of each codimension between the codimension of M and the projective dimension of M . As in the Cohen-Macaulay case, we get an

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¨ MATS BOIJ AND JONAS SODERBERG

algorithm for ?nding the expansion of a given Betti diagram by subtracting as much as possible of the pure diagram corresponding to the lowest shifts. Example 3. For M = k [x, y, z ]/(x2 , xy, xz 2 ) we get the Betti diagram 1 0: 1 1: ? 2: ? which can be expanded into

1 1 1 ? ? ? ? ? ? ? ? ? ? ? ? 40 12 3 1 1 1 6 · ? 6 6 ? + 12 · ? 12 ? ? + 2 · ? ? ? ? + 1 · ? ? ? ? 1 1 1 1 1 ? 3 ? 1 ? ? ? ? ? ? 30 ? ? 8 15 4 3 1 30

3 ? 2 1

3 ? 1 2

1 ? ? 1

The coe?cients can also be obtained by the functionals corresponding to the matrices 5, 6, 8 and 9 from Example 2. We now will go on and prove a generalized version of the Multiplicity Conjecture in terms of the Hilbert series. In order to do this, we ?rst prove that the Hilbert series behaves well with respect to the partial ordering on the normalized pure diagrams. Proposition 6. The Hilbert series is strictly increasing on the partially ordered set of normalized pure diagrams generated in degree zero. Proof. First assume that π ? <π ? ′ is a maximal chain of pure diagrams of codimension s. The Hilbert series can be recovered from the Betti diagrams and we can write 1 H (? π ′ , t) ? H (? π , t) = (S (? π ′ , t) ? S (? π, t)) n (1 ? t)

i j where S (β, t) = n π , t) and S (? π ′ , t) both have i=0 (?1) j βi,j t . Since the polynomials S (? constant term 1, we get that the polynomial S (? π ′ , t) ? S (? π , t), has only s + 1 non-zero terms. Since it also satis?es the Herzog-K¨ uhl equations, we have a unique solution to this up to a scalar multiple, and H (? π , t) ? H (? π ′ , t) is λtd1 times the Hilbert series of a pure module. Since we know that all pure diagrams have positive Hilbert series, we only have to check that that this is a non-negative multiple. The sign of λ can be obtained by looking at the sign of the term of S (? π ′ , t) ? S (? π , t) which comes from S (? π ′, t) and which is not present in S (? π, t). This term comes with the sign (?1)i , if it is in column i, which proves that λ has to by positive. We now consider the case where π ?<π ? ′ is a maximal chain such that the codimension ′ of π ? is s and the codimension of π ? is s ? 1. If we now look at the di?erence of the Hilbert series, 1 H (? π ′ , t) ? H (? π , t) = (S (? π ′ , t) ? S (? π, t)) (1 ? t)n where the polynomial S (? π ′ , t) ? S (? π , t) has zero constant term and s non-zero terms. Since we know that it is divisible by (1 ? t)s?1 , there is again a unique possibility, which is λtd1

BETTI NUMBERS OF GRADED MODULES

13

times the Hilbert series of a module with pure resolution. This time, we can see that the sign of the last term is ?(?1)s = (?1)s?1 , which shows that λ is positive. Theorem 3. For any ?nitely generated module M of projective dimension r and codimension s generated in degree 0, we have that β0 (M )H (? π (0, m1 , m2 , . . . , mr ), t) ≤ H (M, t) ≤ β0 (M )H (? π (0, M1 , M2 , . . . , Ms ), t), where m1 , m2 , . . . , mr are the minimal shifts and M1 , M2 , . . . , Ms are the maximal shifts in a minimal free resolution of M . Equality on either side implies that the resolution is pure. In particular, the right hand inequality implies the Multiplicity Conjecture, i.e., M1 M2 · · · Ms s! with equality if and only if M is Cohen-Macaulay with a pure resolution. e(M ) ≤ β0 (M ) Proof. We can use Theorem 2 to write the Betti diagram of M as a positive linear combination of pure diagrams. Since the Hilbert series by Proposition 6 is increasing on along the chain, we get the ?rst inequalities of the theorem. The multiplicity of M is obtained from the leading coe?cient of the Hilbert polynomial. The coe?cients of the Hilbert series is eventually equal to the Hilbert polynomial, which shows that the multiplicity is increasing with the Hilbert series as long as the degree of the Hilbert polynomial is the same. Since this is the case for M and π (0, M1 , M2 , . . . , Ms ), we get the conclusion of the Multiplicity conjecture. Remark 4. For a Cohen-Macaulay module M , the Hilbert coe?cients are positive linear combinations of the entries of the h-vector. Since we can reduce M modulo a regular sequence and keep the same Betti diagram we get that the Hilbert coe?cients are increasing along chains of normalized pure diagrams. Thus the generalization of the multiplicity conjecture made J. Herzog and X. Zheng [6] is a consequence of Theorem 3. References

[1] Mats Boij and Jonas S¨ oderberg. Betti numbers of graded modules Cohen-Macaulay modules and the Multiplicity Conjecture. Preprint, arXiv:math.AC/0611081 v1, (To appear in J. London Math. Soc.). [2] David Eisenbud, Gunnar Floystad, and Jerzy Weyman. The existence of pure free resolutions. Preprint, arXiv.org:0709.1529. [3] David Eisenbud and Frank-Olaf Schreyer. Betti numbers of graded modules and cohomology of vector bundles. Preprint, arXiv.org:0712.1843. [4] J. Herzog and M. K¨ uhl. On the Betti numbers of ?nite pure and linear resolutions. Comm. Algebra, 12(13-14):1627–1646, 1984. [5] J¨ urgen Herzog and Hema Srinivasan. Bounds for multiplicities. Trans. Amer. Math. Soc., 350(7):2879– 2902, 1998. [6] J¨ urgen Herzog and Xinxian Zheng. Bounds on the Hilbert coe?cients. Preprint, arXiv:0706.0400v1, 2007. [7] Craig Huneke and Matthew Miller. A note on the multiplicity of Cohen-Macaulay algebras with pure resolutions. Canad. J. Math., 37(6):1149–1162, 1985.

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¨ MATS BOIJ AND JONAS SODERBERG

Department of Mathematics, KTH, S–100 44 Stockholm, Sweden E-mail address : boij@kth.se Department of Mathematics, KTH, S–100 44 Stockholm, Sweden E-mail address : jonasso@math.kth.se

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