Castelnuovo-Mumford regularity of deficiency modules
aa r X i v : . [ m a t h . A C ] A p r CASTELNUOVO-MUMFORD REGULARITY OF DEFICIENCYMODULES
MARKUS BRODMANN, MARYAM JAHANGIRI, AND CAO HUY LINH
Abstract.
Let d ∈ N and let M be a finitely generated graded module of dimension ≤ d over a Noetherian homogeneous ring R with local Artinian base ring R . Let beg( M ),gendeg( M ) and reg( M ) respectively denote the beginning, the generating degree and theCastelnuovo-Mumford regularity of M . If i ∈ N and n ∈ Z , let d iM ( n ) denote the R -length of the n -th graded component of the i -th R + -transform module D iR + ( M ) of M andlet K i ( M ) denote the i -th deficiency module of M .Our main result says, that reg( K i ( M )) is bounded in terms of beg( M ) and the ”diagonalvalues” d jM ( − j ) with j = 0 , · · · , d −
1. As an application of this we get a number of furtherbounding results for reg( K i ( M )). Introduction
This paper is motivated by a basic question of projective algebraic geometry, namely:
What bounds cohomology of a projective scheme?
The basic and initiating contributions to this theme are due to Mumford [21] and Kleiman[18] (see also [13]). The numerical invariant which plays a fundamental rˆole in this context,is the Castelnuovo-Mumford regularity, which was introduced in [21]. Besides is founda-tional significance - in the theory of Hilbert schemes for example- this invariant is the basicmeasure of complexity in computational algebraic geometry(s. [1]). This double meaning of(Castelnuovo-Mumford) regularity made it to one of the most studied invariants of algebraicgeometry. Notably a huge number of upper bounds for the regularity have been established.We mention only a few more recent references to such results, namely [1], [4], [7, 2], [10],[11], [12], [19], [22].It is also known, that Castelnuovo-Mumford regularity is closely related to the bounded-ness - or finiteness- of cohomology at all. More precisely, the regularity of deficiency modulesprovides bounds for the so called cohomological postulation numbers, and thus furnishes atool to attack the finiteness problem for (local) cohomology. This relation is investigated byHoa-Hyry [17] and Hoa [16] in the case of graded ideals in a polynomial ring over a field. In[6] it was shown that for coherent sheaves over projective schemes over a field K , cohomologyis bounded by the ”cohomology diagonal”. One challenge is to extend this later result tothe case where the base field K is replaced by an Artinian ring R and hence to replace the October 25, 2018. bounds given in [8] by ”purely diagonal” ones. In the same spirit one could try to generalizethe results of Hoa and Hoa-Hyry. This is what we shall do in the present paper.Our basic result is a ”diagonal bound” for the Castelnuovo-Mumford regularity of defi-ciency modules.To formulate this result we introduce a few notations. By N we denote the set of all non-negative integers, by N the set of all positive integers. Let R := L n ≥ R n be a Noetherianhomogeneous ring with Artinian base ring R and irrelevant ideal R + := L n> R n . Let M be a finitely generated graded R -module. For each i ∈ N consider the graded R -module D iR + ( M ), where D iR + denotes the i -th R + -transform functor, that is the i -th right derivedfunctor of the R + -transform functor D R + ( • ) := lim −→ n Hom R (( R + ) n , • ). In addition, for each n ∈ Z let d iM ( n ) denote the (finite) R -length of the n -th graded component of D iR + ( M ).Moreover, let beg( M ) and reg( M ) respectively denote the beginning and the Castelnuovo-Mumford regularity of M . If the (Artinian) base ring R is local, let K i ( M ) denote the i -thdeficiency module of M . Fix d ∈ N and i ∈ { , · · · , d } and let dim( M ) ≤ d . Then, theannounced bounding result says (s. Theorem 3.6): The beginning beg( M ) of M and the cohomology diagonal ( d iM ( − i )) d − i =0 of M give anupper bound for the regularity of K i ( M ) . This leads to a further bounding result for reg( K i ( M )). To formulate it, let reg ( M )denote the Castelnuovo-Mumford regularity of M at and above level 2 and let p M denotethe Hilbert polynomial of M . Then (s. Theorem 4.2): The invariant reg( K i ( M )) can be bounded in terms of the three invariants beg( M ) , reg ( M ) and p M (reg ( M )) . As a consequence we get (s. Corollary 4.4): If a ⊆ R is a graded ideal, then reg( K i ( a )) and reg( K i ( R/ a )) can be bounded in termsof reg ( a ) , length( R ) , reg ( R ) and the number of generating one-forms of R . Applying this in the case where R = K [ x , · · · , x d ] is a polynomial ring over a field, we getan upper bound for reg( K i ( R/ a )) which depends only on d and reg ( a ). This is a (slightlyimproved) version of a corresponding result found in [17], which uses reg( a ) instead of reg ( a )as a bounding invariant.As an application of Theorem 4.2 (cf. (1.2)) we prove a few more bounding results inthe situation where R = R [ x , · · · , x d ] is a polynomial ring over a local Artinian ring R ,namely (s. Corrolaries 4.6, 4.8, 4.13): If U = 0 is a finitely generated graded R -module and M ⊆ U is a graded submodule,then reg( K i ( M )) and reg( K i ( U/M )) are bounded in terms of d , length( R ) , beg( U ) , reg( U ) , the number of generators of U and the generating degree gendeg( M ) of M . ASTELNUOVO-MUMFORD REGULARITY OF DEFICIENCY MODULES 3 If F p ։ M is an epimorphism of graded R -modules such that F is free and of finiterank, then reg( K i ( M )) is bounded in terms of d , length( R ) , beg( F ) , gendeg( F ) , rank( F ) and gendeg(ker( p )) .Let U and M be as above. Then reg( K i ( M )) and reg( K i ( U/M )) are bounded interms of length( R ) , beg( U ) , reg( U ) , the Hilbert polynomial p U of U and the Hilbertpolynomial p U/M of U/M . For a fixed i ∈ N we consider the i -th cohomological Hilbert function of the secondkind d iM : Z → N given by n d iM ( n ) and the corresponding i -th cohomological Hilbertpolynomial q iM ∈ Q [ x ] so that q iM ( n ) = d iM ( n ) for all n ≪
0. Based on these concepts wedefine the i -th cohomological postulation number of M by: ν iM := inf { n ∈ Z | q iM ( n ) = d iM ( n ) } ( ∈ Z ∪ {∞} ) . Now, let d ∈ N and let D d be the class of all pairs ( R, M ) in which R = L n ∈ N R n isa Noetherian homogeneous ring with Artinian base ring R and M is a finitely generatedgraded R -module of dimension ≤ d . As a first consequence of Theorem 3.6 we get, that forall pairs ( R, M ) ∈ D d and all i ∈ { · · · , d − } the cohomology diagonal ( d jM ( − j )) d − j =0 of M bounds the i -th cohomological postulation number of M (s. Theorem 5.3): There is a function E id : N d → Z such that for all x , · · · , x d − ∈ N and each pair ( R, M ) ∈ D d such that d jM ( − j ) ≤ x j for all j ∈ { · · · , d − } we have ν iM ≥ E id ( x , · · · , x n ) . This is indeed a generalization of the main result of [6] which gives the same conclusionin the case where the base ring R is a field. Moreover, in our present proof, the boundingfunction E id is defined much simpler than in [6].As an application of Theorem 5.3 we show, that there are only finitely many possiblefunctions d iM if the cohomology diagonal is fixed (s. Theorem 5.4): Let x , · · · , x d − ∈ N . Then, the set of functions { d iM | i ∈ N , ( R, M ) ∈ D d : d jM ( − j ) ≤ x j for j = 0 , · · · , d − } is finite. Preliminaries
In this section we recall a few basic facts which shall be used later in our paper. We alsoprove a bounding result for the Castelnuovo-Mumford regularity of certain graded modules.
MARKUS BRODMANN, MARYAM JAHANGIRI, AND CAO HUY LINH
Notation 2.1.
Throughout, let R = ⊕ n ≥ R n be a homogeneous Noetherian ring, so that R is positively graded, R is Noetherian and R = R [ l , · · · , l r ] with finitely many elements l , · · · , l r ∈ R . Let R + denote the irrelevant ideal ⊕ n> R n of R . Reminder 2.2. (Local cohomology and Castelnuovo-Mumford regularity) (A) Let i ∈ N := { , , , · · · } . By H iR + ( • ) we denote the i -th local cohomology functor with respect to R + .Moreover by D iR + ( • ) we denote the i -th right derived functor of the ideal transform functor D R + ( • ) = lim n →∞ (( R + ) n , • ) with respect to R + .(B) Let M := ⊕ n ∈ Z M n be a graded R -module. Keep in mind that in this situation the R -modules H iR + ( M ) and D iR + ( M ) carry natural gradings. Moreover we then have a naturalexact sequence of graded R -modules(i) 0 −→ H R + ( M ) −→ M −→ D R + ( M ) −→ H R + ( M ) −→ R -modules(ii) D iR + ( M ) ∼ = H i +1 R + ( M ) for all i > . (C) If T is a graded R -module and n ∈ Z , we use T n to denote the n -th graded componentof T . In particular, we define the beginning and the end of T respectively by(i) beg( T ) := inf { n ∈ Z | T n = 0 } ,(ii) end( T ) := sup { n ∈ Z | T n = 0 } .with the standard convention that inf ∅ = ∞ and sup ∅ = −∞ .(D) If the graded R -module M is finitely generated, the R -modules H iR + ( M ) n are all finitelygenerated and vanish as well for all n ≫ i ≥
0. So, we have −∞ ≤ a i ( M ) := end( H iR + ( M )) < ∞ for all i ≥ a i ( M ) := −∞ for all i ≥ . If k ∈ N , the Castelnuovo-Mumford regularity of M at and above level k is defined by(i) reg k ( M ) := sup { a i ( M ) + i | i ≥ k } ( < ∞ ),where as the Castelnuovo-Mumford regularity of M is defined by(ii) reg( M ) := reg ( M ) . (E) If M is a graded R -module we denote the generating degree of M by gendeg( M ), thus(i) gendeg( M ) = inf { n ∈ Z | M = L m ≤ n RM m } .Keep in mind the well known relation (s. [9, 15.3.1]) ASTELNUOVO-MUMFORD REGULARITY OF DEFICIENCY MODULES 5 (ii) gendeg( M ) ≤ reg( M ) . Reminder 2.3. (Cohomological Hilbert functions) (A) Let i ∈ N and assume that the basering R is Artinian. Let M be a finitely generated graded R -module. Then, the graded R -modules H iR + ( M ) are Artinian (s. [9, 7.1.4]). In particular for all i ∈ N and all n ∈ Z wemay define the non-negative integers(i) h iM ( n ) := length R ( H iR + ( M ) n ),(ii) d iM ( n ) := length R ( D iR + ( M ) n ),Fix i ∈ N . Then the functions(iii) h iM : Z → N , n h iM ( n ),(iv) d iM : Z → N , n d iM ( n )are called the i -th Cohomological Hilbert functions of the first respectively the second kind of M .(B) Let i ∈ N and let R and M be as in part (A). Then, there is a polynomial p iM ∈ Q [ x ]of degree < i such that (s.[9, 17.1.9])(i) p iM ( n ) = h iM ( n ) for all n ≪ p iM ) ≤ i − , with equality if i = dim( M ) . We call p iM the i -th Cohomological Hilbert polynomial of the first kind of M.
Now, clearlyby the observation made in part (A) we also have polynomials q iM ∈ Q [ x ] such that(iii) q iM ( n ) = d iM ( n ) for all n ≪ . These are called the
Hilbert polynomials of the second kind of M . Observe that(iv) q iM = p i +1 M for all i ∈ N . Finally, for all i ∈ N we define the i -th cohomological postulation number of M as(v) ν iM := inf { n ∈ Z | q iM ( n ) = d iM ( n ) } ( ∈ Z ∪ {∞} ) . Observe that these numbers ν iM differ by 1 from the cohomological postulation numbersintroduced in [8].(C) Let R and M be as in part (A). By p M ∈ Q [ x ] we denote the Hilbert polynomial of M .By p ( M ) we denote the postulation number sup { n ∈ Z | length R ( M n ) = p M ( n ) } of M .Keep in mind that according to the Serre formula we have (s. [9, 17.1.6]) MARKUS BRODMANN, MARYAM JAHANGIRI, AND CAO HUY LINH p M ( n ) = X i ≥ ( − i d iM ( n ) = length R ( M n ) − X j ≥ ( − j h jM ( n ) . Reminder 2.4. (Filter regular linear forms) (A) Let M be a finitely generated graded R -module and let x ∈ R . By NZD R ( M ) resp. ZD R ( M ) we denote the set of non-zerodivisorsresp. of zero divisors of R with respect to M .The linear form x ∈ R is said to be ( R + -) filter regular with respect to M if x ∈ NZD R ( M/ Γ R + ( M )).(B) Finally if x ∈ R is filter regular with respect to M then the graded short exact sequences0 −→ (0 : M x ) −→ M −→ M/ (0 : M x ) −→ , −→ M/ (0 : M x )( − −→ M −→ M/xM −→ ( M ) ≤ reg( M/xM ) ≤ reg( M ) . • The following result will play a crucial role in the proof of our bounding result for theregularity of deficiency modules.
Proposition 2.5.
Assume that the base ring R is Artinian. Let M be a finitely generatedgraded R -module, let x ∈ R be filter regular with respect to M and let m ∈ Z be such that reg( M/xM ) ≤ m and gendeg((0 : M x )) ≤ m . Then reg( M ) ≤ m + h M ( m ) . Proof.
By Reminder 2.4(B) we have reg ( M ) ≤ reg( M/xM ) ≤ m . So, it remains to showthat a ( M ) = end( H R + ( M )) ≤ m + h M ( m ) . The short exact sequence of graded R -modules0 −→ M/ (0 : M x )( − x −→ M −→ M/xM −→ R -modules0 −→ H R + ( M/ (0 : M x )) n −→ H R + ( M ) n +1 −→ H R + ( M/xM ) n +1 −→ H R + ( M/ (0 : M x )) n for all n ∈ Z . As H R + ( M/xM ) n +1 = 0 for all n ≥ m , we thus get H R + ( M/ (0 : M x )) n ∼ = H R + ( M ) n +1 for all n ≥ m. The short exact sequence of graded R -modules0 −→ (0 : M x ) −→ M −→ M/ (0 : M x ) −→ H R + ((0 : M x )) = (0 : M x ) and H R + ((0 : M x )) = 0 induce shortexact sequences of R -modules0 −→ (0 : M x ) n −→ H R + ( M ) n −→ H R + ( M/ (0 : M x )) n −→ ASTELNUOVO-MUMFORD REGULARITY OF DEFICIENCY MODULES 7 for all n ∈ Z .So, for all n ≥ m we get an exact sequence of R -modules0 −→ (0 : M x ) n −→ H R + ( M ) n π n −→ H R + ( M ) n +1 −→ . To prove our claim, we may assume that a ( M ) > m . As end((0 : M x )) = a ( M ) andgendeg((0 : M x )) ≤ m it follows that (0 : M x ) n = 0 for all integers n with m ≤ n ≤ a ( M ).Hence, for all these n , the homomorphism π n is surjective but not injective, so that h M ( n ) >h M ( n + 1). Therefore, for n ≥ m the function n h M ( n ) is strictly decreasing until itreaches the value 0. Thus h M ( n ) = 0 for all n > m + h M ( m ), and this proves our claim. (cid:3) We now recall a few basic facts about deficiency modules and graded local duality.
Reminder 2.6. (Deficiency modules and local duality) (A) We assume that the base ring R is Artinian and local with maximal ideal m . As R is complete it is a homomorphicimage of a complete regular ring A . Factoring out an appropriate system of parametersof A we thus may write R as a homomorphic image of a local Artinian Gorenestein ring( S , n ). Let d ′ be the minimal number of generators of the R -module R and consider thepolynomial ring S := S [ x , ..., x d ′ ]. Then, we have a surjective homomorphism S f ։ R ofgraded rings.For all i ∈ N and all finitely generated graded R -modules, the i -th deficiency moduleof M is defined as the finitely generated graded R -module (cf [23, Section 3.1] for thecorresponding concept for a local Noetherian ring R which is a homomorphic image of alocal Gornestein ring S .)(i) K i ( M ) := Ext d ′ − iS ( M, S ( − d ′ )) . The module(ii) K ( M ) := K dim(M) ( M )is called the canonical module of M . (B) Keep the previous notations and hypotheses. Let E denote the injective envelope of the R -module R / m . Then, by Graded Matlis Duality and the Graded Local Duality Theorem(s. [9, 13.4.5] for example) we havelength R ( K i ( M ) n ) = h iM ( − n )for all i ∈ N and all n ∈ Z .(C) As an easy consequence of the last observation we now get the following relations for all i ∈ N and all n ∈ Z : MARKUS BRODMANN, MARYAM JAHANGIRI, AND CAO HUY LINH (i) d iM ( n ) = length R ( K i +1 ( M ) − n ) , if i > d M ( n ) ≥ length R ( K ( M ) − n ) with equality if n < beg( M );(ii) p iM ( n ) = p K i ( M ) ( − n );(iii) q iM ( n ) = p K i +1 ( M ) ( − n );(iv) a i ( M ) = − beg( K i ( M ));(v) end( K i ( M )) = − beg( H iR + ( M ));(vi) ν iM = − p ( K i +1 ( M )) . • Regularity of Deficiency Modules
We keep the notations introduced in Section 2. Throughout this section we assume inaddition that the Noetherian homogenous ring R = L n ≥ R n has Artinian local base ring( R , m ).The aim of the present section is to show that the Castelnuovo-Mumford regularity of thedeficiency modules K i ( M ) of the finitely generated graded R -module M is bounded in termsof the beginning beg( M ) of M and the ”cohomology diagonal” ( d iM ( − i )) dim( M ) − i =0 of M .We first prove three auxiliary results. Lemma 3.1. depth( K dim( M ) ( M )) ≥ min { , dim( M ) } . Proof.
In the notation introduced in Reminder (2.6) we have K ( M ) m ∼ = K ( M m ). As M and K ( M ) are finitely generated graded R -modules we have dim( M ) = dim( M m ) anddepth( K ( M )) = depth( K ( M ) m ). Now, we conclude by [23, Lemma 3.1.1(C)]. (cid:3) Lemma 3.2.
Let x ∈ R be filter regular with respect to M and the modules K j ( M ) . Then,there are short exact sequences of graded R -modules −→ ( K i +1 ( M ) /xK i +1 ( M ))(+1) −→ K i ( M/xM ) −→ (0 : K i ( M ) x ) −→ Proof.
In the local case, this result is shown in [24, Proposition 2.4]. In our graded situation,one may conclude in the same way. (cid:3)
Lemma 3.3.
Let i ∈ N and n ≥ i . Then length R ( K i +1 ( M ) n ) ≤ i X j =0 (cid:18) n − j − i − j (cid:19)(cid:20) i − j X l =0 (cid:18) i − jl (cid:19) d i − lM ( l − i ) (cid:21) . ASTELNUOVO-MUMFORD REGULARITY OF DEFICIENCY MODULES 9
Proof.
According to [8, Lemma 4.4] we have a corresponding inequality with d iM ( − n ) on thelefthand side. Now, we conclude by Reminder 2.6(C)(i). (cid:3) Next we recursively define a class of bounding functions.
Definition 3.4.
For d ∈ N and i ∈ { , · · · , d } we define the functions F id : N d × Z −→ Z as follows: In the case i = 0 we simply set(i) F d ( x , · · · , x d − , y ) := − y .Concerning the case i = 1 we set(ii) F ( x , y ) := 1 − y and(iii) F d ( x , · · · , x d − , y ) := max { , − y } + P d − i =0 (cid:0) d − i (cid:1) x d − i − , for d ≥ i = d = 2 we define(iv) F ( x , x , y ) := F ( x , x , y ) + 2.If d ≥ ≤ i ≤ d − F i − d − , F id − and F i − d are alreadydefined, we first set(v) m i := max { F i − d − ( x + x , · · · , x d − + x d − , y ) , F i − d ( x , · · · , x d − , y ) + 1 } + 1,(vi) n i := F id − ( x + x , · · · , x d − + x d − , y ),(vii) t i := max { m i , n i } .Then, using this notation we define(viii) F id ( x , · · · , x d − , y ) := t i + P i − j =0 (cid:0) t i − j − i − j − (cid:1) ∆ ij ,where ∆ ij = P i − j − l =0 (cid:0) i − j − l (cid:1) x i − l − .Finally, assuming that d ≥ F d − d − and F d − d are already defined, we set(ix) F dd ( x , · · · , x d − , y ) :=max { F d − d − ( x + x , · · · , x d − + x d − , y ) , F d − d ( x , · · · , x d − , y ) + 1 } + 1. • Remark 3.5. (A) Let d ∈ N and i ∈ { , · · · , d } . Let ( x , · · · , x d − , y ) , ( x ′ , · · · , x ′ d − , y ′ ) ∈ N d × Z such that x i ≤ x ′ i for all i ∈ { , · · · , d − } and y ′ ≤ y. Then it follows easily by induction on i and d that F id ( x , · · · , x d − , y ) ≤ F id ( x ′ , · · · , x ′ d − , y ′ ) . (B) It also follows by induction on i , that the auxiliary numbers m i and t i of Definition 3.4all satisfy the inequality min { m i , t i } ≥ i .(C) Let s, d ∈ N with s ≤ d and let i ∈ N with i ≤ s . Moreover, let ( x , · · · , x s − , y ) ∈ N s × Z . We then easily obtain by induction on i , that F is ( x , · · · , x s − , y ) ≤ F id ( x , · · · , x s − , , · · · , , y ) . • Now we are ready to state the main result of the present section.
Theorem 3.6.
Let d ∈ N , i ∈ { , · · · , d } and let M be a finitely generated graded R -modulesuch that dim( M ) = d . Then reg( K i ( M )) ≤ F id ( d M (0) , d M ( − , · · · , d d − M (1 − d ) , beg( M )) . Proof.
We shall proceed by induction on i and d . As dim( K ( M )) ≤ K ( M )) = end( K ( M )) = − beg( H R + ( M )) ≤ − beg( M )= F d ( d M (0) , · · · , d d − M (1 − d ) , beg( M )) . This proves the case where i = 0.So, let i >
0. We may assume that R / m is infinite. In addition, we may replace M by M/H R + ( M ) and hence assume that depth( M ) > x ∈ R be a filter regular element with respect to M and all the modules K j ( M ).Observe that x ∈ NZD( M ). By Lemma 3.2 we have the exact sequences of graded R -modules0 −→ ( K j +1 ( M ) /xK j +1 ( M ))(+1) −→ K j ( M/xM ) −→ (0 : K j ( M ) x ) −→ j ∈ N .Since depth( M ) > K ( M ) = 0. So, the sequence (1) yields an isomorphism ofgraded R -modules ( K ( M ) /xK ( M ))(+1) ∼ = K ( M/xM ) . (2)As dim( K ( M/xM )) ≤ K ( M ) /xK ( M )) = reg( K ( M/xM )) + 1 = end( K ( M/xM )) + 1= 1 − beg( H R + ( M/xM )) ≤ − beg( M/xM ) ≤ − beg( M ) . Therefore, reg( K ( M ) /xK ( M )) ≤ − beg( M ) . (3) ASTELNUOVO-MUMFORD REGULARITY OF DEFICIENCY MODULES 11
Assume first that d = dim( M ) = 1. Then, by Lemma 3.1 we have depth( K ( M )) ≥ min { , dim( M ) } = 1, whence reg( K ( M )) = reg ( K ( M )). It follows that (cf. Reminder2.4(B)) reg( K ( M )) ≤ reg( K ( M ) /xK ( M )) ≤ − beg( M ) = F ( d M (0) , beg( M )) . This proves our result if d = 1.So, from now on we assume that d ≥
2. We first focus to the case i = 1 and consider theexact sequence (1) for j = 1, hence0 −→ ( K ( M ) /xK ( M ))(+1) −→ K ( M/xM ) −→ (0 : K ( M ) x ) −→ . (4)If d = dim( M ) = 2, we have dim( M/xM ) = 1, and so by the case d = 1 we getreg( K ( M/xM )) ≤ − beg( M/xM ) ≤ − beg( M ) . From (4) and Reminder 2.2(E)(ii) it follows thatgendeg((0 : K ( M ) x )) ≤ reg( K ( M/xM )) ≤ − beg( M ) . Set m := 1 − beg( M ). If m ≤
0, by the inequality (3), Proposition 2.5 (applied with m = 0) and Reminder 2.6(C)(i) we obtainreg( K ( M )) ≤ h K ( M ) (0) ≤ length( K ( M ) )= d M (0) . If m > d M ( − m ) ≤ d M (0). So, by (3), Proposition 2.5 and Reminder 2.6(C)(i)we get reg( K ( M )) ≤ m + h K ( M ) ( m ) ≤ m + length( K ( M ) m )= 1 − beg( M ) + d M ( − m ) ≤ − beg( M ) + d M (0) . So, (cf. Definition 3.4(iii))reg( K ( M )) ≤ max { d M (0) , − beg( M ) + d M (0) }≤ max { , − beg( M ) } + d M (0)= F ( d M (0) , d M ( − , beg( M )) . This proves the case d = 2 , i = 1. If d ≥
3, by induction on d , we have (cf. Definition 3.4(iii))reg( K ( M/xM )) ≤ F d − ( d M/xM (0) , · · · , d d − M/xM (2 − d ) , beg( M/xM ))= max { , − beg( M/xM ) } + d − X i =0 (cid:18) d − i (cid:19) d d − i − M/xM ( i + 3 − d ) ≤ max { , − beg( M ) } + d − X i =0 (cid:18) d − i (cid:19) [ d d − i − M ( i + 3 − d ) + d d − i − M ( i + 2 − d )] . Set t := max { , − beg( M ) } + d − X i =0 (cid:18) d − i (cid:19) [ d d − i − M ( i + 3 − d ) + d d − i − M ( i + 2 − d )] . By the exact sequence (4) and Reminder 2.2(E)(ii) we now getgendeg((0 : K ( M ) x )) ≤ reg( K ( M/xM )) ≤ t . By (3) we also have reg( K ( M ) /xK ( M )) ≤ t . As t ≥
0, we have d M ( − t ) ≤ d M (0). So,by Proposition 2.5 and Reminder 2.6(C)(i) we obtainreg( K ( M )) ≤ t + h K ( M ) ( t ) ≤ t + length( K ( M ) t ) ≤ t + d M ( − t ) ≤ t + d M (0)= max { , − beg( M ) } + d − X i =0 (cid:18) d − i (cid:19) [ d d − i − M ( i + 3 − d ) + d d − i − M ( i + 2 − d )] + d M (0) ≤ max { , − beg( M ) } + d − X i =0 (cid:18) d − i (cid:19) d d − i − M ( i + 2 − d ) . From this we conclude that (cf. Definition 3.4(iii))reg( K ( M )) ≤ F d ( d M (0) , · · · , d d − M (1 − d ) , beg( M )) . So, we have done the case i = 1 for all d ∈ N .We thus attack now the case with i ≥
2. First, let d = 2. Then, in view of the sequence(4), by the fact that x is filter regular with respect to K ( M ) and by what we have alreadyshown in the cases d ∈ { , } and i = 1, we getreg( K ( M ) /xK ( M )) ≤ max { reg( K ( M/xM )) , reg((0 : K ( M ) x )) + 1 } + 1 ≤ max { reg( K ( M/xM )) , reg( K ( M )) + 1 } + 1 ≤ max { − beg( M ) , max { , − beg( M ) } + d M (0) + 1 } + 1 ≤ max { , − beg( M ) } + d M (0) + 2 . ASTELNUOVO-MUMFORD REGULARITY OF DEFICIENCY MODULES 13
As depth( K ( M )) ≥ min { , dim( M ) } (s. Lemma 3.1) we have depth( K ( M )) = 2, thusreg( K ( M )) = reg ( K ( M )). Hence (cf. Reminder 2.4(B) and Definition 3.4(iii),(iv))reg( K ( M )) ≤ reg( K ( M ) /xK ( M )) ≤ max { , − beg( M ) } + d M (0) + 2= F ( d M (0) , d M ( − , beg( M )) . This completes the case d = 2. So, let d > d and in view of Remark 3.5(A) we havereg( K k ( M/xM )) ≤ F kd − ( d M/xM (0) , d M/xM ( − , · · · , d d − M/xM (2 − d ) , beg( M/xM )) ≤ F kd − ( d M (0) + d M ( − , · · · , d d − M (2 − d ) + d d − M (1 − d ) , beg( M )) , for 0 ≤ k ≤ d − K k ( M/xM )) ≤ F kd − ( d M (0) + d M ( − , · · · , d d − M (2 − d ) + d d − M (1 − d ) , beg( M ))for all k ∈ { , · · · , d − } . (5)We first assume that 2 ≤ i ≤ d −
1. Then, by induction on i we havereg( K i − ( M )) ≤ F i − d ( d M (0) , d M ( − , · · · , d d − M (1 − d ) , beg( M )) . (6)If we apply the exact sequence (1) with j = i − x is filter regularwith respect to K i − ( M ) we thus get by (5) and (6):reg( K i ( M ) /xK i ( M )) ≤ max { reg( K i − ( M/xM )) , reg((0 : K i − ( M ) x )) + 1 } + 1 ≤ max { reg( K i − ( M/xM )) , reg( K i − ( M )) + 1 } + 1 ≤ max { F i − d − ( d M (0) + d M ( − , · · · , d d − M (2 − d ) + d d − M (1 − d ) , beg( M )) ,F i − d ( d M (0) , d M ( − , · · · , d d − M (1 − d ) , beg( M )) + 1 } + 1 . If we apply the sequence (1) with j = i , we obtaingendeg((0 : K i ( M ) x )) ≤ reg( K i ( M/xM )) . According to (5) we have the inequalityreg( K i ( M/xM )) ≤ F id − ( d M (0) + d M ( − , · · · , d d − M (2 − d ) + d d − M (1 − d ) , beg( M )) . Set m i : = max { F i − d − ( d M (0) + d M ( − , · · · , d d − M (2 − d ) + d d − M (1 − d ) , beg( M )) ,F i − d ( d M (0) , d M ( − , · · · , d d − M (1 − d ) , beg( M )) + 1 } + 1 ,n i : = F id − ( d M (0) + d M ( − , · · · , d d − M (2 − d ) + d d − M (1 − d ) , beg( M )) , and t i : = max { m i , n i } . Note that by Remark 3.5 (B) we have t i ≥ i . Hence, by Proposition 2.5 and Lemma 3.3reg( K i ( M )) ≤ t i + h K i ( M ) ( t i ) ≤ t i + length( K i ( M ) t i ) ≤ t i + i − X j =0 (cid:18) t i − j − i − j − (cid:19)(cid:20) i − j − X l =0 (cid:18) i − j − l (cid:19) d i − l − M ( l − i + 1) (cid:21) . Thus, we obtain (cf. Definition 3.4(viii))reg( K i ( M )) ≤ F id ( d M (0) , d M ( − , · · · , d d − M (1 − d ) , beg( M )) . This completes the case where i ≤ d −
1. It thus remains to treat the cases with i = d > K d ( M )) ≥
2. So, again by Reminder 2.4(B) and byuse of the sequence (1) we getreg( K d ( M )) ≤ reg( K d ( M ) /xK d ( M )) ≤ max { reg( K d − ( M/xM )) , reg((0 : K d − ( M ) x )) + 1 } + 1 ≤ max { reg( K d − ( M/xM )) , reg( K d − ( M )) + 1 } + 1 . By induction and Remark 3.5(A) it holdsreg( K d − ( M/xM )) ≤ F d − d − ( d M/xM (0) , d M/xM ( − , · · · , d d − M/xM (2 − d ) , beg( M/xM )) ≤ F d − d − ( d M (0) + d M ( − , d M ( −
1) + d M ( − , · · · , d d − M (2 − d ) + d d − M (1 − d ) , beg( M )) . By the case i = d − K d − ( M )) ≤ F d − d ( d M (0) , d M ( − , · · · , d d − M (1 − d ) , beg( M )) . This implies that (cf. Definition 3.4(ix))reg( K d ( M )) ≤ max { F d − d − ( d M (0) + d M ( − , · · · , d d − M (2 − d ) + d d − M (1 − d ) , beg( M )) ,F d − d ( d M (0) , · · · , d d − M (1 − d ) , beg( M )) + 1 } + 1= F dd ( d M (0) , · · · , d d − M (1 − d ) , beg( M )) . So, finally we may conclude thatreg( K i ( M )) ≤ F id ( d M (0) , d M ( − , · · · , d d − M (1 − d ) , beg( M )) , for all d ∈ N and all i ∈ { , · · · , d } . (cid:3) Corollary 3.7.
Let d ∈ N , i ∈ { , · · · , d } , ( x , · · · , x d − , y ) ∈ N d × Z and let M be a finitelygenerated graded R -module such that dim( M ) ≤ d , d jM ( − j ) ≤ x j for all j ∈ { , · · · , d − } and beg( M ) ≥ y . Then reg( K i ( M )) ≤ F id ( x , · · · , x d − , y ) . ASTELNUOVO-MUMFORD REGULARITY OF DEFICIENCY MODULES 15
Proof. If M = 0, we have K i ( M ) = 0 and so our claim is obvious.If dim( M ) = 0, we have K i ( M ) = 0 for all i > K ( M )) ≤ K ( M )) = end( K ( M )) = − beg( H R + ( M )) = − beg( M ) ≤ − y = F ( x , · · · , x d − , y ) . So, it remains to show our claim if dim( M ) >
0. But now, we may conclude by Theorem3.6 and Remark 3.5(A), (C). (cid:3) Bounding reg( K i ( M )) in terms of reg ( M )We keep the notations introduced in section 3. In particular we always assume that thehomogeneous Noetherian ring R = L n ≥ R n has Artinian local base ring ( R , m ). Wehave seen in the previous section, that the Castelnuovo-Mumford regularity of the deficiencymodules K i ( M ) of a finitely generated graded R -module M is bounded in terms of theinvariants d jM ( − j ) ( j = 0 , · · · , dim( M ) −
1) and beg( M ). We shall use this result in order tobound the numbers reg( K i ( M )) in terms of the Castelnuovo-Mumford regularity of M . Thisidea is inspired by Hoa-Hyry [17] who gave similar results for graded ideals in a polynomialring over a field.As an application we shall derive a number of further bounds on the invariants reg( K i ( M )). Definition 4.1.
Let d ∈ N and i ∈ { , · · · , d } . We define a bounding function G id : N × Z → Z by G id ( u, v, w ) := F id ( u, , · · · , , v − w ) − w. Now, we are ready to give a first result of the announced type. It says that the numbersreg( K i ( M )) find upper bounds in terms of reg ( M ) and the Hilbert polynomial of M . • Theorem 4.2.
Let p ∈ N , d ∈ N , i ∈ { , · · · , d } , b, r ∈ Z and let M be a finitely generatedgraded R -module with dim( M ) ≤ d , beg( M ) ≥ b , reg ( M ) ≤ r and p M ( r ) ≤ p . Then reg( K i ( M )) ≤ G id ( p, b, r ) . Proof.
Observe that beg( M ( r )) ≥ b − r . On use of Corollary 3.7 we now getreg( K i ( M )) + r = reg( K i ( M )( − r )) = reg( K i ( M ( r ))) ≤ F id ( d M ( r ) (0) , d M ( r ) ( − , · · · , d d − M ( r ) (1 − d ) , b − r )= F id ( d M ( r ) , d M ( r − , · · · , d d − M ( r + 1 − d ) , b − r ) . For all j ∈ N we have d jM ( r − j ) = h j +1 M ( r − j ) = 0, so that d M ( r −
1) = · · · = d d − M ( r + 1 − d ) = 0 . In addition d jM ( r ) = h j +1 M ( r ) = 0 for all j ∈ N , which implies that d M ( r ) = p M ( r ) ≤ p (s.Reminder 2.3(C)). In view of Remark 3.5(A) the above inequality now inducesreg( K i ( M )) + r ≤ F id ( p, , · · · , , b − r )and this proves our claim. (cid:3) Bearing in mind possible application to Hilbert schemes for example one could ask forbounds which apply uniformly to all graded submodules M of a given finitely generatedgraded R -module U and depend only on basic invariants of M .Our next result gives such a bound which depends only on reg ( M ) and the Hilbertpolynomial p U of the ambient module U . Corollary 4.3.
Let p, d, i, b and r be as in Theorem 4.2. Let U be a finitely generated andgraded R -module such that dim( U ) ≤ d , beg( U ) ≥ b , reg ( U ) ≤ r and p U ( r ) ≤ p . Then, foreach graded submodule M ⊆ U such that reg ( M ) ≤ r we have max { reg( K i ( M )) , reg( K i ( U/M )) } ≤ G id ( p, b, r ) . Proof.
Let M be as above, so that reg ( M ) ≤ r . Then, the short exact sequence0 −→ M −→ U −→ U/M −→ ( U/M ) ≤ r . Now, as previously we get on use of Reminder 2.3(C) d M ( r ) = p M ( r ) , d U ( r ) = p U ( r ) , d U/M ( r ) = p U/M ( r ) . (2)As D R + ( M ) r ∼ = H R + ( M ) r = 0 the sequence (1) implies d M ( r ) + d U/M ( r ) = d U ( r ) . In view of the equalities (2) we thus get p M ( r ) , p U/M ( r ) ≤ p. As dim( M ), dim( U/M ) ≤ d and beg( M ), beg( U/M ) ≥ b we now get the requested inequal-ities by Theorem 4.2. (cid:3) Corollary 4.3 immediately implies a bounding result which is of the type given by Hoa-Hyry [17].
Corollary 4.4.
Let d, m, r ∈ N , i ∈ { , · · · , d } and assume that dim( R ) ≤ d , reg ( R ) ≤ r and dim R / m ( R / m R ) ≤ m . Let γ := G id ( (cid:18) m + r − r − (cid:19) length( R ) , , r ) . Then, for each graded ideal a ⊆ R with reg ( a ) ≤ r we have max { reg( K i ( a )) , reg( K i ( R/ a )) } ≤ γ. ASTELNUOVO-MUMFORD REGULARITY OF DEFICIENCY MODULES 17
Proof.
Let x , · · · , x m be indeterminates. Then, there is a surjective homomorphism ofgraded R -algebras R [ x , · · · , x m ] ։ R , so that p R ( r ) ≤ (cid:0) m + r − r − (cid:1) length( R ).As beg( R ) = 0 we now conclude by corollary 4.3. (cid:3) Remark 4.5. If d ≥ R = K [ x , · · · , x d ] is a standard graded polynomial ring over afield K and a ⊆ R is a graded ideal with reg ( a ) ≤ r , the previous result shows thatreg( K i ( R/ a )) ≤ G id ( (cid:18) d + r − r − (cid:19) , , r ) . This inequality bounds reg( K i ( R/ a )) in terms of reg ( a ). So, our result in a certain wayimproves [17, Theorem 14], which bounds reg( K i ( R/ a )) only in terms of reg( a ) = reg ( a ).On the other hand we do not insist that our bound is sharper from the numerical point ofview. • Recently, ”almost sharp” bounds on the Castelnuovo-Mumford regularity in terms of thegenerating degree have been given by Caviglia- Sbarra [11], Chardin-Fall-Nagel [12] and[4]. Combining these with the previous results of the present section, we get another typeof bounding results for the Castelnuovo-Mumford regularity of deficiency modules. Here,we restrict ourselves to give two such bounds which hold over polynomial rings, as thecorresponding statements get comparatively simple in this case.
Corollary 4.6.
Let d, m ∈ N , let i ∈ { , · · · , d } , let b, r ∈ Z , let R = R [ x , · · · , x d ] be astandard graded polynomial ring and let U = 0 be a graded R -module which is generated by m homogeneous elements and satisfies beg( U ) = b and reg( U ) < r .Set ̺ := [ r + ( m + 1) length( R ) − b ] d − ,π := m (cid:18) d + ̺ − ̺ − (cid:19) length( R ) and δ := G id ( π, b, ̺ + b ) . Then, for each graded submodule M ⊆ U with gendeg( M ) ≤ r we have max { reg( K i ( M )) , reg( K i ( U/M )) } < δ. Proof.
Let U = P mi =1 Ru i with u i ∈ U n i and b = n ≤ n ≤ · · · ≤ n m = gendeg( U ) ≤ reg( U ) < r .As r − b > r < ̺ + b , whence reg( U ) < ̺ + b . Therefore by Reminder 2.3(C)we obtain p U ( ̺ + b ) = length( U ̺ + b ). As there is an epimorphism of graded R -modules m M i =1 R ( − n i ) ։ U we thus obtain p U ( ̺ + b ) ≤ m X i =1 (cid:18) d + ̺ + b − n i − ̺ + b − n i − (cid:19) length( R ) ≤ m (cid:18) d + ̺ − ̺ − (cid:19) length( R ) = π. Finally, by [4, Proposition 6.1] we have reg( M ) ≤ ̺ + b for each graded submodule M ⊆ U with gendeg( M ) ≤ r . Now we conclude by corollary 4.3. (cid:3) Remark 4.7.
Let d, i > R be as in Corollary 4.6 and let a ( R be a graded ideal ofpositive height. Let r := [gendeg( a )(1 + length( R ))] d − ,γ := G id ( (cid:18) d + r − r − (cid:19) length( R ) , , r ) . Then, combining [4, Corollary (5.7)(b)] with Corollary 4.4 we getmax { reg( K i ( a )) , reg( K i ( R/ a )) } < γ. For more involved but sharper bounds of the same type one should combine the boundsgiven in [12] with Corollary 4.3. • Our next bound is in the spirit of the classical ”problem of finitely many steps” (cf. [15],[14]): it bounds reg( K i ( M )) in terms of the discrete data of a minimal free presentation of M . Again we content ourselves to give a bounding result which is comparatively simple andconcerns only the case where R is a polynomial ring. Corollary 4.8.
Let d, m ∈ N , let i ∈ { , · · · , d } , let R = R [ x , · · · , x d ] be a standard gradedpolynomial ring, let p : F ։ N be an epimorphism of finitely generated graded R -modulessuch that F is free of rank m > .Set b := beg( F ) and r := max { gendeg( F )+1 , gendeg(ker( p )) } and define δ as in Corollary4.6. Then reg( K i ( N )) < δ. Proof.
Apply Corollary 4.6 with U = F and with ker( p ) instead of M . (cid:3) Our last application is a bound in the spirit of Mumford’s classical result [21] which usesthe Hilbert coefficients as key bounding invariants. To formulate our result we first introducea few notations.
Reminder 4.9. (Hilbert coefficients) (A) Let d ∈ N and let e := ( e , · · · , e d − ) ∈ Z d \{ } .We introduce the polynomial(i) p e ( x ) := P d − i =0 ( − i e i (cid:0) x + d − i − d − i − (cid:1) ∈ Q [ x ]which satisfies ASTELNUOVO-MUMFORD REGULARITY OF DEFICIENCY MODULES 19 (ii) deg( p e ) = d − − min { i | e i = 0 } . (B) If M is a finitely generated graded R -module of dimension d , we define the Hilbertcoefficients e i ( M ) of M for i = 0 , · · · , d − p M ( x ) = p ( e ( M ) , ··· ,e d − ( M )) ( x ) . In particular e ( M ) ∈ N is the Hilbert-Serre multiplicity of M . In addition we set:(ii) e i ( M ) := 0 for all i ∈ Z \{ , · · · , d − } . • Notation 4.10.
Let m, d ∈ N with d >
1. We define a numerical function H md : Z d → Z ,recursively on d , as follows (cf. Reminder 4.9(A)(i))(i) H m ( e , e ) := 1 − p ( e ,e ) ( − . If d > H md − has already been defined, let e := ( e , · · · , e d − ) ∈ Z d , set(ii) e ′ := ( e , · · · , e d − ) , f := H md − ( e ′ ) , and define (cf. Reminder 4.9(A)(i))(iii) H md ( e ) := length( R ) m (cid:0) f + d − d − (cid:1) − p e ( f −
2) + f, with the convention that (cid:0) td − (cid:1) =: 0 for all t < d − • Remark 4.11.
Let m, d ∈ N be with d > , · · · , H md = F ( d )0 . • The next result is of preliminary nature and extends [9, 17.2.7] which at its turn generalizesMumford bounding result (s. [21, pg.101]).
Proposition 4.12.
Let d, m ∈ N with d > , let r ∈ Z and let U be a finitely generatedgraded R -module with dim( U ) = d , reg( U ) ≤ r and dim R / m ( U r / m U r ) ≤ m . Let M ⊆ U be a graded submodule. Then, setting L := U/M , h := d − dim( L ) and t := H md ( m length( R ) − ( − h e − h ( L ( r )) , ( − h e − h ( L ( r )) , · · · , ( − h e d − − h ( L ( r )) , we have (a) reg ( L ) ≤ max { , t − } + r ; (b) reg ( M ) ≤ max { , t } + r . Proof. If M is R + -torsion, we have reg ( L ) = reg ( U ) ≤ r and reg ( M ) = −∞ so that ourclaim is obvious. Therefore we may assume that M is not R + -torsion.We may assume that R / m is infinite. We may in addition replace R by R/ (0 : R U ) andhence assume that dim( R ) = d . We now find elements a , · · · , a d ∈ R which form a system ofparameters for R . In particular R is a finite integral extension of R [ a , · · · , a d ]. Consider thepolynomial ring R [ x , · · · , x d ] and the homomorphism of R -algebras f : R [ x , · · · , x d ] → R given by x i a i for i = 1 , · · · , d . Then, M is a finitely generated graded module over R [ x , · · · , x d ] and √ R + = p ( x , · · · , x d ) R .So, the numerical invariants of U and M which occur in our statement do not changeif we consider U and M as R [ x , · · · , x d ]-modules by means of f . Therefore, we mayassume that R = R [ x , · · · , x d ]. Now, we have gendeg( U ( r )) ≤ reg( U ( r )) ≤ R / m ( U ( r ) / m U ( r ) ) ≤ m . This implies that the R -module U ( r ) ≥ is generated by(at most) m homogeneous elements of degree 0. Therefore we have an epimorphism ofgraded R -modules R L m g ։ U ( r ) ≥ . Let N := g − ( M ( r ) ≥ ). As M is not R + -torsion we have M ( r ) ≥ = 0 and hence N = 0.As N ⊆ R L m and by our choice of R we thus have dim( N ) = d . Now, the isomorphism ofgraded R -modules R L m /N ∼ = ( L ( r )) ≥ implies m length( R ) (cid:0) x + d − d − (cid:1) − P d − i =0 ( − i e i ( N ) (cid:0) x + d − i − d − i − (cid:1) = p R L m ( x ) − p N ( x ) = p R L m /N ( x ) = p ( L ( r )) ≥ ( x ) = p L ( r ) ( x )= d − h − X j =0 ( − j e j ( L ( r )) (cid:18) x + d − h − j − d − h − j − (cid:19) = d − X i =0 ( − i − h e i − h ( L ( r )) (cid:18) x + d − i − d − i − (cid:19) . Therefore e ( N ) = m length( R ) − ( − h e − h ( L ( r ))and e i ( N ) = ( − h e i − h ( L ( r )) for all i ∈ { , · · · , d − } . So, according to [9, 17.2.7] and Remark 4.11 we obtainreg ( N ) ≤ F ( d )0 ( e ( N ) , · · · , e d − ( N ))= H md ( m length( R ) − ( − h e − h ( L ( r )) , ( − h e − h ( L ( r )) , · · · , ( − h e d − − h ( L ( r ))) =: t. Now, the short exact sequence of graded R -modules0 −→ N −→ R L m −→ ( U ( r ) /M ( r )) ≥ −→ (( U ( r ) /M ( r )) ≥ ) ≤ max { , t − } , whence reg ( U ( r ) /M ( r )) ≤ max { , t − } , sothat finally reg ( L ) = reg ( U ( r ) /M ( r )) + r ≤ max { , t − } + r ASTELNUOVO-MUMFORD REGULARITY OF DEFICIENCY MODULES 21 and reg ( M ) = reg ( M ( r )) + r ≤ max { reg ( U ( r )) , reg ( U ( r ) /M ( r )) + 1 } + r ≤ max { , max { , t − } + 1 } + r ≤ max { , t } + r. This proves our claim. (cid:3)
Now, we may bound the Castelnuovo-Mumford regularity of deficiency modules as follows:
Corollary 4.13.
Let the notations and hypothesis be as in Proposition 4.12. In addition let b ∈ Z and p ∈ N such that beg( U ) ≥ b and p U ( r ) ≤ p .Then, for all i ∈ { , · · · , d } we have max { reg( K i ( M )) , reg( K i ( U/M )) } ≤ G id ( p, b, max { , t } + r ) . Proof.
This is clear by Corollary 4.3 and Proposition 4.12(b). (cid:3)
Applying this to the ”classical” situation of [21] where M = a is a graded ideal of apolynomial ring we finally can say Corollary 4.14.
Let R = R [ x , · · · , x d ] be a standard graded polynomial ring with d > and let a ⊆ R be a graded ideal. Set h := height( a ) and t := H d (length( R ) − ( − h e − h ( R/ a ) , ( − h e − h ( R/ a ) , · · · , ( − h e d − − h ( R/ a )) . Then, for all i ∈ { , · · · , d } we have max { reg( K i ( a )) , reg( K i ( R/ a )) } ≤ G id (1 , , max { , t } ) . Proof.
Choose U := R, M := a , m = 1 , r = 0 , b = 0 , p = 1. Observe also that d − dim( R/ a ) = h and apply Corollary 4.13. (cid:3) Bounding Cohomological Postulation Numbers
In [6, Theorem 4.6] it is shown that the cohomological postulation numbers of a projectivescheme X over a field K with respect to a coherent sheaf of O X -modules F are boundedby the cohomology diagonal ( h i ( X, F ( − i ))) dim( F ) i =0 of F . On use of Theorem 3.6 this ”purelydiagonal bound” now can be generalized to the case where the base field K is replaced byan arbitrary Artinian ring. To do so, we first introduce some appropriate notions. Definition 5.1.
For d ∈ N and i ∈ { , · · · , d − } we define the bounding function E id : N d → Z by E id ( x , · · · , x d − ) := − F i +1 d ( x , · · · , x d − , , where F i +1 d is defined according to Definition 3.4. • Definition 5.2.
Let d ∈ N . By D d we denote the class of all pairs ( R, M ) in which R = ⊕ n ∈ N R n is a Noetherian homogenous ring with Artinian base ring R and M = ⊕ n ∈ Z M n isa finitely generated graded R -module with dim( M ) ≤ d . • Now, we are ready to state the announced ”purely diagonal” bounding result as follows:
Theorem 5.3.
Let d ∈ N , let x , · · · , x d − ∈ N and let ( R, M ) ∈ D d such that d jM ( − j ) ≤ x j for all j ∈ { , · · · , d − } . Then for all i ∈ { , · · · , d − } we have ν iM ≥ E id ( x , · · · , x d − ) . Proof.
On use of standard reduction arguments and the monotonicity statement of Remark3.5(A) we can restrict ourselves to the case where the Artinian base ring R is local. Considerthe graded submodule N := M ≥ = ⊕ n ≥ M n of M . As the module M/N is R + -torsion, thegraded short exact sequence 0 −→ N −→ M −→ M/N −→ R -modules D jR + ( M ) ∼ = D jR + ( N ) and hence equalities d jM = d jN for all j ∈ N . These allowto replace M by N and hence to assume that beg( M ) ≥ ν iM = − p ( K i +1 ( M )) ≥ − reg( K i +1 ( M )) ≥ − F i +1 d ( x , · · · , x d − ,
0) = E id ( x , · · · , x d − ) . (cid:3) As a consequence of Theorem 5.3 we get the following finiteness result which is shown in[6] for the special case of homogeneous rings R whose base rings R are field. Theorem 5.4.
Let d ∈ N and let x , · · · , x d − ∈ N . Then, the set of cohomological Hilbertfunctions { d iM | i ∈ N ; ( R, M ) ∈ D d ; d jM ( − j ) ≤ x j for j = 0 , · · · , d − } is finite.Proof. First, we set D := { ( R, M ) ∈ D d | d jM ( − j ) ≤ x j for j = 0 , · · · , d − } . As d iM ≡ R, M ) ∈ D d and i ≥ d , it suffices to show that the set { d iM | i < d, ( R, M ) ∈ D} is finite.According to [8, Lemma 4.2] we have d iM ( n ) ≤ i X j =0 (cid:18) − n − j − i − j (cid:19)(cid:20) i − j X l =0 (cid:18) i − jl (cid:19) x i − l (cid:21) (1)for all i ∈ N , all n ≤ − i and all ( R, M ) ∈ D . According to Theorem 5.3 there is someinteger c ≤ − d + 1 such that ν iM > c for all ( R, M ) ∈ D and all i < d . So, using the notationof Reminder 2.3(B) we have q iM ( n ) = d iM ( n ) for all i < d and all n ≤ c . ASTELNUOVO-MUMFORD REGULARITY OF DEFICIENCY MODULES 23
As deg( q iM ) ≤ i (s. Reminder 2.3 (B)(ii),(iv)) it follows from (1) that the set { q iM | i < d, ( R, M ) ∈ D} is finite. Consequently, the set { d iM ( n ) | i < d, n ≤ c, ( R, M ) ∈ D} is finite, too. So, in view of (1) the set { d iM ( n ) | i < d, n ≤ − i, ( R, M ) ∈ D} must be finite. It thus remains to show that for each i < d the set S i := { d iM ( n ) | n ≥ − i, ( R, M ) ∈ D} is finite. To this end, we fix i ∈ { , · · · , d − } . According to [7, Corollary (3.11)] there aretwo integers α, β such that d iM ( n ) ≤ α for all n ≥ − i and all ( R, M ) ∈ D , (2)reg ( M ) ≤ β for all ( R, M ) ∈ D . (3)The inequality (3) implies that d iM ( n ) = 0 for all n ≥ β − i + 1 and hence by (2) the set S i is finite.It remains to show that the set S is finite.To do so, we write M := D R + ( M ) ≥ for all pairs ( R, M ) ∈ D . As ( D R + ( M ) /M ) ≥ = 0, H kR + ( D R + ( M )) = 0 for k = 0 , D jR + ( D R + ( M )) ∼ = D jR + ( M ) for all j ∈ N we getΓ R + ( M ) = 0, end( H R + ( M )) < d jM ≡ d jM for all j ∈ N and for all ( R, M ) ∈ D . Inparticular ( R, M ) ∈ D for all ( R, M ) ∈ D . So, writing D := { ( R, M ) ∈ D| Γ R + ( M ) = 0 , end( H R + ( M )) < } it suffices to show that the set S = { d M ( n ) | n ≥ , ( R, M ) ∈ D} is finite.If ( R, M ) ∈ D we conclude by statement (3) that p ( M ) ≤ reg( M ) = reg ( M ) =max { end( H R + ( M )) + 1 , reg ( M ) } ≤ max { , β } := β ′ . As deg( p M ) < d it follows by state-ment (2) that the set of Hilbert polynomials { p M | ( R, M ) ∈ D} is finite. Consequently, theset { d M ( n ) | n > β ′ , ( R, M ) ∈ D} is finite. Another use of statement (2) now implies thefiniteness of S . (cid:3) Corollary 5.5.
Let the notations be as in Theorem 5.4 and Reminder 2.3. Then the sets ofpolynomials { q iM | i ∈ N ; ( R, M ) ∈ D d ; d jM ( − j ) ≤ x j for j = 0 , · · · , d − } , { p M | ( R, M ) ∈ D d ; d jM ( − j ) ≤ x j for j = 0 , · · · , d − } are finite. Proof.
This is clear by Theorem 5.4. (cid:3)
Acknowledgment.
The third author would like to thank the Institute of Mathematicsof University of Z¨urich for financial support and hospitality during the preparation of thispaper.
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University of Z¨urich, Institute of Mathematics, Winterthurerstrasse 190, 8057 Z¨urich.
E-mail address : [email protected] M.Jahangiri, Faculty of Basic Science, Tarbiat Modares University, Tehran, Iran.
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