Generic freeness of local cohomology and graded specialization
aa r X i v : . [ m a t h . A C ] O c t GENERIC FREENESS OF LOCAL COHOMOLOGY AND GRADEDSPECIALIZATION
MARC CHARDIN, YAIRON CID-RUIZ, AND ARON SIMIS
Abstract.
The main focus is the generic freeness of local cohomology modules in a gradedsetting. The present approach takes place in a quite nonrestrictive setting, by solely assumingthat the ground coefficient ring is Noetherian. Under additional assumptions, such as when thelatter is reduced or a domain, the outcome turns out to be stronger. One important applicationof these considerations is to the specialization of rational maps and of symmetric and Reespowers of a module. Introduction
Although the actual strength of this paper has to do with generic freeness in graded localcohomology, we chose to first give an overview of one intended application to specializationtheory.Specialization is a classical and important subject in algebraic geometry and commutativealgebra. Its roots can be traced back to seminal work by Kronecker, Hurwitz ([15]), Krull([19, 20]) and Seidenberg ([25]). More recent papers where specialization is used in the classicalway are [28], [23], [24] and, in a different vein, [9], [18], [13], [14], [29], [26], [7].In the classical setting it reads as follows. Let k be a field and R be a polynomial ring R = k ( z )[ x ] = k ( z )[ x , . . . , x r ] over a purely transcendental field extension k ( z ) = k ( z , . . . , z m )of k . Let I ⊂ R be an ideal of R and α = ( α , . . . , α m ) ∈ k m . The specialization of I withrespect to the substitutions z i → α i is given by the ideal I α := (cid:8) f ( α, x ) | f ( z , x ) ∈ I ∩ k [ z ][ x ] (cid:9) . Setting J := I ∩ k [ z ][ x ], the canonical map π α : k [ z ][ x ] ։ k [ z ][ x ] / ( z − α ) yields the identification(1) I α ≃ π α ( J ) , which is the gist of the classical approach.One aim of this paper is to introduce a notion of specialization on more general settings in thegraded category, whereby k ( z ) will be replaced by a Noetherian reduced ring A and a finitelygenerated graded A -algebra will take the place of R . The emphasis of this paper will be ongraded modules, and more specifically, on the graded parts of local cohomology modules.In order to recover the essential idea behind (1) in our setting, we now explain the notionof specialization used in this work. In the simplest case, let A be a Noetherian ring, let R bea finitely generated positively graded A -algebra and let m ⊂ R denote the graded irrelevantideal m = [ R ] + . Here, for simplicity, let M be a finitely generated torsion-free graded R -modulehaving rank, with a fixed embedding ι : M ֒ → F into a finitely generated graded free R -module F . For any p ∈ Spec( A ), the specialization of M with respect to p will be defined to be S p ( M ) = Im (cid:16) ι ⊗ A k ( p ) : M ⊗ A k ( p ) → F ⊗ A k ( p ) (cid:17) Mathematics Subject Classification.
Primary: 13D45, Secondary: 13A30, 14E05.
Key words and phrases. local cohomology, generic freeness, specialization, rational maps, symmetric algebra,Rees algebra. where k ( p ) = A p / p A p is the residue field of p . If M is not torsion-free, one kills its R -torsion andproceed as above. It can be shown that the definition is independent on the chosen embeddingfor general choice of p ∈ Spec( A ) (see Proposition 5.5).The true impact of the present approach is the following. Theorem A (Corollary 5.6) . Let A be a Noetherian reduced ring and R be a finitely generatedpositively graded A -algebra. Let M be a finitely generated graded R -module having rank. Then,there exists a dense open subset U ⊂
Spec( A ) such that, for all i ≥ , j ∈ Z , the function Spec( A ) −→ Z , p ∈ Spec( A ) dim k ( p ) (cid:2) H i m (cid:0) S p ( M ) (cid:1)(cid:3) j ! is locally constant on U . Alas, although one can control all the graded parts of the specialization of M , not so much forall higher symmetric and Rees powers, whereby the results will only be able to control certaingraded strands.Anyway, one has enough to imply the local constancy of numerical invariants such as dimen-sion, depth, a -invariant and regularity under a general specialization (see Proposition 5.13).The main tool in this paper is the behavior of local cohomology of graded modules undergeneric localization with a view towards generic freeness (hence its inclusion in the title). Thisis a problem of great interest in its own right, having been addressed earlier by several authors.We will approach the matter in a quite nonrestrictive setting, by assuming at the outset that A is an arbitrary Noetherian ring. When A is a domain or, sometimes, just a reduced ring, onerecovers and often extends some results by Hochster and Roberts [12, Section 3] and by Smith[27].An obstruction for local freeness of local cohomology of a finitely generated graded R -module M is here described in terms of certain closed subsets of Spec( A ) defined in terms of M and itsExt modules. To wit, it can be shown that the set U M = (cid:8) p ∈ Spec( A ) | [ M ] µ ⊗ A A p is A p -free for every µ ∈ Z (cid:9) is an open set of Spec( A ), and that is dense when A is reduced. Its complement T M = Spec( A ) \ U M will play a central role in this regard.For convenience, set ( M ) ∗ A := ∗ Hom A ( M, A ) = L ν ∈ Z Hom A (cid:0) [ M ] − ν , A (cid:1) . The following theorem encompasses the main results in this direction, with a noted differenceas to whether A is a domain or just reduced. Theorem B (Theorem 3.6) . Let A be a Noetherian ring and R be a positively graded polynomialring R = A [ x , . . . , x r ] over A . Set δ = deg( x )+ · · · +deg( x r ) ∈ N . Let M be a finitely generatedgraded R -module. (I) If p ∈ Spec( A ) \ (cid:16) T M ∪ S ∞ j =0 T Ext jR ( M,R ) (cid:17) , then the following statements hold for any ≤ i ≤ r : (a) (cid:2) H i m ( M ⊗ A A p ) (cid:3) ν is free over A p for all ν ∈ Z . (b) For any A p -module N , the natural map H i m ( M ) ⊗ A p N → H i m ( M ⊗ A p N ) is an iso-morphism. (c) For any A p -module N , there is an isomorphism H i m (cid:0) M ⊗ A p N (cid:1) ≃ (cid:16) Ext r − iR ⊗ A A p ( M ⊗ A A p , R ( − δ ) ⊗ A A p ) (cid:17) ∗ A p ⊗ A p N. (II) Let F • : · · · → F k → · · · → F → F → be a graded free resolution of M by mod-ules of finite rank. If p ∈ Spec( A ) \ (cid:16) T M ∪ T D r +1 M ∪ S rj =0 T Ext jR ( M,R ) (cid:17) where D r +1 M = ENERIC FREENESS OF LOCAL COHOMOLOGY AND GRADED SPECIALIZATION 3
Coker (Hom R ( F r , R ) → Hom R ( F r +1 , R )) , then the same statements as in (a), (b), (c) of (I) hold. (III) If A is reduced, then there exists an element a ∈ A avoiding the minimal primes of A suchthat, for any ≤ i ≤ r , the following statements hold: (a) H i m ( M ⊗ A A a ) is projective over A a . (b) For any A a -module N , the natural map H i m ( M ) ⊗ A a N → H i m ( M ⊗ A a N ) is an iso-morphism. (c) For any A a -module N , there is an isomorphism H i m ( M ⊗ A a N ) ≃ (cid:16) Ext r − iR ⊗ A A a ( M ⊗ A A a , R ( − δ ) ⊗ A A a ) (cid:17) ∗ Aa ⊗ A a N. (IV) If A is a domain, then there exists an element = a ∈ A such that (cid:2) H i m ( M ⊗ A A a ) (cid:3) ν isfree over A a for all ν ∈ Z . Above, the fact that when A is a reduced ring, but not a domain, the local cohomology modulesH i m ( M ⊗ A A a ) are projective but not necessarily free over A a , seems to be a hard knuckle. Asif to single out this difficulty, we note that [27, Corollary 1.3] might not be altogether correct –we give a counter-example in Example 3.8.Having briefly accounted for the main results on the generic localization of local cohomologyand how it affects the problem of specialization, we turn to the question of specializing the powersof a module. We consider both symmetric powers as Rees powers, leaving out wedge powersfor future consideration. Then, one is naturally led to focus on a bigraded setting as treated inSection 4. In this setting, we will be able to control certain graded strands, but unfortunatelynot all graded parts. This impairment is not due to insufficient work, rather mother nature ashas been proved by Katzman ([17]; see Example 4.6).For the main result of the section, one assumes that A is a Noetherian reduced ring and R = A [ x , . . . , x r , y , . . . , y s ] is a bigraded polynomial ring with bideg( x ) = ( δ i ,
0) with δ i > y i ) = ( − γ i ,
1) with γ i ≥
0. Set m := ( x , . . . , x r ) R , the extension to R of the irrelevantgraded ideal of the positively graded polynomial ring A [ x , . . . , x r ].Then: Theorem C. (Theorem 4.5)
Let M be a finitely generated bigraded R -module. For a fixed integer j ∈ Z , there exists a dense open subset U j ⊂ Spec( A ) such that, for all i ≥ , ν ∈ Z , the function Spec( A ) −→ Z , p ∈ Spec( A ) dim k ( p ) (cid:2) H i m (cid:0) M ⊗ A k ( p ) (cid:1)(cid:3) ( j,ν ) ! is locally constant on U j . The rationale of the paper is that the first four sections deal with the algebraic tools re-garding exactness of fibered complexes, local cohomology of general fibers and generic freenessof graded local cohomology, whereas the last section Section 5 contains the applications of themain theorems so far to various events of specialization. For the sake of visibility, we organizedthis section in three subsections, each about the specialization of objects of different nature, soto say. Thus, the first piece concerns as to how the local cohomology of the specialized powersof a graded module behaves. To skip the technical preliminaries, we refer the reader directlyto the corresponding results Theorem 5.3 and Theorem 5.7. Some of these should be comparedwith the results of [23], though the techniques are different.The second part of Section 5 concerns the problem of specializing rational maps, pretty muchin the spirit of the recent paper [7]. Namely, one gives an encore of the fact, previously shownin loc. cit., that the (topological) degree of the rational map and the degree (multiplicity) of thecorresponding image remain stable under a general specialization of the coefficients involved in
MARC CHARDIN, YAIRON CID-RUIZ, AND ARON SIMIS the given data. Here, the outcome shapes up as a consequence of Theorem 5.7 and [3, Corollary2.12]. Given the known relations between rational maps and both the saturated special fiber ringsand the j -multiplicities, it seems only natural to consider the latter under general specialization.The last part is a short account of typical numerical module invariants, such as dimension,depth, a -invariant and regularity, showing that they are locally constant when tensoring with ageneral fiber and under a general specialization. Taking a more geometric view one shows that,for a coherent sheaf, the dimension of the cohomology of a general fiber is locally constant forevery twisting of the sheaf, which can be looked upon as a slight improvement on the well-knownupper semi-continuity theorem.2. Exactness of the fibers of a complex
In this section one studies how the process of taking tensor product with a fiber affects thehomology of a complex. In the main result of the section one shows that, under a nearlyunrestrictive setting, an exact complex remains exact after taking tensor product with a generalfiber. This result can be seen as a vast generalization and an adaptation of [23, Theorem 1.5,Proposition 2.7].Since one is interested in certain naturally bigraded algebras – such as the symmetric orthe Rees algebra of graded modules – and there is no significant difference between a bigradedsetting or a general graded one, one will deal with the following encompassing setting.
Setup 2.1.
Let A be a ring – always assumed to be commutative and unitary. Let R be thepolynomial ring R = A [ x , . . . , x r ] graded by an Abelian group G . Assume that deg( a ) = 0 ∈ G for a ∈ A and that there is a Z -linear map ψ : G → Z such that ψ (deg( x i )) > ≤ i ≤ r .Under the above setup, which is assumed throughout, for any finitely generated graded R -module M , the graded strands [ M ] µ ( µ ∈ G ) are finitely generated A -modules.The notation below will be used throughout the paper. Notation 2.2.
For a complex of A -modules P • : · · · φ i +1 −−−→ P i φ i −→ · · · φ −→ P φ −→ P , one setsZ i ( P • ) := Ker( φ i ), B i ( P • ) := Im( φ i +1 ), H i ( P • ) := Z i ( P • ) / B i ( P • ), and C i ( P • ) := P i / B i ( P • ) ⊃ H i ( P • ) for all i ∈ Z . Remark 2.3.
One of the few general assertions at this point is the following: for a complex of A -modules P • and an A -module N , one has a four-term exact sequence0 → H i (cid:0) P • ⊗ A N (cid:1) → C i ( P • ) ⊗ A N → P i − ⊗ A N → C i − ( P • ) ⊗ A N → A -modules. Lemma 2.4.
Under
Setup 2.1 , let F • be a graded complex of finitely generated free R -modules.Then, for every integer i , there exists a finite set D ( i ) ⊆ G such that for any homomorphism φ : A → k from A to a field k , the shifts in the minimal free graded resolution of H i ( F • ⊗ A k ) belong to D ( i ) .Proof. For the sake of clarity we divide the proof into three shorter steps.
Step 1.
First, assume that G = Z and deg( x i ) = 1. From [6, Lemma 2.2 (2)] applied tothe complex F • ⊗ A k of finitely generated free R ⊗ A k ( ≃ k [ x , . . . , x r ])-modules, we obtain aconstant C ( i ) which is independent of the field k and that bounds the Castelnuovo-Mumfordregularity of H i ( F • ⊗ A k ), i.e., reg (H i ( F • ⊗ A k )) ≤ C ( i ). Therefore, by using the definition ofCastelnuovo-Mumford regularity in terms of minimal free resolutions, we get the existence ofsuch finite set D ( i ) (that is independent of k ). Step 2.
Next, assume that G is generated by the elements deg( x i ) ∈ G . Let g i := deg( x i ) ∈ G and d i := ψ ( g i ) ∈ Z > . If f ∈ R is G -homogeneous, it is also Z -homogeneous for the induced ENERIC FREENESS OF LOCAL COHOMOLOGY AND GRADED SPECIALIZATION 5 Z -grading deg Z ( F ) := ψ (deg G ( F )). Then, since the degrees of H i ( F • ⊗ A k ) (in the Z -gradinginduced by G ) are bounded below by the smallest shift in F i (in the Z -grading induced by G ),the condition deg Z ( x i ) > D µ ( i ) ⊂ G (independent of k ) such that n ν ∈ G | h Tor jR ⊗ A k (H i ( F • ⊗ A k ) , k ) i ν = 0 for some j ≥ o ∩ ψ − ( µ ) ⊆ D µ ( i )for every µ ∈ Z .It is then sufficient to prove the result for the Z -grading induced by G .Set S := R [ x ], where x is a new variable. Then S has two gradings, one is the standardgrading as a polynomial ring over A , the other comes as an extension of the Z -grading of R bysetting deg Z ( x ) = 0. Now, given a G -homogeneous element f = P α c α x α ∈ R , consider thepolynomial f ′ := X α c α x deg Z ( x α ) −| α | x α ∈ S. The latter is homogeneous in the above two gradings of S . Setbideg( f ′ ) := (deg( f ′ ) , deg Z ( f ′ )) = deg Z ( f ) · (1 , M = ( f i,j ) of G -homogeneous elements, one sets M ′ := ( f ′ i,j ).Now, as bideg( f ′ ) = deg Z ( f ) · (1 ,
1) for any G -homogeneous element f , homogenizing all themaps in F • in this way provides a complex F ′• of standard graded free S -modules relative to thefirst component of the grading, with shifts controlled in terms of the initial ones. It then followsfrom the standard graded case (treated in Step 1 ) that the minimal bigraded free ( S ⊗ A k )-resolution of H i ( F ′• ⊗ A k ) has shifts ( a, b ) with a bounded above by an integer K ( i ) that does notdepend on the field k . Specializing x to 1 provides a (possibly non minimal) deg Z -graded free( R ⊗ A k )-resolution of H i ( F • ⊗ A k ) (see, e.g., [8, proof of Corollary 19.8]). But for any monomialdeg Z ( x α ) ≤ max j { d j } deg( x α ). Hence, all shifts in the minimal free ( R ⊗ A k )-resolution ofH i ( F • ⊗ A k ) are bounded above by K ( i ) max j { d j } , and the claim follows, since the initial degreeof H i ( F • ⊗ A k ) is bounded below by the smallest shift in F i . Step 3.
Finally, let G ′ be the subgroup of G generated by the g i ’s. If h , . . . h s are repre-sentatives of the different classes modulo G ′ of the shifts appearing in F i − , F i and F i +1 , the( R ⊗ A k )-module H i ( F • ⊗ A k ) is the direct sum of the homology of the strands correspond-ing to summands whose shifts belong to these classes. Again, each of these gives rise, by theabove proof, to only finitely many options for the shifts in the minimal free resolution of thecorresponding strand of H i ( F • ⊗ A k ). (cid:3) The gist of Lemma 2.4 is the ability of reducing the vanishing of the fiber homology of a freegraded complex of R -modules to a finite number of degrees. This will be transparent in thefollowing result.Recall the usual notation by which, for any p ∈ Spec( A ), k ( p ) denotes the residue field k ( p ) := A p / p A p = Quot( A/ p ) . Lemma 2.5.
Under
Setup 2.1 , let F • be a graded complex of finitely generated free R -modules. (i) For every i , there exists a finite set of degrees D ( i ) such that, for any prime ideal p ∈ Spec( A ) , the following are equivalent: (a) H i ([ F • ] µ ⊗ A k ( p )) = 0 , for every µ ∈ D ( i ) , (b) H i ( F • ⊗ A k ( p )) = 0 . (ii) For every i , the set (cid:8) p ∈ Spec( A ) | H i ( F • ⊗ A k ( p )) = 0 (cid:9) is open in Spec( A ) . MARC CHARDIN, YAIRON CID-RUIZ, AND ARON SIMIS (iii)
Assume that A is locally Noetherian, F i = 0 for i < and H i ( F • ) = 0 for i > . Set M := H ( F • ) . Then, the set { p ∈ Spec( A ) | [ M ] µ ⊗ A A p is A p -free for all µ ∈ G } is openin Spec( A ) .Proof. (i) It is clear that (b) implies (a). For the converse, it follows from Lemma 2.4 that[H i (( F • ) ⊗ A k ( p ))] µ is generated by elements whose G -degree belong to a finite set D ( i ) ⊆ G .In particular, if [H i (( F • ) ⊗ A k ( p ))] µ vanishes for µ ∈ D ( i ), it follows that H i ( F • ⊗ A k ( p )) = 0.(ii) For a given µ , [H i (( F • ) ⊗ A k ( p ))] µ = 0 is equivalent to the condition rank[( d i +1 ⊗ A k ( p ))] µ +rank[( d i ⊗ A k ( p ))] µ < rank[ F i ] µ , a closed condition in terms of ideals of minors of matricesrepresenting these graded pieces of the differentials d i +1 and d i of F • . So, the result follows frompart (i).(iii) By the local criterion for flatness (see, e.g., [8, Theorem 6.8]), for any µ , the followingare equivalent:(a) µ Tor A p ([ M ] µ ⊗ A A p , k ( p )) = [H ( F • ⊗ A k ( p ))] µ = 0,(b) µ [ M ] µ ⊗ A A p is A p -flat,(c) µ [ M ] µ ⊗ A A p is A p -free,where the last three conditions coincide since [ M ] µ ⊗ A A p is a finitely presented A p -module forany µ . So, the conclusion follows from part (ii). (cid:3) Notation 2.6.
For any finitely generated graded R -module M , one denotes by T M the comple-ment in Spec( A ) of the open set U M := { p ∈ Spec( A ) | [ M ] µ ⊗ A A p is A p -free for all µ ∈ G } introduced in Lemma 2.5(iii). Remark 2.7.
When A is Noetherian and M is a finitely generated graded R -module, T M isa closed subset of Spec( A ) by Lemma 2.5(iii). Furthermore, if p is a minimal prime of A suchthat A p is a field, then p T M . In particular, if A is generically reduced then U M is dense inSpec( A ). This is in particular the case when A is reduced – a frequent assumption in this paper. Lemma 2.8.
Let P • be a complex of A -modules and let s ≥ denote an integer. Assume that: (a) P i is A -flat for every ≤ i ≤ s . (b) H i ( P • ) is A -flat for every ≤ i ≤ s .Then, for any A -module N and any ≤ i ≤ s , one has that H i ( P • ⊗ A N ) ≃ H i ( P • ) ⊗ A N. Proof.
Let F • be a free A -resolution of N . The two spectral sequences associated to the dou-ble complex with components P p ⊗ A F q have respective second terms Tor Aq (H p ( P • ) , N ) andH p (Tor Aq ( P • , N )). As Tor Aq ( P • , N ) = 0 for q > Aq (H p ( P • ) , N ) for q > (cid:3) For a graded R -module M , denote for brevity( M ) ∗ A = ∗ Hom A ( M, A ) := M ν ∈ G Hom A (cid:0) [ M ] − ν , A (cid:1) . Note that ( M ) ∗ A has a natural structure of graded R -module. Lemma 2.9.
Let P • be a co-complex of finitely generated graded R -modules. Assume that: (a) P i is A -flat for all i ≥ . (b) H i ( P • ) is A -flat for all i ≥ . (c) A is Noetherian. ENERIC FREENESS OF LOCAL COHOMOLOGY AND GRADED SPECIALIZATION 7
Then, for all i ≥ , one has that H i (cid:0) ( P • ) ∗ A (cid:1) ≃ (cid:0) H i ( P • ) (cid:1) ∗ A . Proof.
First notice that, since [ P p ] µ is finitely generated over A for any µ and A is Noetherian,the modules P p , ( P p ) ∗ A , H p ( P • ) and (H p ( P • )) ∗ A are direct sums of finitely presented A -modules.Hence, each one of these is A -flat if and only if is A -projective.Let I • be an injective A -resolution of A . The two spectral sequences associated to the secondquadrant double complex G • , • with components G − p,q = L µ ∈ G Hom A ([ P p ] − µ , I q ) have respec-tive second terms II E p, − q = M µ ∈ G Ext pA (cid:0) [H q ( P • )] − µ , A (cid:1) and I E − p,q = M µ ∈ G H − p (cid:0) Ext qA ([ P • ] − µ , A ) (cid:1) . From (a) and (b) we obtain that Ext qA ([ P • ] − µ , A ) = 0 for q > pA ([H q ( P • )] − µ , A ) = 0for p >
0, respectively. ThereforeH p (cid:0) ( P • ) ∗ A (cid:1) = H − p (cid:0) ( P • ) ∗ A (cid:1) ≃ (H p ( P • )) ∗ A , ∀ p ≥ , and so the result follows. (cid:3) The following local version of the classical generic freeness lemma will be used over and over.
Corollary 2.10.
Under
Setup 2.1 , let A be a reduced Noetherian ring and let M be a finitelygenerated graded R -module. Then, there exists an element a ∈ A avoiding the minimal primesof A such that M a is a projective A a -module.Proof. From Lemma 2.5(iii) and the prime avoidance lemma one can find a ∈ A avoiding theminimal primes of A such that D ( a ) ⊂ Spec( A ) \ T M . (cid:3) In the sequel, given p ∈ Spec( A ), an R -module M and an A p -module N , the ( R ⊗ A A p )-module M p ⊗ A p N = ( M ⊗ A A p ) ⊗ A p N will as usual be denoted by M ⊗ A p N . By the same token, given a ∈ A , M ⊗ A a N will denote M a ⊗ A a N .Next is the main result of the section. For a complex of finitely generated graded R -modules,the following theorem gives an explicit closed subset of Spec( A ) outside which homology com-mutes with tensor product. Theorem 2.11.
Under
Setup 2.1 , let A be a Noetherian ring and let P • be a complex of finitelygenerated graded R -modules with P i = 0 for i < . Given an integer s ≥ and a prime p ∈ Spec( A ) \ S si =0 (cid:0) T P i ∪ T H i ( P • ) (cid:1) , then H i ( P • ⊗ A p N ) ≃ H i ( P • ) ⊗ A p N, for any A p -module N and for every ≤ i ≤ s .Proof. Let P •• be a Cartan–Eilenberg graded free R -resolution of P • with finitely generatedsummands. The totalization T • of P •• is a complex of finitely generated graded free R -moduleswith H i ( T • ) = H i ( P • ) for all i . On the one hand,H i ( T • ⊗ A p N ) ≃ H i ( T • ) ⊗ A p N = H i ( P • ) ⊗ A p N, ∀ ≤ i ≤ s, by Lemma 2.5(iii) and Lemma 2.8 since p S si =0 T H i ( T • ) = S si =0 T H i ( P • ) .On the other hand, the spectral sequence from P •• ⊗ A p N with first term E p,q = Tor A p q ( P p ⊗ A A p , N ), shows that H i ( T • ⊗ A p N ) ≃ H i ( P • ⊗ A p N ) , ∀ ≤ i ≤ s, by Lemma 2.5(iii) since p
6∈ ∪ si =0 T P i . (cid:3) MARC CHARDIN, YAIRON CID-RUIZ, AND ARON SIMIS Generic freeness of graded local cohomology modules
In this section one is concerned with the generic freeness of graded local cohomology modules.Here one extends the results of [27] and [12, Section 3] to a graded environment, adding a fewgeneralizations.The following setup will hold throughout the section.
Setup 3.1.
Keep the notation introduced in Setup 2.1, so that R = A [ x , . . . , x r ] is a G -gradedpolynomial ring. Assume in addition that A is Noetherian and set m = ( x , . . . , x r ) ⊂ R and δ = deg( x ) + · · · + deg( x r ) ∈ G . Recall that H r m ( R ) ≃ x ··· x r A [ x − , . . . , x − r ]. Remark 3.2.
Let M be a finitely generated graded R -module. Since one is assuming that ψ (deg( x i )) >
0, it follows that (cid:2) H i m ( M ) (cid:3) ν is a finitely generated A -module for all i ≥ ν ∈ G (see [4, Theorem 2.1]).Consider the canonical perfect pairing of free A -modules in “top” cohomology[ R ] ν ⊗ A [H r m ( R )] − δ − ν → [H r m ( R )] − δ ≃ A inducing a canonical graded R -isomorphism H r m ( R ) ≃ ( R ( − δ )) ∗ A = ∗ Hom A ( R ( − δ ) , A ) . The functor ( • ) ∗ A has been introduced in the previous section. It can be regarded as a relativeversion (with respect to A ) of the graded Matlis dual. Lemma 3.3.
Let F • be a complex of finitely generated graded free R -modules. Then, one hasthe isomorphism of complexes H r m ( F • ) ≃ (cid:16) Hom R ( F • , R ( − δ )) (cid:17) ∗ A . Proof.
This is well-known (see, e.g., [16, Section 2.15], [5, Corollary 1.4]). (cid:3)
Lemma 3.4.
Let F • stand for a graded free resolution of a finitely generated graded R -module M by modules of finite rank. If M is A -flat, then H i m ( M ⊗ A N ) ≃ H r − i (cid:0) H r m ( F • ) ⊗ A N (cid:1) for any A -module N .Proof. Consider the double complex C • m F • ⊗ A N obtained by taking the ˇCech complex on F • ⊗ A N .Since M is A -flat, F • ⊗ A N is acyclic and H ( F • ⊗ A N ) ≃ M ⊗ A N . Therefore, as localizationis exact and H i m ( R ⊗ A N ) ≃ ( H r m ( R ) ⊗ A N if i = r , by analyzing the spectral sequences coming from the double complex C • m F • ⊗ A N , the isomor-phism H i m ( M ⊗ A N ) ≃ H r − i (cid:0) H r m ( F • ) ⊗ A p N (cid:1) follows. (cid:3) For the proof of the main theorem of the section, we need a version of the celebratedGrothendieck’s generic freeness lemma in a more encompassing graded environment. We takeverbatim the basic assumptions of the non-graded version first stated in [11, Lemma 8.1], makingthe needed adjustment in the graded case. The standard assumption on the ring A is that it bea domain, but we also give a generic projectivity counterpart if A is just assumed to be reduced.In order to state a bona fide graded version, we make the following convention. First, A isalso considered as a G -graded ring, with (trivial) grading [ A ] ν = 0 for ν = 0 ∈ G . In addition, an A -module H is said to be G -graded if it has a direct summands decomposition H = L µ ∈ G [ H ] ν indexed by G , where each [ H ] ν is an A -module. For a G -graded A -algebra B and a G -graded B -module M , one says that a G -graded A -module H is a G -graded A -submodule of M if onehas [ H ] ν ⊆ [ M ] ν for all ν ∈ G . ENERIC FREENESS OF LOCAL COHOMOLOGY AND GRADED SPECIALIZATION 9
Theorem 3.5.
Assume
Setup 3.1 . In addition, let B ⊃ R be a finitely generated G -graded R -algebra. Let M be a finitely generated G -graded B -module. Let E be a finitely generated G -graded R -submodule of M and H be a finitely generated G -graded A -submodule of M . Set M = M/ ( E + H ) , which is a G -graded A -module. (i) If A is reduced, then there is an element a ∈ A avoiding the minimal primes of A such that M a is A a -projective. (ii) If A is a domain, then there is an element = a ∈ A such that each graded component [ M a ] ν , ν ∈ G of M a is A a -free.Proof. The proof follows along the same lines of [11, Lemma 8.1] (see also [22, Theorem 24.1]).When A is reduced, one draws upon Corollary 2.10 in order to start an appropriate inductiveargument. (cid:3) The ground work having been carried through the previous results so far, we now collect theessential applications in the main theorem of the section.
Theorem 3.6.
Under
Setup 3.1 , let M be a finitely generated graded R -module. (I) If p ∈ Spec( A ) \ (cid:16) T M ∪ S ∞ j =0 T Ext jR ( M,R ) (cid:17) , then the following statements hold for any ≤ i ≤ r : (a) (cid:2) H i m ( M ⊗ A A p ) (cid:3) ν is free over A p for all ν ∈ G . (b) For any A p -module N , the natural map H i m ( M ) ⊗ A p N → H i m ( M ⊗ A p N ) is an iso-morphism. (c) For any A p -module N , there is an isomorphism H i m (cid:0) M ⊗ A p N (cid:1) ≃ (cid:16) Ext r − iR ⊗ A A p ( M ⊗ A A p , R ( − δ ) ⊗ A A p ) (cid:17) ∗ A p ⊗ A p N. (II) Let F • : · · · → F k → · · · → F → F be a graded free resolution of M by mod-ules of finite rank. If p ∈ Spec( A ) \ (cid:16) T M ∪ T D r +1 M ∪ S rj =0 T Ext jR ( M,R ) (cid:17) where D r +1 M =Coker (Hom R ( F r , R ) → Hom R ( F r +1 , R )) , then the same statements as in (a), (b), (c) of (I) hold. (III) If A is reduced, then there exists an element a ∈ A avoiding the minimal primes of A suchthat, for any ≤ i ≤ r , the following statements hold: (a) H i m ( M ⊗ A A a ) is projective over A a . (b) For any A a -module N , the natural map H i m ( M ) ⊗ A a N → H i m ( M ⊗ A a N ) is an iso-morphism. (c) For any A a -module N , there is an isomorphism H i m ( M ⊗ A a N ) ≃ (cid:16) Ext r − iR ⊗ A A a ( M ⊗ A A a , R ( − δ ) ⊗ A A a ) (cid:17) ∗ Aa ⊗ A a N. (IV) If A is a domain, then there exists an element = a ∈ A such that (cid:2) H i m ( M ⊗ A A a ) (cid:3) ν isfree over A a for all ν ∈ G .Proof. (I) Let F • : · · · → F k → · · · → F → F be a graded free resolution of M by modules offinite rank.(I)(a) From the fact that p T M , Lemma 3.4 and Lemma 3.3 yield the isomorphisms(2) H i m ( M ⊗ A A p ) ≃ H r − i (H r m ( F • ) ⊗ A A p ) ≃ H r − i (cid:16)(cid:16) Hom R ( F • , R ( − δ )) (cid:17) ∗ A ⊗ A A p (cid:17) , for all i . One has thatH r − i (Hom R ( F • , R ( − δ )) ⊗ A A p ) ≃ (cid:16) Ext r − iR ⊗ A A p ( M ⊗ A A p , R ( − δ ) ⊗ A A p ) (cid:17) . Since p T Ext jR ( M,R ) for all j ≥
0, Lemma 2.5(iii) and Lemma 2.9 applied to the co-complexHom R ( F • , R ( − δ )) ⊗ A A p give the following isomorphismsH i m ( M ⊗ A A p ) ≃ H r − i (cid:16)(cid:16) Hom R ( F • , R ( − δ )) (cid:17) ∗ A ⊗ A A p (cid:17) ≃ (cid:16) Ext r − iR ⊗ A A p ( M ⊗ A A p , R ( − δ ) ⊗ A A p ) (cid:17) ∗ A p (3)for all 0 ≤ i ≤ r . Again, as p T Ext jR ( M,R ) , the result is obtained from Lemma 2.5(iii).(I)(b) The natural map H r − i (H r m ( F • )) ⊗ A p N → H r − i (cid:0) H r m ( F • ) ⊗ A p N (cid:1) is an isomorphismbecause each H r − i (H r m ( F • ) ⊗ A A p ) ≃ H i m ( M ⊗ A A p ) is a free A p -module (from part (I)(a)) andH r m ( F • ) is a complex of free A -modules (see Lemma 2.8). So, the result follows from Lemma 3.4.(I)(c) From part (I)(b) and (3) we get the isomorphismsH i m ( M ⊗ A p N ) ≃ H i m ( M ) ⊗ A p N ≃ (cid:16) Ext r − iR ⊗ A A p ( M ⊗ A A p , R ( − δ ) ⊗ A A p ) (cid:17) ∗ A p ⊗ A p N, which gives the result.(II) We basically repeat the same steps in the proof of (I), but instead of considering the freeresolution F • , we now analyze the truncated complex F ≤ r +1 • : 0 → F r +1 → F r → · · · → F → F . Note that H j (Hom R ( F ≤ r +1 • , R )) ≃ Ext jR ( M, R ) for 0 ≤ j ≤ r and H r +1 (Hom R ( F ≤ r +1 • , R )) ≃ D r +1 M . As p ∈ Spec( A ) \ (cid:16) T M ∪ T D r +1 M ∪ S rj =0 T Ext jR ( M,R ) (cid:17) , after applying Lemma 2.5(iii) andLemma 2.9 to the co-complex Hom( F ≤ r +1 • , R ) ⊗ A A p and invoking (2) we obtain the followingisomorphisms H i m ( M ⊗ A A p ) ≃ H r − i (cid:16)(cid:16) Hom R ( F • , R ( − δ )) (cid:17) ∗ A ⊗ A A p (cid:17) ≃ H r − i (cid:16)(cid:16) Hom R ( F ≤ r +1 • , R ( − δ )) (cid:17) ∗ A ⊗ A A p (cid:17) ≃ (cid:16) Ext r − iR ⊗ A A p ( M ⊗ A A p , R ( − δ ) ⊗ A A p ) (cid:17) ∗ A p for all 0 ≤ i ≤ r . Since p T Ext jR ( M,R ) for all 0 ≤ j ≤ r , the conclusion of (I)(a) follows fromLemma 2.5(iii).The arguments in the proofs of (I)(b) and (I)(c) only depend on the conclusion of (I)(a) andthe isomorphisms given in (3). Note that we have proved the last two results in the current part(II). Therefore, the conclusions of parts (I)(b) and (I)(c) also follow under the assumptions ofpart (II).(III) The proof is nearly verbatim the one of the part (II). By using Corollary 2.10, one canchoose a ∈ A avoiding the minimal primes of A such that M a , D r +1 M ⊗ A A a and Ext jR ( M, R ) ⊗ A A a are projective A a -modules for 0 ≤ j ≤ r . Then, Lemma 3.4 and Lemma 3.3 give theisomorphismsH i m ( M ⊗ A A a ) ≃ H r − i (H r m ( F • ) ⊗ A A a ) ≃ H r − i (cid:16)(cid:16) Hom R ( F • , R ( − δ )) (cid:17) ∗ A ⊗ A A a (cid:17) for all i . Again, by applying Lemma 2.9 to the co-complex Hom( F ≤ r +1 • , R ) ⊗ A A a we obtain thefollowing isomorphism(4) H i m ( M ⊗ A A a ) ≃ (cid:16) Ext r − iR ⊗ A A a ( M ⊗ A A a , R ( − δ ) ⊗ A A a ) (cid:17) ∗ Aa for all 0 ≤ i ≤ r . So, the result of part (III)(a) also follows, and parts (III)(a) and (III)(b) areobtained from (III)(a) and (4). ENERIC FREENESS OF LOCAL COHOMOLOGY AND GRADED SPECIALIZATION 11 (IV) Take a ′ ∈ A from part (III)(c) such that the isomorphismsH i m ( M ⊗ A A a ′ ) ≃ (cid:16) Ext r − iR ⊗ A A a ′ ( M ⊗ A A a ′ , R ( − δ ) ⊗ A A a ′ ) (cid:17) ∗ Aa ′ hold, for all 0 ≤ i ≤ r .Now, let 0 ≤ j ≤ r . For each such j apply Theorem 3.5(ii) with B = R and M = Ext jR ( M, R );since there are finitely many j ’s, there exists an a ′′ = 0 in A such that h Ext jR ( M, R ) ⊗ A A a ′′ i ν is a free A a ′′ -module for all 0 ≤ j ≤ r and ν ∈ G . So, the result follows by setting a = a ′ a ′′ . (cid:3) The theorem has an important consequence, as follows.
Proposition 3.7.
Under
Setup 3.1 , assume in addition that A is reduced. Given a finitelygenerated graded R -module M , there exists a dense open subset U ⊂
Spec( A ) such that, for all i ≥ , ν ∈ G , the function Spec( A ) −→ Z , p ∈ Spec( A ) dim k ( p ) (cid:2) H i m (cid:0) M ⊗ A k ( p ) (cid:1)(cid:3) ν ! is locally constant on U .Proof. By Theorem 3.6(II), there is an element a ∈ A avoiding the minimal primes of A suchthat for all i ≥
0, H i m ( M ) ⊗ A a k ( p ) ≃ H i m ( M ⊗ A a k ( p )) is an isomorphism and (cid:2) H i m ( M ⊗ A A a ) (cid:3) ν is a finitely generated projective module over A a for all ν ∈ G . Then, by setting U = D ( a ) ⊂ Spec( A ), the result follows from the fiberwise characterization of projective modules (see [8,Exercise 20.13]). (cid:3) Closing the section, we thought it appropriate to provide a counter-example to the resultstated in [27, Corollary 1.3]. The example shows that when A is only reduced and not a domain,generic freeness of the local cohomology modules H i m ( M ) may fail to hold. Example 3.8.
Let k be a field and A be the reduced Noetherian ring A = k [ t ]( t ( t − . Let R bethe polynomial ring R = A [ x ] and let m be the graded irrelevant ideal m = ( x ) ⊂ R .(i) Take M as M = R ( x,t ) = A ( t ) . It is clear that M = H m ( M ). Then, for any g ∈ A avoidingthe minimal primes of A , 0 = (cid:16) A ( t ) (cid:17) g is a projective A g -module but not a free A g -module.(ii) Take M as M = R ( t ) = A ( t ) [ x ] . One has that H m ( M ) = x (cid:16) A ( t ) [ x − ] (cid:17) . Then, for any g ∈ A avoiding the minimal primes of A , 0 = H m ( M ) g is a projective A g -module but not a free A g -module.4. Local cohomology of general fibers: bigraded case
The following setup will be used throughout the section.
Setup 4.1.
Let A be a reduced Noetherian ring. Consider the ( Z × Z )-bigraded polynomialring R = A [ x , . . . , x r , y , . . . , y s ], where bideg( x i ) = ( δ i ,
0) with δ i > y i ) = ( − γ i , γ i ≥
0. Consider m = ( x , . . . , x r ) R ⊂ R as a ( Z × Z )-bigraded ideal and recall thatH r m ( R ) ≃ x · · · x r A [ x − , . . . , x − r , y , . . . , y s ] . Let S be the standard graded polynomial ring given by S := A (cid:2) y i | ≤ i ≤ s and γ i = 0 (cid:3) ⊂ A [ y , . . . , y s ] ⊂ R . If M is a ( Z × Z )-bigraded module over R , then, for any i ≥
0, the local cohomology moduleH i m ( M ) has a natural structure of bigraded R -module. Also, denote by [ M ] j the Z -graded S -module(5) [ M ] j = M ν ∈ Z [ M ] ( j,ν ) . Remark 4.2.
As a particular, but important case, take M = R . Let { y i , . . . , y i l } ⊂ { y , . . . , y s } stand for the subset of variables with strictly negative x -degree, that is, bideg( y i t ) = ( − γ i t , − γ i t <
0. Then, for a fixed j ∈ Z , [H r m ( R )] j = L ν ∈ Z [H r m ( R )] ( j,ν ) is a finitely generated Z -graded S -module with a finite set of generators given by ( x α ··· x αrr y β i · · · y β l i l α ≥ , . . . , α r ≥ , β ≥ , . . . , β l ≥ , − ( α δ + · · · + α r δ r + β γ i + · · · + β l γ i l ) = j ) . Fix the following additional notation for the section.
Notation 4.3.
Let M be a finitely generated bigraded R -module and choose a bigraded freeresolution F • : · · · φ −→ F φ −→ F → M → F i is a finitely generated bigraded free R -module. Let L • = H r m ( F • ) : · · · Ψ −−→ L −−→ L be the induced complex in local cohomologywhere L i = H r m ( F i ) and Ψ i = H r m ( φ i ) : L i → L i − for i ≥ Lemma 4.4.
Under
Setup 4.1 and with the above notation, the following statements hold: (i)
There is an isomorphism H i m ( M ) ≃ H r − i (cid:0) L • (cid:1) of bigraded R -modules for i ≥ . (ii) (cid:2) H i m ( M ) (cid:3) j ≃ (cid:2) H r − i (cid:0) L • (cid:1)(cid:3) j is a finitely generated graded S -module for i ≥ and j ∈ Z . (iii) There is a dense open subset V ⊂ Spec( A ) such that, for every p ∈ V , there is an isomor-phism H i m (cid:0) M ⊗ A k ( p ) (cid:1) ≃ H r − i (cid:0) L • ⊗ A k ( p ) (cid:1) of bigraded (cid:0) R ⊗ A k ( p ) (cid:1) -modules for i ≥ . (iv) Fix an integer j ∈ Z . Then, there exists an element a ∈ A avoiding the minimal primes of A such that [( C i ( L • )) a ] j is a projective module over A a for ≤ i ≤ r .Proof. For (i) and (ii) see [4, Theorem 2.1].(iii) The argument is similar to the one in Lemma 3.4.(iv) Fix 0 ≤ i ≤ r . From Remark 4.2, one has that [ C i ( L • )] j is a finitely generated graded S -module. Therefore, Corollary 2.10 yields the existence of an element a i ∈ A avoiding theminimal primes of A such that (cid:2) ( C i ( L • )) a i (cid:3) j is a projective A a i -module. The required resultfollows by taking a = a a · · · a r . (cid:3) Next is the main result of this section. Its proof is very short as it is downplayed by theprevious lemma and its predecessors.
Theorem 4.5.
Under
Setup 4.1 , let M be a finitely generated bigraded R -module and fix aninteger j ∈ Z . Then, there exists a dense open subset U j ⊂ Spec( A ) such that, for all i ≥ , ν ∈ Z ,the function Spec( A ) −→ Z , p ∈ Spec( A ) dim k ( p ) (cid:2) H i m (cid:0) M ⊗ A k ( p ) (cid:1)(cid:3) ( j,ν ) ! is locally constant on U j .Proof. By Lemma 4.4(iii) one can choose a dense open subset U ⊂ Spec( A ) such that H i m (cid:0) M ⊗ A k ( p ) (cid:1) ≃ H r − i (cid:0) L • ⊗ A k ( p ) (cid:1) . By Remark 2.3, one has an exact sequence0 → H i m (cid:0) M ⊗ A k ( p ) (cid:1) → C r − i ( L • ) ⊗ A k ( p ) → L r − i − ⊗ A k ( p ) → C r − i − ( L • ) ⊗ A k ( p ) → p ∈ U . Therefore, the result is clear by setting U j = U ∩ V j with V j ⊂ Spec( A ) a denseopen subset as in Lemma 4.4(iv). (cid:3) ENERIC FREENESS OF LOCAL COHOMOLOGY AND GRADED SPECIALIZATION 13
The following example shows that in the current setting one can only hope to control certaingraded parts, as in the result of Theorem 4.5. It also shows the crucial importance of the choiceof bigrading in Setup 4.1 for the correctness of Theorem 4.5.
Example 4.6 ([17, Theorem 1.2]) . Since the example is slightly long, for organizational pur-poses, we divide it into four different parts. The first three parts are intended to stress strangephenomena that can happen if we weaken some of the previous settings.
Part 1:
This part shows that the result of Theorem 3.6(III) may fail if we consider a moregeneral situation (this part should be compared with
Part 4 below).Let k be a field and R be the graded k -algebra R = k [ s, t, x, y, u, v ]( xsx v − ( t + s ) xyuv + ty u )with grading deg( s ) = deg( t ) = deg( x ) = deg( y ) = 0 and deg( u ) = deg( v ) = 1. Then, for every d ≥
2, one has that h H R + ( R ) i − d has τ d − -torsion where τ d − = ( − d − ( t d − + st d − + · · · + s d − t + s d − ) ∈ k [ s, t ] . By [17, Lemma 1.1(ii)], it gives rise to infinitely many irreducible homogeneous polynomials { p i ∈ k [ s, t ] | i ≥ } such that H R + ( R ) has p i -torsion. Furthermore, from [17, proof of Theorem1.2], each ( p i ) yields an associated prime of H R + ( R ) in k [ s, t ]. Therefore, one cannot find anelement 0 = a ∈ k [ s, t ] such that H R + ( R ) ⊗ k [ s,t ] k [ s, t ] a is a projective k [ s, t ] a -module. Part 2:
In this part, we specify the current example in a bigraded setting that agrees withSetup 4.1, and we show that Theorem 4.5 cannot be extended to control all the possible gradedparts.Suppose that A = k [ s, t ] and that R is the standard bigraded A -algebra R = A [ u, v, x, y ] / (cid:0) sx v − ( t + s ) xyuv + ty u (cid:1) with bideg( u ) = bideg( v ) = (1 ,
0) and bideg( x ) = bideg( y ) = (0 , U ⊂
Spec( A ) such that the function p ∈ Spec( A ) dim k ( p ) (cid:18)h H u,v ) ( R ) i ( j,ν ) ⊗ A k ( p ) (cid:19) is constant on U for all j, ν ∈ Z , then there exists an element 0 = a ∈ A such that the A a -module h H u,v ) ( R ) i ( j,ν ) ⊗ A A a is projective for all j, ν ∈ Z (see [8, Exercise 20.13]). But, this conclusioncontradicts the assertion shown in Part 1 . Part 3:
In this part, we provide a bigrading that does not agree with Setup 4.1 and for whichthe statement of Theorem 4.5 would be incorrect (this bigrading is deduced for the bigradingused in [17, proof of Theorem 1.2]).First, we need to recall some details from the construction made in [17, proof of Theorem1.2]. Consider R as a Z -graded ring by setting trideg( s ) = trideg( t ) = (0 , , x ) =trideg( y ) = (1 , ,
0) and trideg( u ) = trideg( v ) = (0 , , d ≥ k [ s, t ]-module(6) h H u,v ) ( R ) i ( d, ∗ , − d ) has τ d − -torsion(see the argument made for “Coker A d − ” considered in [17, Proof of Theorem 1.2], and notethat “(Coker A d − ) ( d,d ) = Coker B d − ” is a k [ s, t ]-submodule of h H u,v ) ( R ) i ( d, ∗ , − d ) above). As in
Part 2 , consider R as a bigraded algebra over A = k [ s, t ], but now set bideg( u ) =bideg( v ) = (1 ,
0) and bideg( x ) = bideg( y ) = (1 , Z -grading via the map π : Z → Z , ( n , n , n ) ( n + n , n ).In other words, under this new bigrading the graded part h H u,v ) ( R ) i ( a,b ) can be described withthe following direct sum(7) h H u,v ) ( R ) i ( a,b ) = M a ,a ∈ Z a + a = a h H u,v ) ( R ) i ( a , b, a ) . Combining (6) and (7) it follows that h H u,v ) ( R ) i (0 , ∗ ) has τ d − -torsion for all d ≥
2. Therefore,we obtain that Theorem 4.5 fails when setting j = 0 and M = R with the current bigrad-ing. The fact that Theorem 4.5 is not valid in this case is somehow not surprising because h H u,v ) ( k [ s, t, x, y, u, v ]) i (0 , ∗ ) is then an infinitely generated A -module with a set generators n u α v α s β t β x γ y γ | γ + γ − α − α = 0 o ;this is an opposite situation to Remark 4.2. Part 4:
On the other hand, similarly to the previous Section 3, set S = k [ s, t, x, y ] andsuppose that R is the standard graded S -algebra R = S [ u, v ] / (cid:0) sx v − ( t + s ) xyuv + ty u (cid:1) .Then, Theorem 3.6(IV) implies the existence of an element 0 = b ∈ S such thatH u,v ) ( R ) ⊗ S S b is a free S b -module. Additionally, Proposition 3.7 gives a dense open subset V ⊂
Spec( S ) suchthat the function q ∈ Spec( S ) dim k ( q ) (cid:18)h H u,v ) ( R ) i j ⊗ S k ( q ) (cid:19) is constant on V for all j ∈ Z . 5. Specialization
In this section, we focus on various specialization environments, where the main results areobtained as an application of the previous sections.5.1.
Powers of a graded module.
In this part we look at the situation of a given gradedmodule and its symmetric and Rees powers. More precisely, we consider the problem of thelocal behavior of the following gadgets:(I) Local cohomology of a general fiber for all the symmetric powers of a module.(II) Local cohomology of a general specialization for all the Rees powers of a module.The main results in this regard turn out to be obtainable as an application of Theorem 4.5.Throughout this section the following simplified setup will be assumed.
Setup 5.1.
Let A be a Noetherian reduced ring. Let R be a finitely generated graded A -algebrawhich is positively graded (i.e., N -graded). Let m be the graded irrelevant ideal m = [ R ] + . ENERIC FREENESS OF LOCAL COHOMOLOGY AND GRADED SPECIALIZATION 15
Symmetric powers.
Quite generally, if M is a finitely generated R -module with a freepresentation F ϕ −→ F → M → , associated to a set of generators of M with s elements, then the symmetric algebra of M over R has a presentation Sym R ( M ) ≃ B / L , where B := R [ y , . . . , y s ] is a polynomial ring over R and L = I (cid:16) [ y , . . . , y s ] · ϕ (cid:17) .Now, if M is moreover graded, one has a presentation which is graded, where, say, F = L sj =1 R ( − µ j ). Fix an integer b ≥ max { µ , . . . , µ s } , consider the shifted module M ( b ) withcorresponding graded free presentation F ( b ) ϕ −→ s M j =1 R ( b − µ j ) → M ( b ) → . Then, the symmetric algebra Sym R ( M ( b )) is naturally a bigraded A -algebra with the same sortof presentation as above, only now B has a bigraded structure with bidegrees bideg( x ) = ( ν, x ∈ [ R ] ν ⊂ B and bideg( y j ) = ( µ j − b,
1) for 1 ≤ j ≤ s .Clearly, then(8) [Sym R ( M ( b ))] ( j,k ) ≃ h Sym kR ( M ) i j + kb for k ≥ , j ∈ Z , where Sym kR ( M ) denotes the k -th symmetric power of M .Let T = A [ x , . . . , x r ] be a standard graded polynomial ring mapping onto R , set in addition A = T [ y , . . . , y s ], with a bigrading given in the same way as for B . Therefore, one has thefollowing surjective bihomogeneous homomorphisms(9) A ։ B ։ Sym R ( M ( b )) . Notation 5.2. If M is a finitely generated graded R -module, let β ( M ) denote the maximaldegree of an element in a minimal set of generators of M . Thus, by the graded version ofNakayama’s lemma one has β ( M ) := max { k ∈ Z | [ M/ m M ] k = 0 } . One has the following theorem as an application of Theorem 4.5 and the above considerations.
Theorem 5.3.
Under
Setup 5.1 , let M be a finitely generated graded R -module and let j be afixed integer. Given b ∈ Z such that b ≥ β ( M ) , there exists a dense open subset U j ⊂ Spec( A ) such that, for all i ≥ , k ≥ , the function Spec( A ) −→ Z , p ∈ Spec( A ) dim k ( p ) (cid:18)h H i m (cid:16) Sym kR ⊗ A k ( p ) ( M ⊗ A k ( p )) (cid:17)i j + kb (cid:19) is locally constant on U j .Proof. Drawing on the assumption that b ≥ β ( M ) and (9), one applies the statement ofTheorem 4.5 by taking the bigraded module there to be Sym R ( M ( b )). Since one has the iso-morphism Sym R ( M ) ⊗ A k ( p ) ≃ Sym R ⊗ A k ( p ) ( M ⊗ A k ( p )) , the result follows from (8). (cid:3) Rees powers.
Here the notation and terminology are the ones of [26]. In particular, the
Rees algebra R R ( M ) of a finitely generated R -module M having rank is defined to be thesymmetric algebra modulo its R -torsion. With this definition, R R ( M ) inherits from Sym R ( M )a natural bigraded structure.There are a couple of ways to introduce the k -th power of M : M k := [ R R ( M )] ( ∗ ,k ) = M j ∈ Z [ R R ( M )] ( j,k ) ≃ Sym kR ( M ) /τ R (Sym kR ( M )) , where τ R denotes R -torsion.In addition, there is an R -embedding ι k : M k = (cid:2) R R ( M ) (cid:3) ( ∗ ,k ) ֒ → (cid:2) R [ t , . . . , t m ] (cid:3) ( ∗ ,k ) out ofan embedding(10) R R ( M ) ֒ → Sym R ( F ) ≃ R [ t , . . . , t m ] , induced by a given embedding of M = M/τ R ( M ) into a free R -module F of rank equal to therank of M . Definition 5.4.
Let M be a finitely generated graded R -module having rank. For p ∈ Spec( A )and k ≥
0, the specialization of M k with respect to p is the following R ⊗ A k ( p )-module S p ( M k ) := Im (cid:16) ι k ⊗ A k ( p ) : M k ⊗ A k ( p ) → (cid:2) ( R ⊗ A k ( p ))[ t , . . . , t m ] (cid:3) ( ∗ ,k ) (cid:17) . If no confusion arises, one sets S p ( M ) := S p ( M ). Proposition 5.5.
Let M be a finitely generated graded R -module having rank. Then, there isa dense open subset V ⊂ Spec( A ) such that, for all p ∈ V and k ≥ , one has S p ( M k ) ≃ M k ⊗ A k ( p ) . In particular, S p ( M k ) is independent of the chosen embedding M ֒ → F .Proof. From (10), consider the short exact sequence0 → R R ( M ) → R [ t , . . . , t m ] → R [ t , . . . , t m ] R R ( M ) → . By using Theorem 3.5(i) (as applied in the notation there with M = B = R [ t , . . . , t m ], E = R R ( M ) and H = 0) choose a ∈ A avoiding the minimal primes of A such that R [ t ,...,t m ] R R ( M ) ⊗ A A a is a projective A a -module. So, the result follows by setting V = D ( a ) ⊂ Spec( A ). (cid:3) Corollary 5.6.
Under
Setup 5.1 , let M be a finitely generated graded R -module having rank.Then, there exists a dense open subset U ⊂
Spec( A ) such that, for all i ≥ , j ∈ Z , the function Spec( A ) −→ Z , p ∈ Spec( A ) dim k ( p ) (cid:2) H i m (cid:0) S p ( M ) (cid:1)(cid:3) j ! is locally constant on U .Proof. It follows from Proposition 3.7 and Proposition 5.5. (cid:3)
Next is the principal result about the specialization of the Rees powers of a graded module.The proof is again short because it is downplayed by the use of previous theorems.
Theorem 5.7.
Under
Setup 5.1 , let M be a finitely generated graded R -module having rank.Fix an integer j ∈ Z and let b ∈ Z be an integer such that b ≥ β ( M ) . Then, there exists a denseopen subset U j ⊂ Spec( A ) such that, for all i ≥ , k ≥ , the function Spec( A ) −→ Z , p ∈ Spec( A ) dim k ( p ) (cid:18)h H i m (cid:16) S p ( M k ) (cid:17)i j + kb (cid:19) is locally constant on U j .Proof. One extends (9) to the following surjective bihomogeneous homomorphisms A ։ B ։ Sym R ( M ( b )) ։ R R ( M ( b )) . Note that [ R R ( M ( b ))] ( j,k ) ≃ (cid:2) M k (cid:3) j + kb for all j ∈ Z , k ≥
0. One sets R R ( M ( b )) to be the bi-graded module in the statement of Theorem 4.5. Then, let U j ⊂ Spec( A ) be a dense open subsetobtained from Theorem 4.5. Let V ⊂ Spec( A ) be a dense open subset from Proposition 5.5.Therefore, the result follows by setting U j = U j ∩ V . (cid:3) ENERIC FREENESS OF LOCAL COHOMOLOGY AND GRADED SPECIALIZATION 17
Rational maps and the saturated special fiber.
In this section one revisits the prob-lem of specialization of rational maps, as studied in [7]. We recover some of the results thereas a consequence of Theorem 5.7 and [3, Corollary 2.12]. Quite naturally, one also studies thesaturated special fiber ring and the j -multiplicity of a general specialization of an ideal.For the basics of rational maps with source and target projective varieties defined over anarbitrary Noetherian domain, the reader is referred to [7, Section 3].Throughout this section the following setup is used. Setup 5.8.
Let A be a Noetherian domain and R be the standard graded polynomial ring R = A [ x , . . . , x r ]. Fix homogeneous elements { g , . . . , g s } ⊂ R of the same degree d > G : P rA P sA denote the corresponding rational map given by the representative g = ( g : · · · : g s ). Set m = [ R ] + = ( x , . . . , x r ) ⊂ R .We specialize this rational map as follows. Given p ∈ Spec( A ), take the rational map G ( p ) : P rk ( p ) P sk ( p ) with representative π p ( g ) = ( π p ( g ) : · · · : π p ( g s )) , where π p ( g i ) is the image of g i under the canonical map π p : R → R ⊗ A k ( p ).Set I = ( g , . . . , g s ) ⊂ R and note that S p ( I ) = ( π p ( g ) , . . . , π p ( g s )) ⊂ R ⊗ A k ( p ) and that S p ( I k ) = S p ( I ) k ⊂ R ⊗ A k ( p ), for p ∈ Spec( A ) , k ≥ p ∈ Spec( A ), let Y ⊂ P sA and Y ( p ) ⊂ P sk ( p ) denote the respective closed images of G and of G ( p ).Recall that the rational map G ( p ) is generically finite if one of the following equivalent con-ditions is satisfied:(i) The field extension K ( Y ( p )) ֒ → K ( P rk ( p ) ) is finite, where K ( P rk ( p ) ) and K ( Y ( p )) denote thefields of rational functions of P rk ( p ) and Y ( p ), respectively.(ii) dim( Y ( p )) = dim( P rk ( p ) ) = r .(iii) The analytic spread ℓ (cid:0) S p ( I ) (cid:1) := dim (cid:16) R R ⊗ A k ( p ) ( S p ( I )) / m R R ⊗ A k ( p ) ( S p ( I )) (cid:17) of S p ( I ) at-tains the maximum possible value dim ( R ⊗ A k ( p )) = r + 1 . The degree of G ( p ) is defined as deg( G ( p )) := h K ( P rk ( p ) ) : K ( Y ( p )) i . Definition 5.9 ([1]) . For any p ∈ Spec( A ) and any homogeneous ideal J ⊂ R ⊗ A k ( p ), the j -multiplicity of J is given by j ( J ) := r ! lim n →∞ dim k ( p ) (cid:16) H m (cid:0) J n /J n +1 (cid:1) (cid:17) n r . Definition 5.10 ([3]) . For any p ∈ Spec( A ) and any homogeneous ideal J ⊂ R ⊗ A k ( p ) generatedby elements of the same degree d >
0, the saturated special fiber ring of J is given by ^ F R ⊗ Ak ( p ) ( J ) := ∞ M n =0 (cid:2)(cid:0) J n : m ∞ (cid:1)(cid:3) nd . Next is the main result of this section.
Theorem 5.11.
Under
Setup 5.8 , assume in addition that G ((0)) is generically finite. Then,there exists a dense open subset U ⊂
Spec( A ) such that G ( p ) is generically finite for any p ∈ U and the functions (i) p ∈ Spec( A ) deg ( G ( p )) , (ii) p ∈ Spec( A ) deg P sk ( p ) ( Y ( p )) , (iii) p ∈ Spec( A ) e (cid:16) ^ F R ⊗ Ak ( p ) ( S p ( I )) (cid:17) and (iv) p ∈ Spec( A ) j ( S p ( I )) are constant on U .Proof. We first argue for (i) and (ii). By Proposition 5.5 there exists a dense open subset U ⊂ Spec( A ) such that R R ⊗ Ak ( p ) (cid:0) S p ( I ) (cid:1) = ∞ M k =0 S p ( I ) k ≃ R R ( I ) ⊗ A k ( p ) ≃ ∞ M k =0 I k ⊗ A k ( p )for all p ∈ U . One has an isomorphism Y ( p ) ≃ Proj (cid:16) k ( p ) (cid:2) π p ( g ) , . . . , π p ( g s ) (cid:3)(cid:17) (see, e.g., [7,Definition-Proposition 3.12]). By restricting to the zero graded part in the R -grading, we obtainthe following isomorphisms of graded k ( p )-algebras k ( p ) (cid:2) π p ( g ) , . . . , π p ( g s ) (cid:3) ≃ h R R ⊗ Ak ( p ) (cid:0) S p ( I ) (cid:1)i ≃ [ R R ( I )] ⊗ A k ( p )for any p ∈ U (as before in (5), one uses the notation [ R R ( I )] = L ∞ ν =0 [ R R ( I )] (0 ,ν ) ).By Theorem 3.5(ii), as applied with M := R R ( I ), there is an element 0 = a ∈ A such thatall the graded components of [ R R ( I )] ⊗ A A a are free A a -modules. Set V = D ( a ) ⊂ Spec( A ).Since G ((0)) is generically finite, one has dim (cid:0) [ R R ( I )] ⊗ A k ((0)) (cid:1) = dim ( R ⊗ A k ((0))), and soit follows thatdim (cid:0) [ R R ( I )] ⊗ A k ( p ) (cid:1) = dim (cid:0) [ R R ( I )] ⊗ A k ((0)) (cid:1) = dim ( R ((0))) = dim ( R ⊗ A k ( p ))and thatdeg P sk ( p ) ( Y ( p )) = e (cid:0) [ R R ( I )] ⊗ A k ( p ) (cid:1) = e (cid:0) [ R R ( I )] ⊗ A k ((0)) (cid:1) = deg P sk ((0)) (cid:0) Y (0) (cid:1) for any p ∈ U ∩ V .For any p ∈ U ∩ V , [3, Corollary 2.12] yields the formuladeg P sk ( p ) ( Y ( p )) (deg ( G ( p )) −
1) = r ! lim k →∞ dim k ( p ) (cid:16)h H m (cid:16) S p ( I k ) (cid:17)i kd (cid:17) k r . Let W ⊂ Spec( A ) be a dense open subset obtained from Theorem 5.7 with M := I ( d ). It thenfollows that the function p ∈ Spec( A ) deg P sk ( p ) ( Y ( p )) (deg ( G ( p )) − U ∩ V ∩ W. So, the result follows by taking U = U ∩ V ∩ W .(iii) It follows from parts (i), (ii) and [3, Theorem 2.4].(iv) It follows from parts (i), (ii) and [21, Theorem 5.3]. (cid:3) Numerical invariants.
The goal is to show that dimension, depth, a -invariants and reg-ularity of a module are locally constant under tensor product with a general fiber and generalspecialization. As a side-result, we provide a slight improvement of the upper semi-continuitytheorem (see [10, Chapter III, Theorem 12.8])) for the dimension of sheaf cohomology of ageneral fiber.For a finitely generated graded R -module M the i -th a -invariant is defined as(11) a i ( M ) := ( max (cid:8) n | (cid:2) H i m ( M ) (cid:3) n = 0 (cid:9) if M = 0 −∞ if M = 0and the Castelnuovo–Mumford regularity is given by(12) reg( M ) := max (cid:8) a i ( M ) + i | i ≥ (cid:9) . We first state the local behavior of the numerical invariants for the fibers.
ENERIC FREENESS OF LOCAL COHOMOLOGY AND GRADED SPECIALIZATION 19
Proposition 5.12.
Under
Setup 5.1 , let M be a finitely generated graded R -module. Then,there exists a dense open subset U ⊂
Spec( A ) such that the functions (i) p ∈ Spec( A ) dim ( M ⊗ A k ( p )) , (ii) p ∈ Spec( A ) depth ( M ⊗ A k ( p )) , (iii) p ∈ Spec( A ) a i ( M ⊗ A k ( p )) for i ≥ , and (iv) p ∈ Spec( A ) reg ( M ⊗ A k ( p )) are locally constant on U .Proof. It follows from Proposition 3.7, [2, Corollary 6.2.8], (11) and (12). (cid:3)
Next is the local behavior of the numerical invariants for the specialization.
Proposition 5.13.
Under
Setup 5.1 , let M be a finitely generated graded R -module having rank.Then, there exists a dense open subset U ⊂
Spec( A ) such that the functions (i) p ∈ Spec( A ) dim ( S p ( M )) , (ii) p ∈ Spec( A ) depth ( S p ( M )) , (iii) p ∈ Spec( A ) a i ( S p ( M )) for i ≥ and (iv) p ∈ Spec( A ) reg ( S p ( M )) are locally constant on U .Proof. It follows from Corollary 5.6, [2, Corollary 6.2.8], (11) and (12). (cid:3)
An additional outcome is a slight improvement of the upper semicontinuity theorem.
Proposition 5.14.
Let A be denote a reduced Noetherian ring. Let R be a standard gradedfinitely generated A -algebra and X := Proj( R ) . Given a finitely generated graded R -module M ,there exists a dense open subset U ⊂
Spec( A ) such that, for all i ≥ , n ∈ Z , the function Spec( A ) −→ Z , p ∈ Spec( A ) dim k ( p ) (cid:16) H i (cid:0) X × A k ( p ) , ^ M ( n ) ⊗ A k ( p ) (cid:1)(cid:17) is locally constant on U .Proof. For i ≥
1, one has that H i (cid:0) X × A k ( p ) , ^ M ( n ) ⊗ A k ( p ) (cid:1) ≃ (cid:2) H i +1 m ( M ⊗ A k ( p )) (cid:3) n (see, e.g.,[8, Theorem A4.1]), and so in this case the result is obtained directly from Proposition 3.7.For i = 0, one has the short exact sequence0 → (cid:2) H m ( M ⊗ A k ( p )) (cid:3) n → [ M ⊗ A k ( p )] n → H (cid:0) X × A k ( p ) , ^ M ( n ) ⊗ A k ( p ) (cid:1) → (cid:2) H m ( M ⊗ A k ( p )) (cid:3) n → U ⊂ Spec( A )such that dim k ( p ) ([ M ⊗ A k ( p )] n ) is locally constant for all p ∈ U . Take a dense open subset V ⊂ Spec( A ) given as in Proposition 3.7. So, the result follows in both cases by setting U = U ∩ V . (cid:3) Acknowledgments
The authors are indebted to the referee for a constructive criticism of the paper.
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Email address : [email protected] (Cid-Ruiz) Department of Mathematics: Algebra and Geometry, 9000 Ghent, Belgium. (Cid-Ruiz)
Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig,Germany.
Email address : [email protected] URL : https://ycid.github.io (Simis) Departamento de Matem´atica, CCEN, Universidade Federal de Pernambuco, 50740-560Recife, PE, Brazil
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