arXiv:2108.13986v2 [math.AG] 1 Sep 2021
YAIRON CID-RUIZ AND RITVIK RAMKUMAR
ABSTRACT. We introduce the fiber-full scheme which can be seen as the parameter space that general- izes the Hilbert and Quot schemes by controlling the entire cohomological data. The fiber-full scheme FibhF/X/Sis a fine moduli space parametrizing all quotientsGof a fixed coherent sheafFon a projective morphismf:X⊂PrS→Ssuch thatRif∗(G(ν))is a locally freeOS-module of rank equal tohi(ν), where h= (h0, . . . ,hr) :Zr+1→Nr+1is a fixed tuple of functions. In other words, the fiber-full scheme controls the dimension of all cohomologies of all possible twistings, instead of just the Hilbert polynomial. We show that the fiber-full scheme is a quasi-projectiveS-scheme and a locally closed subscheme of its correspond- ing Quot scheme. In the context of applications, we demonstrate that the fiber-full scheme provides the natural parameter space for arithmetically Cohen-Macaulay and arithmetically Gorenstein schemes with fixed cohomological data, and for square-free Gr¨obner degenerations.
1. INTRODUCTION
The Hilbert and Quot schemes were introduced by Grothendieck [17] and are probably among the most important constructions in algebraic geometry. The Quot scheme QuotPF/X/S is a fine moduli space parametrizing all quotients of a fixed coherent sheafFon a projective morphismf:X⊂PrS→S that are flat overSand have a fixed Hilbert polynomialP(m)∈Q[m]along all fibers; the Hilbert scheme is a special case withF=OX. These schemes can be used to construct other important moduli spaces such as the moduli of curves and the Picard scheme [26,34]. They are used to study deformations of curves which is an essential part of birational geometry [31,33]. They also feature prominently in other areas of mathematics. The Hilbert schemes of points on a singular plane curve and a surface are related to knot invariants [15,37]. The Hilbert scheme of points on a surface appears in representation theory [36], in combinatorics [18] and, if the surface is a K3-surface, this Hilbert scheme provides an important class of examples of hyperk¨ahler varieties [4]. The main goal of this paper is to introduce a far-reaching generalization of the Hilbert and Quot schemes that controls all the cohomological data of the quotients of the coherent sheafFinstead of just the Hilbert polynomial.
We start with the classical example of the Hilbert scheme compactification of the space of twisted cubics that was studied by Piene and Schlessinger [39]. The motivating example below shows how this well-studied Hilbert scheme decomposes into locally closed subschemes that have constant cohomolog- ical data.
Example 1.1 (Example 6.2). In [39], it was shown that Hilb3m+1
P3k =H∪H′ is a union of two smooth irreducible components such that the general member ofHparametrizes a twisted cubic, and the general
Date: September 2, 2021.
2010Mathematics Subject Classification. 14C05, 14D22, 13D02, 13D07, 13D45.
Key words and phrases. Hilbert scheme, Quot scheme, fiber-full scheme, fiber-full sheaf, flattening stratification, local cohomology, Ext module, arithmetically Cohen-Macaulay, arithmetically Gorenstein, square-free Gr¨obner degeneration.
1
member ofH′parametrizes a plane cubic union an isolated point. It is also known thatH−H∩H′is the locus of arithmetically Cohen-Macaulay curves of degree 3 and genus 0. We then have a decomposition
Hilb3m+1
P3k = (H−H∩H′)⊔H′. Furthermore, one can show that the functions
hiX:Z→N, ν7→dimk Hi(X,OX(ν))
are the same for any element[X]∈H−H∩H′ and the same for any element [X]∈H′(for an explicit computation, seeExample 6.2). It then follows that Hilb3m+1
P3k can be decomposed into two locally closed subschemes where the cohomological functionshiXare constant. It should also be noted that one might be quite interested in studying H−H∩H′ as it gives all the closed subschemes ofP3k with the same cohomological data as that of a twisted cubic.
As presented below, the scheme we introduce allows us to provide a unified and systematic treatment of the decomposition seen in Example 1.1. Let S be a locally Noetherian scheme, f:X⊂PrS →S be a projective morphism and Fbe a coherent sheaf onX. We follow Grothendieck’s general idea of considering a contravariant functor whose representing scheme (if it exists) is the parameter space one is interested in.
We define thefiber-full functorwhich for anS-schemeT parametrizes all coherent quotientsFT ։G such that all higher direct images ofGand its twistings are locally free overT. More precisely, for any (locally Noetherian)S-schemeT we define
FibF/X/S(T) =
coherent quotientFT ։G Rif(T)∗(G(ν)) is locally free overT for all 06i6r,ν∈Z
, whereFT is the pull-back sheaf onXT =X×ST andf(T):XT ⊂PrT →T is the base change morphism f(T)=f×ST. We have that
FibF/X/S : (Sch/S)opp→(Sets)
is a contravariant functor from the category of (locally Noetherian) S-schemes to the category of sets (seeLemma 5.2). We stratify this functor in terms of “Hilbert functions” for all the cohomologies. Let h= (h0, . . . ,hr) :Zr+1→Nr+1be a tuple of functions. Then, we define the following functor depending onh:
FibhF/X/S(T) =
G∈FibF/X/S(T) dimκ(t) Hi(Xt,Gt(ν))
=hi(ν) for all 06i6r,ν∈Z,t∈T
,
whereκ(t)denotes the residue field of the pointt∈T,Xt=XT×TSpec(κ(t))is the fiber overt∈T, and Gtis the pull-back sheaf onXt. The idea of this functor is to measure the dimension ofall cohomologies of all possible twistings. We easily obtain the stratificationFibF/X/S(T) = F
h:Zr+1→Nr+1FibhF/X/S(T) when T is connected, and so it follows that FibF/X/S(T) is a representable functor if all the functors FibhF/X/S(T)are representable. WhenF=OX, we simplify the notation by writingFibhX/Sinstead of FibhOX/X/S.
For any numerical polynomialP∈Q[m], we have Grothendieck’s definition of the Quot functor QuotPF/X/S: (Sch/S)opp→(Sets)
which for anS-schemeT parametrizes all coherent quotientsFT→Gthat are flat overTand have Hilbert polynomial equal toPalong all fibers. The Hilbert functorHilbPF/X/Sis the special case ofQuotPF/X/S with F=OX. Then, the fiber-full functor can be thought of as a refinement of the Hilbert and Quot functors due to the following inclusions. From the tuple of functions h= (h0, . . . ,hr) :Zr+1→Nr+1, we define the functionPh=Pr
i=0(−1)ihi. WhenPh(m)∈Q[m]is a numerical polynomial, since the Hilbert polynomial of a sheaf coincides with its Euler characteristic, we automatically get the inclusions
FibhX/S(T) ⊂ HilbPX/Sh (T) and FibhF/X/S(T) ⊂ QuotPF/X/Sh (T)
for any (locally Noetherian)S-schemeT. IfPhis not a numerical polynomial, thenFibhF/X/S(T) =∅for anyS-schemeT.
The following is the main theorem of this article. Here, we show that the functor FibhF/X/Sis rep- resented by a quasi-projective S-scheme that we call thefiber-full scheme and we write as FibhF/X/S. From the definition of FibhF/X/S, it follows that the fiber-full scheme FibhF/X/S is the finest possible generalization of the Quot scheme QuotPF/X/Sh if one is interested in controlling all the cohomological data.
Theorem A (Theorem 5.4). Let S be a locally Noetherian scheme, f:X⊂PrS →S be a projective morphism and Fbe a coherent sheaf onX. Leth= (h0, . . . ,hr) :Zr+1→Nr+1be a tuple of functions and suppose thatPh(m)∈Q[m]is a numerical polynomial. Then, there is a quasi-projectiveS-scheme FibhF/X/S that represents the functor FibhF/X/S and that is a locally closed subscheme of the Quot schemeQuotPF/X/Sh .
Our main tool for constructing the fiber-full scheme is given in Theorem 3.3 where we provide a flattening stratification theorem that deals with all the direct images of a sheaf and its possible twistings.
To prove this technical theorem we utilize some techniques previously developed in the papers [6,8].
In a related direction, we also introduce the notion of fiber-full sheaves and we give three equivalent possible definitions inTheorem 4.2. Under the above notation, we say thatF is afiber-full sheaf over SifRif∗(F(ν))is locally free overSfor all 06i6randν∈Z. Fiber-full sheaves serve as a sheaf- theoretic extension of the notions ofalgebras having liftable local cohomology[30] andcohomologically full rings[10].
It turns out there has been previous interest in stratifying the Hilbert scheme in terms of the whole cohomological data.
In the work of Martin-Deschamps and Perrin [32], they were able to control the cohomologies of a sheaf, but not all the possible twistings, as their method would yield the intersection of infinitely many (not necessarily closed) subschemes (see [32, Chapitre VI, Proposition 1.9 and Corollaire 1.10]); their approach is based on classical techniques related to the Grothendieck complex which are covered, e.g., in [21, §III.12].
In the thesis of Fumasoli [13,14], he stratified the Hilbert scheme by bounding below the cohomo- logical functions of the points of the Hilbert scheme, which is a consequence of the classical upper semicontinuity theorem (see [21, Theorem III.12.8]).
Our main resultTheorem Avastly generalizes the two aforementioned approaches and shows that one can indeed stratify the Hilbert and Quot schemes by taking into account all the cohomological data. In
this regard, one important part of our work is to develop the necessary tools that allow us to prove the general stratification result ofTheorem 3.3.
Next, we describe some applications that follow from the existence of the fiber-full scheme.
There is a large literature on the study of the loci ofarithmetically Cohen-Macaulay(ACM for short) schemes and the loci ofarithmetically Gorensteinscheme within the Hilbert scheme (see [12,22,27–29, 32] and the references therein). As a result of considering the fiber-full scheme, we can provide a finer description of these loci and parametrize ACM and AG schemes with a fixed cohomological data. Let d∈Nandh0,hd:Z→Nbe two functions, and consider the tuple of functionsh:Zr+1→Nr+1given by h= (h0, 0, . . . , 0,hd, 0, . . . , 0) where 0:Z→Ndenotes the zero function. To study ACM and AG schemes, since all the intermediate cohomologies vanish in these cases, it becomes natural to consider the following two functors. For any (locally Noetherian)S-schemeT, we have
ACMhX/S0,hd(T) =
closed subschemeZ⊂XT Z∈FibhX/S(T)andZtis ACM for allt∈T
and
AGhX/S0,hd(T) =
closed subschemeZ⊂XT Z∈FibhX/S(T)andZt is AG for allt∈T
. The following theorem shows the two functors above are representable, and so it provides the natural parameter spaces for ACM and AG schemes with fixed cohomological data.
Theorem B(Theorem 5.8). LetSbe a locally Noetherian scheme andf:X⊂PrS→S be a projective morphism. Let d∈N and h0,hd:Z→Nbe two functions, and consider the tuple of functions h= (h0, 0, . . . , 0,hd, 0, . . . , 0) :Zr+1→Nr+1. Suppose thatPh(m)∈Q[m]is a numerical polynomial. Then, there exist openS-subschemesACMhX/S0,hdandAGhX/S0,hdofFibhX/Sthat represent the functorsACMhX/S0,hd andAGhX/S0,hd, respectively.
We study several examples of Hilbert schemes that we stratify in terms of fiber-full schemes. The list includes the Hilbert scheme of points and classical examples such as the Hilbert scheme of twisted cubics and the Hilbert scheme of skew lines (seeSection 6). Furthermore, by using the recent classification of Skjelnes and Smith [46], we show inProposition 7.5that smooth Hilbert schemes coincide with a fiber- full scheme (i.e., cohomological data is constant for points in a smooth Hilbert scheme).
On top of being a generalization of the Hilbert scheme, the fiber-full scheme is also the natural pa- rameter space for square-freeGr¨obner degenerations, an important class of degenerations in algebraic geometry and commutative algebra.
Letkbe a field,S=k[x0, . . . ,xr]be a polynomial ring and>be a monomial order onS. Given a closed subschemeV(I)⊆Prk, a Gr¨obner degeneration is a flat family over Spec(k[t])with general fiber isomor- phic to Proj(S/I)and special fiber isomorphic to Proj(S/in>(I)). In particular, the natural parameter space to study Gr¨obner degenerations is the Hilbert scheme of projective space, and this point of view has been very fruitful in the study of these Hilbert schemes. For example, one proves the connectedness of Hilbert schemes, and more generally Quot schemes, by proving that any two points are connected by a chain of rational curves, each of which is a Gr¨obner degeneration [20,38]. Square-free Gr¨obner de- generations, those degenerations where in>(I)is square-free, have always been an important subclass of
Gr¨obner degenerations, especially since they are amenable to combinatorial techniques (Stanley-Reisner theory). By the recent work of Conca and Varbaro [9] the cohomological data is constant along the fibers of a square-free Gr¨obner degeneration. Thus, the fiber-full scheme is the natural parameter space to study square-free Gr¨obner degenerations, as evidenced byCorollary 8.4. To phrase this in another way, when the special fiber of a Gr¨obner degeneration is square-free, the whole degeneration lies inside the same fiber-full scheme.
Our construction of the fiber-full scheme generates many interesting questions on the structure and geometry of these schemes. One that is particularly interesting, and which we plan to address in a forthcoming paper, is to construct areasonablemodular compactification of the fiber-full scheme. Unlike the Hilbert scheme, fixing the cohomological data, in many cases, ensures that the fiber-full scheme does not contain extraneous components. Ideally, one would desire it to be the closure of the fiber-full scheme inside the Hilbert scheme. Not only would this have quite a few enumerative applications, it would also provide a modular description for various interesting loci such as arithmetically Cohen-Macaulay schemes (cf. [23,24]), arithmetically Gorenstein schemes, twisted quartic curves and so on.
Organization. InSection 2andSection 3, we provide flattening stratifications for various modules and complexes; the most important being the local cohomology modules (Theorem 2.16) and the higher direct image sheaves (Theorem 3.3). In Section 4 we develop the notion of a fiber-full sheaf and in Section 5we construct the fiber-full scheme (Theorem 5.4). InSection 6andSection 7, we provide sev- eral examples of fiber-full schemes. Finally, inSection 8, we consider square-free Gr¨obner degenerations and we relate them to the fiber-full scheme.
2. SOME FLATTENING STRATIFICATION THEOREMS IN A GRADED CATEGORY OF MODULES
In this section, we provide several flattening stratification theorems in a graded category modules;
the list includes: the case of modules, cohomology of complexes of modules, Ext modules and local cohomology modules. For organizational purposes, we divide the section into four different subsections.
2.1. Flattening stratification of modules. In this subsection, we concentrate on an extension for mod- ules of the flattening stratification theorem given in [2] (also, see [35, §8]). Throughout this subsection, we shall use the following setup.
Setup 2.1. LetAbe ring (always assumed to be commutative and unitary) andRbe a finitely generated gradedA-algebra. For anyp∈Spec(A), letκ(p) :=Ap/pApbe the residue field ofp.
For a gradedR-module M, we say thatMhas aHilbert function over Aif for allν∈Zthe graded part [M]ν is a finitely generated locally freeA-module of constant rank on Spec(A); and in this case, the Hilbert function is hM:Z→N, hM(ν) =rankA([M]ν). If a graded R-module Mhas a Hilbert function overA, thenM⊗ABhas the same Hilbert function over anyA-algebraB.
Remark 2.2. Let 0→L→M→N→0 be a short exact sequence of gradedR-modules.
(i) IfLandNhave Hilbert functions overA, thenMhas a Hilbert function overAgiven byhM(ν) = hL(ν) +hN(ν).
(ii) IfMandNhave Hilbert functions overA, thenLhas a Hilbert function overAgiven byhL(ν) = hM(ν) −hN(ν).
For completeness, we recall the following flatness result.
Lemma 2.3. Assume thatAis Noetherian. LetMbe a finitely generated gradedA-module. Then, the following locus
UM:=
p∈Spec(A)|M⊗AApis a flatAp-module is an open subset ofSpec(A).
Proof. For a proof, see [2, Lemma 2.1] or [6, Lemma 2.5].
For a given gradedR-moduleMand a functionh:Z→N, we consider the following functor for any ringB,
FhM(B) :=
morphism Spec(B)→Spec(A) [M⊗AB]ν is a locally freeB-module of rankh(ν)for allν∈Z
. We now describe our first flattening stratification theorem.
Theorem 2.4. AssumeSetup 2.1 withANoetherian. LetM be a finitely generated gradedR-module andh:Z→Nbe a function. Then, the following statements hold:
(i) The functorFhMis represented by a locally closed subschemeFhM⊂Spec(A). In other words, for any morphismg:Spec(B)→Spec(A), M⊗ABhas a Hilbert function overBequal to hif and only ifgcan be factored as
Spec(B) → FhM → Spec(A).
(ii) There is only a finite number of different functionsh1, . . . ,hm:Z→Nsuch thatFhMi 6=∅, and so Spec(A)is set-theoretically equal to the disjoint union of the locally closed subschemesFhMi. Proof. (i) For any morphism Spec(B)→Spec(A), one has that[M⊗AB]νis locally free of rankh(ν) if and only if Fitth(ν)−1([M⊗AB]ν) =0 and Fitth(ν)([M⊗AB]ν) =B, and that Fittj([M⊗AB]ν) =
Fittj([M]ν)
B(for more details on Fitting ideals, see [11, §20.2]).
LetZhM⊂Spec(A)be the closed subscheme given byZhM:=Spec A/ P
ν∈ZFitth(ν)−1([M]ν) . We have that Fitth(ν)−1([M⊗AB]ν) =0 for allν∈Zif and only if Spec(B)→Spec(A)factors through ZhM. Therefore, we can substitute Spec(A)byZhM, and we do so.
Letp∈UMand suppose thatM⊗AAphas a Hilbert functionhM⊗AAp=hoverAp. ByLemma 2.3, there is an affine open neighborhoodp∈Spec(Aa)⊂UM ofpfor somea∈A. Thus [47,Tag 00NX]
implies that for allν∈Zthe function Spec(Aa)→N,q7→dimκ(q)([M⊗Aaκ(q)]ν)is locally constant.
Consequently, there is an open connected neighborhoodp∈V⊂Spec(Aa)ofpsuch thathM⊗AAq =h for allq∈V. It then follows that the following locus
UhM:=
p∈Spec(A)|M⊗AAphas a Hilbert functionhM⊗AAp=hoverAp
is an open subset of Spec(A).
Note that Fitth(ν)([M⊗AB]ν) =B if and only if P∩A6⊃Fitth(ν)([M]ν) for all P∈Spec(B).
Hence, under the condition Fitth(ν)−1([M⊗AB]ν) =0 for allν∈Z, it follows that Fitth(ν)([M⊗A
B]ν) =Bfor allν∈Zif and only if Spec(B)→Spec(A)factors throughUhM. So, after having changed Spec(A)byZhM, we have thatFhMis represented by the open subscheme UhM⊂Spec(A). This com- pletes the proof of this part.
(ii) For eachp∈Spec(A), lethp be the functionhp(ν) :=dimκ(p)([M⊗Aκ(p)]ν). As we have a natural morphism Spec(κ(p))→Spec(A), it clearly follows thatp∈FhMp. Therefore, by the Noetherian hypothesis, we can show that there is a finite number of distinct functions h1, . . . ,hm such that set- theoretically we have the equality Spec(A) =Fm
i=1FhMi.
2.2. Flattening stratification of the cohomologies of a complex. Here we study how the process of taking tensor product with another ring affects the cohomology of a bounded complex. In this subsection, we continue usingSetup 2.1. The notation below will be used throughout the paper.
Notation 2.5. For a (co-)complex ofA-modulesK•:· · · →Ki−1−−−→φi−1 Ki φ−→i Ki+1→ · · ·, one defines Zi(K•) :=Ker(φi), Bi(K•) :=Im(φi−1), Hi(K•) :=Zi(K•)/Bi(K•), and Ci(K•) :=Ki/Bi(K•) ⊃ Hi(K•)for alli∈Z. We use analogous notation with lower indices for a complexK•.
Remark 2.6. A basic result that we shall use several times is the following: for a complex ofA-modules K•and anA-moduleN, we have a four-term exact sequence
0→Hi K•⊗AN
→Ci(K•)⊗AN→Ki+1⊗AN→Ci+1(K•)⊗AN→0 ofA-modules.
The following lemma transfers the burden of studying the cohomologies of a bounded complex to considering the cokernels of the maps.
Lemma 2.7. Let K• :0→K0→ K1→ · · · →Kp→ 0 be a bounded complex of graded R-modules.
Suppose that eachKihas a Hilbert function overA. LetSpec(B)→Spec(A)be a morphism. Then, the following two conditions are equivalent:
(1) Hi(K•⊗AB)has a Hilbert function overBfor all06i6p.
(2) Ci(K•)⊗ABhas a Hilbert function overBfor all06i6p.
Moreover, if any of the above conditions are satisfied, we have
hHi(K•⊗AB) = hCi(K•)⊗AB+hCi+1(K•)⊗AB−hKi+1⊗AB
and
hCi(K•)⊗AB = Xp
j=i
(−1)j−i
hHj(K•⊗AB)+hKj+1⊗AB
. Proof. We have the four-term exact sequence 0→ Hi K•⊗AB
→Ci(K•)⊗AB→Ki+1⊗AB→ Ci+1(K•)⊗AB→0, which can be broken into short exact sequences
0→Hi K•⊗AB
→Ci(K•)⊗AB→Li→0 and 0→Li→Ki+1⊗AB→Ci+1(K•)⊗AB→0 whereLiis some gradedR-module.
ByRemark 2.2, if all Ci(K•)⊗ABhave a Hilbert function overBthen allLihave a Hilbert function over B and, by the same token, it follows that all Hi(K•⊗AB) have a Hilbert function overB. This establishes the implication (2)⇒(1).
Suppose that all Hi(K•⊗AB) have a Hilbert function overB. As a consequence of Remark 2.2, if Ci+1(K•)⊗ABhas a Hilbert function overB, we obtain thatLiand, subsequently, Ci(K•)⊗ABhave
Hilbert functions over B. Since Cp(K•)⊗AB=Hp(K•⊗AB), by descending induction oni, we get that all Ci(K•)⊗ABhave a Hilbert function overB. So, the other implication (1)⇒(2) also holds.
The additional equations relating the Hilbert functions of Ci(K•)⊗ABand Hi(K•⊗AB)are straight-
forwardly checked.
For a given bounded complex of gradedR-modulesK•:0→K0→K1→ · · · →Kp→0 such that each Kihas a Hilbert function overAand a given tuple ofp+1 functionsh= (h0, . . . ,hp) :Zp+1→Np+1, we consider the following functor for any ringB,
FhK•(B) :=
morphism Spec(B)→Spec(A)
Hi(K•⊗AB)
ν is a locally freeB-module of rankhi(ν)for all 06i6p,ν∈Z
. For completeness, we include a lemma which shows that, in our setting, flatness is equivalent to being locally free.
Lemma 2.8. Let K• :0→K0→ K1→ · · · →Kp→ 0 be a bounded complex of graded R-modules.
Suppose that[Ki]νis a finitely generated locally freeA-module for all06i6p,ν∈Z. LetSpec(B)→ Spec(A)be a morphism. Then, the following two conditions are equivalent:
(1) [Hi(K•⊗AB)]νis a flatB-module for all06i6p,ν∈Z.
(2) [Hi(K•⊗AB)]νis a locally freeB-module for all06i6p,ν∈Z.
Proof. The implication (2)⇒(1) is clear. So, we assume that each[Hi(K•⊗AB)]νis a flatB-module.
As in the proof ofLemma 2.7, we consider the short exact sequences 0→Hi K•⊗AB
→Ci(K•)⊗A B→Li→0 and 0→Li→Ki+1⊗AB→Ci+1(K•)⊗AB→0. Note that each[Cp(K•)⊗AB]ν= [Hp(K•⊗AB)]νis a locally freeB-module since it is flat of finite presentation as aB-module. Similarly toLemma 2.7, by descending induction oni, we can show that[Hi(K•⊗AB)]νand[Ci(K•)⊗AB]νare
locally freeB-modules for all 06i6p,ν∈Z.
The following theorem deals with the stratification of the cohomologies of bounded complexes.
Theorem 2.9. Assume Setup 2.1 with A Noetherian. Let K•:0→K0 →K1→ · · · →Kp →0 be a bounded complex of finitely generated gradedR-modules andh= (h0, . . . ,hp) :Zp+1→Np+1be a tuple of functions. Suppose that eachKihas a Hilbert function overA. Then, the functorFhK•is represented by a locally closed subschemeFhK• ⊂Spec(A). In other words, for any morphismg:Spec(B)→Spec(A), eachHi(K•⊗AB)has a Hilbert function overBequal tohiif and only ifgcan be factored as
Spec(B) → FhK• → Spec(A).
Proof. For any morphism Spec(B)→Spec(A),Lemma 2.7implies that each Hi(K•⊗AB)has a Hilbert function overBequal tohiif and only if each Ci(K•)⊗ABhas a Hilbert function overBequal tohi′, where hi′:=Pp
j=i(−1)j−i hj+hKj+1⊗AB
. Therefore, by Theorem 2.4, FhK• is represented by the locally closed subschemeFhK• ⊂Spec(A)given byFh0′
C0(K•)∩Fh1′
C1(K•)∩ · · · ∩Fh
p′
Cp(K•).
2.3. Flattening stratification of Ext modules. We now focus on a flattening stratification result for certain Ext modules. During this subsection, we shall use the following setup.
Setup 2.10. LetAbe a Noetherian ring andRbe a positively graded polynomial ringR=A[x1, . . . ,xr] overA.
First, we recall the following result from [8].
Lemma 2.11. LetMbe a finitely generated gradedR-module and suppose that Mis a flat A-module.
LetF•:· · · →Fi→ · · · →F1→F0be a graded freeR-resolution ofMby modules of finite rank. Let DiM:=Coker HomR(Fi−1,R)→HomR(Fi,R)
for eachi>0. Then, the following statements hold:
(i) ExtiR(M,R) =0for alli>r+1.
(ii) DiMis a flatA-module for alli>r+1.
(iii) IfExtiR(M,R)is a flatA-module for all06i6r, then
ExtiR(M,R)⊗AB−→=∼ ExtiR⊗AB(M⊗AB,R⊗AB) for alli>0anyA-algebraB.
Proof. It follows directly from [8, Lemma 2.10].
For a given finitely generated gradedR-moduleMthat isA-flat and a tuple of functionsh= (h0, . . . ,hr) : Zr+1→Nr+1, we consider the following functor for any ringB,
FExthM(B) :=
morphism Spec(B)→Spec(A)
ExtiR⊗AB(M⊗AB,R⊗AB)
ν is a locally free B-module of rankhi(ν)for all 06i6r,ν∈Z
. Note that this functor controls all the Ext modules ofMbecause, as a consequence ofLemma 2.11, ifM isA-flat then ExtiR⊗AB(M⊗AB,R⊗AB) =0 for alli>r+1. The next theorem provides a flattening stratification for all the Ext modules.
Theorem 2.12. Assume Setup 2.10. LetM be a finitely generated graded R-module that is a flat A- module, and h= (h0, . . . ,hr) :Zr+1 →Nr+1 be a tuple of functions. Then, the functor FExthM is represented by a locally closed subscheme FExthM⊂Spec(A). In other words, for any morphism g: Spec(B)→Spec(A), eachExtiR⊗
AB(M⊗AB,R⊗AB)has a Hilbert function overBequal tohiif and only ifgcan be factored as
Spec(B) → FExthM → Spec(A).
Proof. Let F• :· · · →Fi→ · · · →F1→F0 be a graded free R-resolution of M by modules of finite rank. Consider the complex F6r+1• given as the truncation F6r+1• :0→Fr+1→Fr→ · · · →F1→F0, and P•:=HomR(F6r+1• ,R). ByLemma 2.11,Dr+1M =Hr+1(P•) =Cr+1(P•) is a flatA-module and so each [Dr+1M ]ν(being finitely presented over A) is a locally free A-module. Hence [47, Tag 00NX]
implies that for allν∈Zthe function Spec(A)→N,p7→dimκ(p)([Dr+1M ⊗Aκ(p)]ν)is locally constant.
As a consequence, hDr+1
M is a constant function on each connected component of Spec(A).
Consider the bounded complexK•given by
K•: 0→P0→ · · · →Pr→Pr+1→Dr+1M →0.
Note that Hi(K•⊗AB) =Hi(P•⊗AB)=∼ ExtiR⊗
AB(M⊗AB,R⊗AB) for all 06i6r(since M is A-flat), and that Hr+1(K•⊗AB) =Hr+2(K•⊗AB) =0.
To show that the functorFExthMis representable, we can simply restrict Spec(A)to one of its con- nected components. Thus, we now assume that Spec(A)is connected, and soDr+1M has a Hilbert func- tion overA. Leth′= (h0, . . . ,hr, 0, 0) :Zr+3→Nr+3 be obtained by concatenating two zero functions 0:Z→N toh. Finally, by Theorem 2.9, it follows that FExthM is represented by the locally closed subscheme FExthM:=FhK′•⊂Spec(A). This settles the proof of the theorem.
2.4. Flattening stratification of local cohomology modules. Next, we provide a flattening stratifica- tion theorem for local cohomology modules. The main idea is that, by using some techniques from [6,8], we can obtain a flattening stratification of local cohomology modules from the one of Ext modules given inTheorem 2.12.
We start with the following lemma that gives a base change of local cohomology modules over a base which is not necessarily Noetherian.
Lemma 2.13. LetAbe a ring, R=A[x1, . . . ,xr]be a positively graded polynomial ring overA,m = (x1, . . . ,xr)⊂Rbe the graded irrelevant ideal, andMbe a gradedR-module. IfMisA-flat andHim(M) isA-flat for all06i6r, thenHim(M)⊗AB−→=∼ Him(M⊗AB)for all06i6rand anyA-algebraB.
Proof. By using [45] and the fact thatx1, . . . ,xris a regular sequence inR, even ifAis not Noetherian, we can compute Him(M)as thei-th cohomology ofC•m⊗RMwhereC•mdenotes the ˇCech complex with respect to m = (x1, . . . ,xr). Let L•:· · · →Li→ · · · →L1→L0 be a graded free R-resolution of M.
By considering the spectral sequences coming from the double complexC•m⊗SL•⊗AB, we obtain the isomorphisms
Him(M⊗AB)=∼Hr−i Hrm(L•)⊗AB
for anyA-algebraBand all integersi(see [6, Lemma 3.4]). By the flatness condition and standard base change results (see [6, Lemma 2.8]), we obtain
Him(M)⊗AB=∼Hr−i(Hrm(L•))⊗AB−=→∼ Hr−i(Hrm(L•)⊗AB)=∼Him(M⊗AB),
and so the result follows.
The following setup is now set in place for the rest of the subsection.
Setup 2.14. LetAbe a Noetherian ring,Rbe a positively graded polynomial ringR=A[x1, . . . ,xr]over A,m = (x1, . . . ,xr)⊂Rbe the graded irrelevant ideal, andδ:=deg(x1) +· · ·+deg(xr)∈Z+.
For a graded R-module Mand a morphism Spec(B)→Spec(A), we consider the graded(R⊗AB)- moduleM⊗ABand we denote theB-relative graded Matlis dualby
(M⊗AB)∗B=∗HomB(M⊗AB,B) :=M
ν∈Z
HomB [M⊗AB]−ν,B .
Note that(M⊗AB)∗B has a natural structure of graded(R⊗AB)-module. From the canonical perfect pairing of free A-modules in “top” local cohomology [R]ν⊗A[Hrm(R)]−δ−ν→[Hrm(R)]−δ =∼ A we obtain a canonical gradedR-isomorphism Hrm(R)= (R(−δ))∼ ∗A =∗HomA(R(−δ),A). Then, for a mor- phism Spec(B)→Spec(A)and a complex F•:· · · →Fi→ · · · →F1→F0 of finitely generated graded freeR-modules, we obtain the isomorphisms of complexes
Hrm(F•⊗AB)=∼Hrm(F•)⊗AB=∼ HomR(F•,R(−δ))∗A
⊗AB=∼ HomR(F•,R(−δ))⊗AB∗B
.
The next proposition gives a sort of local duality theorem (see [8, Proposition 2.11]).
Proposition 2.15. LetMbe a finitely generated gradedR-module and suppose thatMis a flatA-module.
LetSpec(B)→Spec(A)be a morphism. Then, the following two conditions are equivalent:
(1) Him(M⊗AB)has a Hilbert function overBfor all06i6r.
(2) ExtiR⊗
AB(M⊗AB,R⊗AB)has a Hilbert function overBfor all06i6r.
Moreover, when any of the above equivalent conditions is satisfied, we have that hHi
m(M⊗AB)(ν) = hExtr−i
R⊗AB(M⊗AB,R⊗AB)(−ν−δ) for alli,ν∈Z.
Proof. Let F• :· · · →Fi→ · · · →F1→F0 be a graded free R-resolution of M by modules of finite rank. AsMisA-flat,F•⊗ABis a resolution ofM⊗AB. Then, by using the isomorphism of complexes Hrm(F•⊗AB)=∼ HomR(F•,R(−δ))⊗AB∗B
and the same proof of [8, Proposition 2.11], we obtain that conditions (1) and (2) are equivalent, and that in the case they are satisfied, we have the isomorphism Him(M⊗AB)=∼ Extr−iR⊗
AB(M⊗AB,R(−δ)⊗AB)∗B
.
For a given finitely generated gradedR-moduleMthat isA-flat and a tuple of functionsh= (h0, . . . ,hr) : Zr+1→Nr+1, we consider the following functor for any ringB,
FLochM(B) :=
morphism Spec(B)→Spec(A)
Him(M⊗AB)
ν is a locally freeB-module of rankhi(ν)for all 06i6r,ν∈Z
. Finally, we have below a theorem that gives a flattening stratification for local cohomology modules.
Theorem 2.16. Assume Setup 2.14. LetM be a finitely generated graded R-module that is a flat A- module, and h= (h0, . . . ,hr) :Zr+1 → Nr+1 be a tuple of functions. Then, the functor FLochM is represented by a locally closed subscheme FLochM⊂Spec(A). In other words, for any morphism g:Spec(B)→Spec(A), eachHim(M⊗AB) has a Hilbert function overBequal to hi if and only if gcan be factored as
Spec(B) → FLochM → Spec(A).
Proof. Leth′= (h0′, . . . ,hr′) :Zr+1→Nr+1 be a tuple of functions defined byhi′(ν) :=hr−i(−ν−δ).
So, it follows directly fromProposition 2.15andTheorem 2.12thatFLochMis represented by the locally
closed subscheme FLochM:=FExthM′ ⊂Spec(A).
3. FLATTENING STRATIFICATION OF THE HIGHER DIRECT IMAGES OF A SHEAF AND ITS TWISTINGS
In this section, we provide a flattening stratification theorem that deals with all the direct images of a sheaf and its possible twistings. This result is the core of our approach to show that the fiber-full scheme exists.
For completeness, we start with a base change result which is probably well-known to the experts, but we could not find it in the generality we need (cf. [19, Lemma 4.1]). LetSbe a scheme andf:X⊂PrS→S be a projective morphism. Let g:T →Sbe a morphism of schemes and t∈T be a point. We use the
notationXT:=X×ST,f(T):=f×ST:XT →T,Xt:=XT×TSpec(κ(t))andf(t):=f(T)×TSpec(κ(t)) : Xt→Spec(κ(t)), and we consider the commutative diagram
Xt=XT×TSpec(κ(t)) XT =X×ST X=X×SS
Spec(κ(t)) T S.
1×Tιt
ιt
f(t)
1×Sg
g f(T) f
For a quasi-coherent sheaf FonX, letFT := (1×Sg)∗Fbe the sheaf onXT obtained by the pull-back induced bygandFt:= (1×T ιt)∗FT be the sheaf onXtobtained by taking the fiber overt. Recall that in this setting, we have the base change mapg∗Rif∗F→Rif(T)∗(FT)for alli>0.
Proposition 3.1. LetSbe a scheme,f:X⊂PrS→Sbe a projective morphism andFbe a quasi-coherent OX-module. Suppose thatRif∗(F(ν))is a flatOS-module for all06i6r,ν∈Z. Letg:T →Sbe a morphism of schemes. Then,Fis flat overSand we have a base change isomorphism
g∗Rif∗(F(ν))−→=∼ Rif(T)∗(FT(ν)) for all06i6r,ν∈Z.
Proof. Since the first consequence is local onSand the second one is local onT, we may assume that T =Spec(B)andS=Spec(A)are affine schemes. Then, we have the identifications
Rif∗(F(ν))=∼ Hi(X,F(ν))∼=∼Hi(PrA,F(ν))∼ and
Rif(T)∗(FT(ν))=∼ Hi(XT,FT(ν))∼=∼ Hi(PrB,FT(ν))∼
(see [47,Tag 01XK], [21, Proposition 8.5]). LetR:=A[x0, . . . ,xr]withPrA=Proj(R),m = (x0, . . . ,xr), and Mbe the graded R-module given byM:=L
ν∈ZH0(PrA,F(ν)). Note that F ∼=M∼ and FT =∼ (M⊗AB)∼. Thus, it is clear thatFis flat overS. We have the exact sequence
0→H0m(M⊗AB)→M⊗AB→M
ν∈Z
H0(PrB,FT(ν))→H1m(M⊗AB)→0 and the isomorphism Hi+1m (M⊗AB)=∼L
ν∈ZHi(PrB,FT(ν))for alli>1. In the special caseB=A, since M=L
ν∈ZH0(PrA,F(ν)), we obtain that H0m(M) =H1m(M) =0. Finally, Lemma 2.13implies that Him(M)⊗AB−=→∼ Him(M⊗AB)for all 06i6r+1, and so the proof of the proposition is complete.
We fix the following setup for the rest of this section.
Setup 3.2. LetSbe a locally Noetherian scheme andf:X⊂PrS→Sbe a projective morphism.
When we take the fiberXt=XT×TSpec(κ(t))off(T)overt∈T, we get the isomorphism Rif(t)∗(Ft) =∼ Hi(Xt,Ft)∼
for alli>0. Our main object of study is the following functor. For a given coherent sheafFonXthat isS-flat and a tuple of functionsh= (h0, . . . ,hr) :Zr+1→Nr+1, we consider the following functor for
any schemeT,
FDirhF(T) :=
morphismT→S
Rif(T)∗(FT(ν))is locally free overT and dimκ(t) Hi(Xt,Ft(ν))
=hi(ν) for all 06i6r,ν∈Z,t∈T
.
Note that, as a consequence ofProposition 3.1, a morphismT →Sbelongs to the setFDirhF(T)if and only ifRif(T)∗(FT(ν))is a locally freeOT-module of rankhi(ν)for all 06i6r,ν∈Z. The following theorem yields the representability of the functorFDirhF. This result will be our main tool.
Theorem 3.3. AssumeSetup 3.2. LetFbe a coherent sheaf onXthat is flat overS, andh= (h0, . . . ,hr) : Zr+1 →Nr+1 be a tuple of functions. Then, the functor FDirhF is represented by a locally closed subschemeFDirhF⊂S. In other words, for any morphismg:T→Sof schemes, eachRif(T)∗(FT(ν))is a locally freeOT-module of rankhi(ν)if and only ifgcan be factored as
T → FDirhF → S.
Proof. LetS=S
j∈JSjbe an open covering ofSwhere eachSjis a Noetherian affine scheme. Note that the functorFDirhFis a Zariski sheaf and it has a Zariski covering by the open subfunctors{Gj}j∈Jwhere
Gj(T) :=
morphismT→Sj
Rif(T)∗(FT(ν))is locally free overT and dimκ(t) Hi(Xt,Ft(ν))
=hi(ν) for all 06i6r,ν∈Z,t∈T
(see [16, §8.3]). Therefore, due to [16, Theorem 8.9], in order to show thatFDirhFis representable by a locally closed subscheme of S, it suffices to show that each Gj is representable by a locally closed subscheme ofSj.
As a consequence of the above reductions, we assume thatAis a Noetherian ring andS=Spec(A).
Since all the conditions that we consider onRif(T)∗(FT(ν))are local onT, we may restrict to an affine morphismT=Spec(B)→Spec(A), and we do so.
LetR:=A[x0, . . . ,xr]withPrA=Proj(R) and m = (x0, . . . ,xr)⊂R. By known arguments, we can choose an integerm∈Zsuch that the following conditions are satisfied:
(i) M:=L
ν>mH0(PrA,F(ν))is a finitely generated gradedR-module that is flat overA, (ii) M∼=∼Fand(M⊗AB)∼=∼ FT,
(iii) M⊗AB=∼ L
ν>mH0(PrB,FT(ν)), and (iv) Hi(PrA,F(ν)) =0 for all 16i6r,ν>m
(see, e.g., [21, §III.9]). Therefore, we obtain a short exact sequence 0→M⊗AB→M
ν∈Z
H0(PrB,FT(ν))→H1m(M⊗AB)→0 that splits into the isomorphisms
M⊗AB=∼ M
ν>m
H0(PrB,FT(ν)) and M
ν<m
H0(PrB,FT(ν))=∼ H1m(M⊗AB), and we get the isomorphism Hi+1m (M⊗AB)=∼ L
ν∈ZHi(PrB,FT(ν))for alli>1.
We have obtained that Hi(PrB,FT(ν))is a locally freeB-module of rankhi(ν)for alli>0,ν∈Zif and only if the following three conditions hold:
– [M⊗AB]νis a locally freeB-module of rankh0(ν)for allν>m,
– [H1m(M⊗AB)]νis a locally freeB-module of rankh0(ν)for allν < m, and – [Him(M⊗AB)]νis a locally freeB-module of rankhi−1(ν)for alli>2,ν∈Z.
Leth0′,h0′′:Z→Nbe the functions h0′(ν) :=
h0(ν) ifν < m
0 otherwise
and h0′′(ν) :=
h0(ν) ifν>m
0 otherwise,
and h′:Zr+2→Nr+2 be the tuple of functions defined byh′:= (0,h0′,h1, . . . ,hr), where 0:Z→N denotes the zero function.
Finally, by Theorem 2.4and Theorem 2.16, we obtain that each Hi(PrB,FT(ν))is a locally freeB- module of rankhi(ν)if and only if the morphismg:T =Spec(B)→S=Spec(A)factors through the locally closed subschemeFh
′′
0
M∩FLochM′ ⊂S=Spec(A). This concludes the proof of the theorem.
4. FIBER-FULL SHEAVES
In this short section, we introduce the notion of fiber-full sheaf that extends the concept of fiber-full modules from [8]. LetSbe a locally Noetherian scheme,f:X⊂PrS→Sbe a projective morphism, and Fbe a coherent sheaf onX.
Definition 4.1. We say that F is afiber-full sheaf over S if Rif∗(F(ν))is locally free over Sfor all 06i6randν∈Z.
For everys∈Sandq>1, letgs,qbe the natural mapgs,q:Spec(OS,s/mqs)→Swheremsdenotes the maximal ideal of the local ringOS,s,Xs,qbe the schemeXs,q:=X×SSpec(OS,s/mqs), andFs,q:=
(1×Sgs,q)∗Fbe the sheaf onXs,qobtained by the pull-back induced bygs,q. For the caseq=1 (i.e., when we take the fiber at a point s∈S), we simply write gs=gs,1, Xs=Xs,1 and Fs=Fs,1. The following theorem gives two further equivalent definitions for the notion of a fiber-full sheaf. The name
“fiber-full” is inspired by condition (3) below.
Theorem 4.2. Under the above notations, the following three conditions are equivalent:
(1) Fis a fiber-full sheaf overS.
(2) Fis a locally freeOS-module andHi(Xs,q,Fs,q(ν))is a freeOS,s/mqs-module for alls∈S, 06 i6r,ν∈Zandq>1.
(3) Fis a locally freeOS-module and the natural mapHi(Xs,q,Fs,q(ν))→Hi(Xs,Fs(ν))is surjective for alls∈S,06i6r,ν∈Zandq>1.
Proof. Since the three conditions are local onS, we can choose a points∈Sand assume that(B,b) = (OS,s,ms)is a Noetherian local ring andS=Spec(B). Moreover, in each of the three above conditions one is assuming thatFis flat overS. LetR:=B[x0, . . . ,xr]withPrB=Proj(R)andm = (x0, . . . ,xr)⊂R.
Then, we can choose an integerm∈Zsuch that the following conditions are satisfied:
(i) M:=L
ν>mH0(PrB,F(ν))is a finitely generated gradedR-module that is flat overB, (ii) M∼=∼Fand(M⊗BB/bq)∼=∼Fs,q, and
