Regularity, singularities and h -vector of graded algebras
aa r X i v : . [ m a t h . A C ] J a n REGULARITY, SINGULARITIES AND h -VECTOR OF GRADEDALGEBRAS HAILONG DAO, LINQUAN MA, AND MATTEO VARBARO
Abstract.
Let R be a standard graded algebra over a field. We investigatehow the singularities of Spec R or Proj R affect the h -vector of R , which is thecoefficients of the numerator of its Hilbert series. The most concrete consequenceof our work asserts that if R satisfies Serre’s condition ( S r ) and have reasonablesingularities (Du Bois on the punctured spectrum or F -pure), then h , . . . , h r ≥ R is at least h + h + · · · + h r − . We also provethat equality in many cases forces R to be Cohen-Macaulay. The main technicaltools are sharp bounds on regularity of certain Ext modules, which can be viewedas Kodaira-type vanishing statements for Du Bois and F -pure singularities. Manycorollaries are deduced, for instance that nice singularities of small codimensionmust be Cohen-Macaulay. Our results build on and extend previous work by deFernex-Ein, Eisenbud-Goto, Huneke-Smith, Murai-Terai and others. Introduction
Let R = k [ x , . . . , x n ] /I be a standard graded algebra over a field k . Suppose R is a domain. How many independent quadrics can I contain? Such a question haslong been studied in algebraic geometry, and some beautiful answers are known. Forexample, if k is algebraically closed, and I is prime and contains no linear forms,then the number of quadratic minimal generators of I is at most (cid:0) n − d +12 (cid:1) where d = dim R . Furthermore, equality happens if and only if R is a variety of minimaldegree (see Theorem 5.6 for a generalization of this result to non-reduced schemes).In another vein, a striking result by Bertram-Ein-Lazarsfeld ([3, Corollary 2])states that if Proj R is smooth and the sum of n − d highest degrees among theminimal generators of I is less than n , then R is Cohen-Macaulay. We will alsoextend this result to mild singularities.The common theme in the aforementioned statements is how the singularities of R affect its Hilbert function. This article grew out of an attempt to better understandthis phenomenon. It turns out that such numerical information is most convenientlyexpressed via the h -vector. Recall that we can write the Hilbert series of RH R ( t ) = X i ∈ N (dim k R i ) t i = h + h t + · · · + h s t s (1 − t ) d . Date : January 30, 2019.2000
Mathematics Subject Classification.
Primary 13-02, 13F55, 05C10.
We will call the vector of coefficients ( h , h , . . . , h s ) the h -vector of R . The boundon the number of quadrics mentioned above can be easily seen to be equivalent tothe statement that h ≥
0. Furthermore, it is well-known and easy to prove that if R is Cohen-Macaulay then h i ≥ i . A very concrete motivation for our workcomes from the following question: Question 1.1.
Suppose R satisfies Serre’s condition ( S r ) . When is it true that h i ≥ for i = 0 , . . . , r ? An answer for this question when I is a square-free monomial ideal is containedin a beautiful result of Murai-Terai (the h -vector of a simplicial complex ∆ equalsthe h -vector of its Stanley-Reisner ring k [∆]). Theorem ([25]) . Let ∆ be a simplicial complex with h -vector ( h , h , . . . , h s ) . If k [∆] satisfies ( S r ) for some field k , then h i ≥ for i = 0 , . . . , r . In general, some additional assumptions are needed as shown in Remark 2.13where, for any n ≥
3, it is constructed a standard graded k -algebra R satisfy-ing ( S n − ) and having h -vector (1 , n, − R hasCastelnuovo-Mumford regularity 1 and is Buchsbaum. Contrary to this, we cansurprisingly show the following, which is our main result: Theorem 1.2.
Let R be a standard graded algebra over a field and ( h , . . . , h s ) bethe h -vector of R . Assume R satisfies ( S r ) and either:(1) char k = 0 and X = Proj R is Du Bois.(2) char k = p > and R is F -pure.Then h i ≥ for i = 0 , . . . , r . Also, h r + h r +1 + · · · + h s ≥ , or equivalently R hasmultiplicity at least h + h + · · · + h r − .If furthermore R has Castelnuovo-Mumford regularity less than r or h i = 0 forsome i ≤ r , then R is Cohen-Macaulay. Note that Theorem 1.2 generalizes the work of Murai-Terai, since a Stanley-Reisner ring is Du Bois in characteristic 0 and F -pure in characteristic p . The keypoint is to establish good bounds on certain Ext modules, just like what Murai andTerai did, but for ideals that are not necessarily square-free.One can deduce other surprising consequences from our results. We discuss oneof them now. There is a well-known but mysterious theme in algebraic geometrythat “nice singularities of small codimension” should be Cohen-Macaulay. Perhapsthe most famous example is Hartshorne’s conjecture that a smooth subvariety in P n k of codimension e has to be a complete intersection provided e < n/
3. Although thisconjecture is wide-open, numerous partial results has been obtained, and many ofthem assert that such variety has to be projectively Cohen-Macaulay. One or ourcorollaries can be viewed as another confirmation of this viewpoint (extending [7,Corollary 1.3]).
EGULARITY, SINGULARITIES AND h -VECTOR OF GRADED ALGEBRAS 3 Corollary 1.3 (Corollary 5.4) . Let R = k [ x , . . . , x n ] /I be a standard graded algebraover a field k of characteristic with e = ht I and d = dim R . Let d ≥ d ≥ . . . bethe degree sequence of a minimal set of generators for I . Assume that R is unmixed,equidimensional, Cohen-Macaulay in codimension l and X = Proj R has only MJ-log canonical singularities. If e + l ≥ d + · · · + d e , then R is Cohen-Macaulay. For instance, the Corollary asserts that if I is generated by quadrics, then R is Cohen-Macaulay provided that it is Cohen-Macaulay in codimension e . Understronger assumptions, it follows that R is Gorenstein or even complete intersection,see Corollary 5.5.We now briefly describe the content and organization of the paper. Section 2establishes all the important preparatory results. It contains a careful analysis ofa property which we call ( M T r ). This is a condition on regularity of certain Extmodules studied by Murai-Terai (these modules are sometimes called deficiencymodules in the literature, see [27]). It turns out that this regularity condition isprecisely what we need to prove all the statements of the main theorem. We givea self-contained and independent proof that ( M T r ) implies non-negativity of h i for i ≤ r , when equalities occur, and also that ( M T r ) is preserved under generichyperplane section. Our proofs work also for modules.What remains, which is our main technical contribution, is to establish the con-dition ( M T r ) for algebras with nice singularities. We achieve this goal in Sections3 (characteristic 0) and 4 (positive characteristic). Our results there give strongbounds on regularity of the Matlis dual of the local cohomology modules of suchalgebras, and are perhaps of independent interests. As mentioned above, they canbe viewed as Kodaira-vanishing type statements.In Section 5 we put everything together to prove Theorem 1.2 and give severalapplications. Our bounds imply stringent conditions on Hilbert functions of algebrassatisfying assumptions of 1.2, see Corollary 5.2. One can also show that certainalgebras with nice singularities and small codimension must be Cohen-Macaulay,see Corollary 5.4. We also write down perhaps the strongest possible statement for h ( R ), which can be viewed as generalization of the classical bound e ( R ) ≥ e where e is the embedding codimension, and the equality case which forces R to berings of minimal multiplicity.In the last section, we propose an open question which can be viewed as a verystrong generalization of our result and prove a first step toward such problem, whichgeneralized one of the main results of Huneke and Smith in [15]. Acknowledgement.
The first author is partially supported by Simons FoundationCollaboration Grant 527316. The second author is partially supported by an NSFGrant
H. DAO, L. MA, AND M. VARBARO to thank Kangjin Han for useful discussions on h -vectors and singularities, whichstarted this project. We also thank Tommaso de Fernex and S´andor Kov´acs forsome helpful email exchanges. 2. Preliminaries
Set up 2.1.
We will use the following notations and conventions throughout thepaper:(i) n is a positive integer;(ii) k is a field;(iii) S is the standard graded polynomial ring in n variables over k ;(iv) m is the irrelevant ideal of S ;(v) I ⊂ S is a homogeneous ideal of height e ;(vi) R = S/I is the quotient of dimension d = n − e ;(vii) X = Proj R is the projective scheme associated to R ( dim X = d − ).(viii) For a k -scheme Y we say that Y has P singularities if and only if Y ⊗ k ¯ k has P singularities, where ¯ k is the algebraic closure of k . Here P can be non-singular,Du Bois, M J -log canonical, F -pure, etc.(ix) We say that R has P singularities if and only if Spec R has P singularities.(x) For a nonzero finitely generated graded S -module M of dimension d , let H M ( t ) denote the Hilbert series of M , and ( h a ( M ) , h a +1 ( M ) , . . . , h s ( M )) denote the h -vector of M , given by: H M ( t ) = X i ∈ Z (dim k M i ) t i = h a t a + h a +1 t a +1 + · · · + h s t s (1 − t ) d , We also write p M ( t ) for the numerator and c r ( M ) for the quantity h r ( M ) + h r +1 ( M ) + · · · + h s ( M ) . Note that c r ( M ) = e ( M ) − P i Theorem 2.2. If reg(Ext n − iS ( R, ω S )) ≤ i − r ∀ i = 0 , . . . , d − , then h i ( R ) ≥ for i ≤ r . Motivated by this result, we make the following definition. Definition 2.3. Consider a graded module M over S of dimension d . We say that M satisfies condition ( M T r ) if reg(Ext n − iS ( M, ω S )) ≤ i − r ∀ i = 0 , . . . , d − . The analysis of this notion led us to a more direct proof of Theorem 2.2. Wefirst note a few preparatory results. The following facts are trivial consequences ofgraded duality: Proposition 2.4. (1) The condition ( M T r ) depends only on R and does notdepend on the presentation R = S/I (we won’t need this). EGULARITY, SINGULARITIES AND h -VECTOR OF GRADED ALGEBRAS 5 (2) If M is Cohen-Macaulay, then it is ( M T r ) for any r .(3) If N = H i m ( M ) has finite length, then reg(Ext n − iS ( M, ω S )) ≤ i − r if and onlyif N Suppose k is infinite. Let M be a graded S -module. For a genericlinear form l : reg( M ) ≥ max { reg 0 : M l, reg M/lM } Proof. This is well-known and easy, but we give a proof for convenience. Let N = H m ( M ). Then for generic l , 0 : M l ⊆ N , so it’s regularity is at most reg( N ) ≤ reg( M ). Consider the short exact sequence 0 → N → M → M ′ → 0. We canassume l is regular on M ′ , which leads to 0 → N/lN → M/lM → M ′ /lM ′ → 0. Since reg( N/lN ) ≤ reg( N ) and reg( M ′ /lM ′ ) = reg( M ′ ) ≤ reg( M ), the resultfollows. (cid:3) Proposition 2.6. Suppose k is infinite. Let M be a graded S -module. Assume that M is ( M T r ) , then so are the following modules: M ′ = M/H m ( M ) , M ′ /lM ′ , M/lM ,for a generic linear form l .Proof. Let K i ( M ) denote the module Ext n − iS ( M, ω S ). The short exact sequence0 → H m ( M ) → M → M ′ → K ( M ′ ) = 0 and K i ( M ′ ) = K i ( M ) for i ≥ 1. So M ′ is ( M T r ). For the rest, we assume d = dim M ≥ 1. As l is generic, it isregular on M ′ , so from the short exact sequence 0 → M ′ ( − → M ′ → M ′ /lM ′ → i ≤ d − M ′ /lM ′ :0 → K i +1 ( M ′ ( − /lK i +1 ( M ′ ( − → K i ( M ′ /lM ′ ) → K i ( M ′ ) l → K i ( M ′ /lM ′ ) follows from the property ( M T r ) for M ′ andProposition 2.5. For the module M/lM , note that we have the short exact sequence0 → N/lN → M/lM → M ′ /lM ′ → N = H m ( M ). Another application ofthe long exact sequence of Ext and the fact that M ′ /lM ′ is ( M T r ) proves what weneed. (cid:3) The next proposition tracks the behavior of h -vector modulo a generic linear formfor a module satisfying ( M T r ). Let c r ( M ) be the quantity: h r ( M ) + h r +1 ( M ) + ... = e ( M ) − P r − i =0 h i ( M ). Proposition 2.7. Suppose k is infinite. Let M be a graded S -module of dimension d ≥ . Let l be a generic linear form. Let N = H m ( M ) and M ′ = M/N . Assumethat N Proof. First of all M is ( M T r ) implies N 0, which gives: • h i ( M/lM ) = h i ( M ′ /lM ′ ) = h i ( M ′ ) = h i ( M ) for i < r . • h r ( M/lM ) = h r ( M ′ /lM ′ ) + h r ( N/lN ) ≤ h r ( M ). • c r ( M ) = c r ( M ′ ) = c r ( M ′ /lM ′ ). • c r ( M/lM ) = c r ( M ′ /lM ′ ) = c r ( M ) if d > (cid:3) We can now give a direct and simple proof of Theorem 2.2, extended to the modulecase. Theorem 2.8. Let M be a module satisfying ( M T r ) . Then h i ( M ) ≥ for i ≤ r and also c r ( M ) ≥ .Proof. We can extend k if necessary and hence assume it is infinite. Since M is( M T r ), we have N Proposition 2.9. If M is ( M T r ) and either:(1) reg( M ) < r , or(2) h i ( M ) = 0 for some i ≤ r and M is generated in degree .Then M is Cohen-Macaulay.Proof. We can extend k if necessary and hence assume it is infinite. We use inductionon d = dim M and therefore assume d ≥ 1. Assume first that reg( M ) < r . Let N = H m ( M ), then N 1. By Propositions 2.5 and 2.6 as well as induction hypotheses, M/lM is Cohen-Macaulay for a generic l , which implies that M is Cohen-Macaulay.Now assume that h i ( M ) = 0 for some i ≤ r . Again, we shall show that N = H m ( M ) = 0. Let M ′ = M/N . Then as in the proof of Proposition 2.6 and 2.7, M ′ is still ( M T r ) and h i ( M ′ ) = h i ( M ). Passing to M ′ /lM ′ preserves everything, so byinduction M ′ /lM ′ is Cohen-Macaulay, and so is M ′ . It follows that h j ( M ′ ) = 0 for j ≥ i and reg M ′ ≤ i − M = F/P where P is the first syzygy of M . Then N = Q/P for Q ⊆ F , and M ′ = F/Q . Since reg( M ′ ) ≤ i − Q is generated in degree at most i . But N ≤ i = 0(we already know that N 1, thendim Ext h − iS ( R, S ) = dim Ext n − ( n − h + i ) S ( R, S ) ≤ n − h + i − ℓ < n − h for all h = e, . . . , n (because i < b ≤ r ). (cid:3) Remark 2.12. In Proposition 2.11, the equidimensionality of R is needed only inthe case “ r = 1 ”. Indeed, if R satisfies S r with r ≥ then it has no embeddedprime, and if R satisfies S r with r ≥ then it is equidimensional by [12, Remark2.4.1] .On the other hand, an argument similar to the one used to prove Proposition2.11 shows that dim R/ p = dim R for all associated prime ideals if and only if dim Ext n − iS ( R, S ) < i ∀ i < dim R . We conclude this section with the following remark that shows that we cannothope to answering “always!” to Questions 1.1. Remark 2.13. Let S = k [ x i , y i : i = 1 , . . . , n ] and I ⊆ S the following ideal: I = ( x , . . . , x n ) + ( x y + . . . + x n y n ) . Setting R = S/I , we claim the following properties: (i) R is an n -dimensional ring satisfying ( S n − ) ; (ii) reg R = 1 ; (iii) R is not Cohen-Macaulay; (iv) H R ( t ) = 1 + nt − t (1 − t ) n ; (v) H n − m ( R ) = H n − m ( R ) − n = k , thus R is Buchsbaum.For it, let us notice that, for any term order on S , the set { x i x j : 1 ≤ i ≤ j ≤ n } ∪ { x y + . . . + x n y n } is a Gr¨obner basis by Buchberger’s criterion. So we canchoose a term order such that the initial ideal of I is J = ( x , . . . , x n ) + ( x y ) . So depth R ≥ depth S/J = n − and H R ( t ) = H S/J ( t ) = H P/J ∩ P ( t ) / (1 − t ) n − ,where P = k [ x , . . . , x n , y ] . Note that dim k ( P/J ∩ P ) = 1 , dim k ( P/J ∩ P ) = n + 1 and dim k ( P/J ∩ P ) d = n for any d > . So H P/J ∩ P ( t ) = (1 + nt − t ) / (1 − t ) ,yielding H R ( t ) = 1 + nt − t (1 − t ) n . In particular, R is not Cohen-Macaulay, therefore depth R = n − . We have ≤ reg R ≤ reg S/J . Since it is a monomial ideal, it is simple to check that J haslinear quotients, so reg S/J = 1 . Finally, take a prime ideal p ∈ Proj S containing I and consider the ring S p /IS p . Notice that there exists i such that y i / ∈ p , so IS p = ( x , . . . , x n ) + ( x i + 1 /y i X j = i x j y j ) . EGULARITY, SINGULARITIES AND h -VECTOR OF GRADED ALGEBRAS 9 Therefore, by denoting S ′ = k [ x j , y k : j = i ] and p ′ = p ∩ S ′ , we get S p IS p ∼ = S ′ p ′ ( x j , : j = i ) , that is certainly Cohen-Macaulay.For the last point, notice that we have a short exact sequence: → S ( x , . . . , x n ) ( − · ( x y + ... + x n y n ) −−−−−−−−−−→ S ( x , . . . , x n ) → SI → which induces → H n − m ( S/I ) → H n m (cid:18) S ( x , . . . , x n ) ( − (cid:19) · ( x y + ... + x n y n ) −−−−−−−−−−→ H n m (cid:18) S ( x , . . . , x n ) (cid:19) So H n − m ( S/I ) is the kernel, but it is easy to compute that, up to scalar, the only ele-ment in the kernel is the element y ··· y n , which has degree − n in H n m ( S ( x ,...,x n ) ( − .Thus we are done. Regularity bounds in characteristic 0 In this section we establish the key vanishing result in the characteristic 0 thatis needed in Theorem 1.2 stated in the introduction. As explained in Section 2, weneed to establish condition ( M T r ) (Definition 2.3) for nice singularities satisfyingSerre’s condition ( S r ). The following gives a bound on regularity of Ext iS ( R, ω S )which is crucial. We begin by recalling the definition of Du Bois singularities.Suppose that X is a reduced scheme essentially of finite type over a field ofcharacteristic 0. Associated to X is an object Ω X ∈ D b ( X ) with a map O X → Ω X .Following [28, 2.1], if X ⊆ Y is an embedding with Y smooth (which is always thecase if X is affine or projective), then Ω X ∼ = Rπ ∗ O E where E is the reduced pre-image of X in a strong log resolution of the pair ( Y, X ). X is Du Bois if O X → Ω X is an isomorphism. Proposition 3.1. If X = Proj R is Du Bois, then H j m (Ext n − iS ( R, ω S )) > = 0 for all i, j . In particular, reg Ext iS ( R, ω S ) ≤ dim Ext iS ( R, ω S ) ∀ i. To this purpose, we will use ω • R to denote the normalized dualizing complex of R , thus h − i ( ω • R ) ∼ = Ext n − iS ( R, ω S ). We use ω • R to denote R Hom R (Ω R , ω • R ). Similarlywe have ω • X , ω • X for X . Note that when R (resp. X ) is Du Bois, we have ω • R = ω • R (resp. ω • X = ω • X ). We begin by recalling some vanishing/injectivity result that wewill need: This means π is a log resolution of ( Y, X ) that is an isomorphism outside of X . We note thatthis is not the original definition of Ω X but it is equivalent by the main result of [28]. Theorem 3.2 (Lemma 3.3 in [18]) . h − i ( ω • R ) ֒ → h − i ( ω • R ) is injective for all i . Theorem 3.3 (Lemma 3.3 in [19] and Theorem 3.2 in [1]) . h − i ( ω • X ) satisfies theKodaira vanishing: H j ( X, h − i ( ω • X ) ⊗ L m ) = 0 for all i , j ≥ , m ≥ and L ampleline bundle on X . Theorem 3.4 (Proposition 4.4 and Theorem 4.5 in [23]) . Suppose X is Du Bois,then:(1) h j (Ω R ) ∼ = H j +1 m ( R ) > for all j ≥ , and h (Ω R ) /R ∼ = H m ( R ) > .(2) We have a degree-preserving injection Ext n − iS ( R, ω S ) ≥ ֒ → H n − iI ( ω S ) ≥ forall i . We are now ready to prove Proposition 3.1: Proof. First we note that h − i ( ω • R ) ∼ = Ext n − iS ( R, ω S ). When j ≥ 2, we have H j m ( h − i ( ω • R )) m = H j − ( X, ^ h − i ( ω • R ) ⊗ L m ) = H j − ( X, h − i +1 ( ω • X ) ⊗ L m ) = 0for all m > L = O X (1) is very ample, and since X has DuBois singularities ω • X = ω • X ). Thus H j m ( h − i ( ω • R )) > = 0 for all i and j ≥ j = 0. We note that by Theorem 3.4 we have H m (Ext n − iS ( R, ω S )) > ֒ → Ext n − iS ( R, ω S ) > ֒ → H n − iI ( ω S ) > ֒ → H n − iI ( ω S ) . Because H m (Ext n − iS ( R, ω S )) > is clearly m -torsion, the above injection factors through H m H n − iI ( ω S ). Thus we have an induced injection H m (Ext n − iS ( R, ω S )) > ֒ → H m H n − iI ( ω S ) > . Now we note that H m H n − iI ( ω S ) > ∼ = H m H n − iI ( S ) > − n . But by the main result of [24],we have H m H n − iI ( S ) > − n = 0 and thus the injection above implies H m ( h − i ( ω • R )) > = H m (Ext n − iS ( R, ω S )) > = 0for all i .It remains to prove the case j = 1. By Theorem 3.2 we have(1) 0 → h − i ( ω • R ) → h − i ( ω • R ) → C → C = { m } because X is Du Bois. It follows from this sequence that H m ( h − i ( ω • R )) > = 0 for all i since it injects into H m ( h − i ( ω • R )) > = 0.At this point, we note that the long exact sequence of local cohomology of (1)implies that H j m ( h − i ( ω • R )) ∼ = H j m ( h − i ( ω • R )) = 0 for all j ≥ H m ( h − i ( ω • R )) ։ H m ( h − i ( ω • R )). Thus it suffices to show H m ( h − i ( ω • R )) > = 0. Since ω • R is the normal-ized dualizing complex, by the definition of ω • R , we haveΩ R ∼ = R Hom R ( ω • R , ω • R ) . Thus we have a spectral sequence E ij = h − j ( R Hom R ( h − i ( ω • R ) , ω • R )) ⇒ h i − j (Ω R ) . EGULARITY, SINGULARITIES AND h -VECTOR OF GRADED ALGEBRAS 11 By graded local duality, ( E ij ) ∨ ∼ = H j m ( h − i ( ω • R )), and we already know the latter onevanishes in degree > i and j = 1, thus ( E ij ) < = 0 for all i and j = 1.Therefore the spectral sequence in degree < E -page. It followsthat h − ( R Hom R ( h − i ( ω • R ) , ω • R )) < = ( E i ) < ∼ = h i − (Ω R ) < . By Theorem 3.4, h i − (Ω R ) < = 0 for all i . This implies h − ( R Hom R ( h − i ( ω • R ) , ω • R )) < = 0and thus by graded local duality H m ( h − i ( ω • R )) > = 0. (cid:3) Regularity bounds in positive characteristic The purpose of this section is to prove the analog of Proposition 3.1 in character-istic p > 0. The natural analogous assumption would be to assume X = Proj R is F -injective. Albeit we do not know any counterexample to the thesis of Theorem1.2 under the assumption that X is F -injective, we know that Proposition 3.1 is nottrue in positive characteristic (as Kodaira vanishing fails even if X is smooth). Asusual for this business, we can settle the problem assuming that R itself is F -pure. Proposition 4.1. If char( k ) = p > and R is F -pure, then H j m (Ext iS ( R, ω S )) > = 0 ∀ i, j In particular, reg(Ext iS ( R, ω S )) ≤ dim Ext iS ( R, ω S ) for all i .Proof. Let q = p t for some t ∈ N . Given an R -module M , we denote by q M the R -module that is M as an abelian group and has R -action given by r · m = r q m . Thestatement allows to enlarge the field, thus we can assume that k is perfect. Under theassumptions, by [13, Corollary 5.3] the t th iterated Frobenius power F t : R → q R splits as a map of R -modules, so for any j = 0 , . . . , n the induced map on localcohomology H j m ( R ) H j m ( F t ) −−−−→ H j m ( q R )is an injective splitting of R -modules as well. Notice that a degree s element in H j m ( R ) is mapped by H j m ( F t ) to a degree sq element of H j m ( q R ). By abuse of notation,from now on we will write F t for H j m ( F t ). Let us apply to such map the functor − ∨ = Hom k ( − , k ), obtaining the surjective splitting of R -modules: H j m ( R ) ∨ ( F t ) ∨ ←−−− H j m ( q R ) ∨ . Furthermore, notice that if η ∗ is a degree u element in H j m ( q R ) ∨ , then ( F t ) ∨ ( η ∗ ) = η ∗ ◦ F t is 0 if q does not divide u , while it has degree s if u = qs . By gradedGrothendieck’s duality, we therefore have a surjective splitting of R -modulesExt n − jS ( R, ω S ) ( F t ) ∨ ←−−− Ext n − jS ( q R, ω S ) ∼ = q Ext n − jS ( R, ω S ) which “divides” the degrees by q . By applying the local cohomology functor H k m ( − )to the above splitting, we get a surjective map of graded R -modules H k m (Ext n − jS ( R, ω S )) H k m (( F t ) ∨ ) ←−−−−−− H k m ( q Ext n − jS ( R, ω S )) ∼ = q H k m (Ext n − jS ( R, ω S ))dividing the degrees by q . Hence, because q can be chosen arbitrarily high, thegraded surjection of R -modules H k m (( F t ) ∨ ) yields that H k m ((Ext n − jS ( R, ω S ))) s is ac-tually 0 whenever s > (cid:3) Applications and examples In this section, we first put together our technical results to prove main Theorem1.2. Proof of Theorem 1.2. Assume R satisfies ( S r ) and one of the assumptions: X =Proj R is Du Bois (characteristic 0) or R is F -pure (characteristic p ). We canextend k if necessary and hence assume it is algebraically closed. By Theorem 2.8and Proposition 2.9, all the conclusions follow if we can establish that R is ( M T r ).But Propositions 2.11 tells us that dim Ext n − iS ( R, ω S ) ≤ i − r , and Propositions 3.1and 4.1 assert that reg Ext n − iS ( R, ω S ) ≤ dim Ext n − iS ( R, ω S ), which gives preciselywhat we want. (cid:3) The h -vector inequalities can be translated to information about the Hilbert func-tions. Here is the precise statement. Recall that R = S/I , e = ht I, d = dim R . Let s i = dim k I i . Proposition 5.1. For each l ≥ , h l = (cid:18) e + l − l (cid:19) − l X j =0 ( − j s l − j (cid:18) dj (cid:19) Proof. The Hilbert series of R can be written as: H R ( t ) = X r i t i = h + h t + · · · + h l t l (1 − t ) d · · · ( ∗ )where r i = dim k R i . By comparing coefficients in ( ∗ ) we always have: h l = l X j =0 ( − j r l − j (cid:18) dj (cid:19) Since s i = (cid:0) n + i − i (cid:1) − r i , what we need to prove amount to: l X j =0 ( − j (cid:18) n + l − j − l − j (cid:19)(cid:18) dj (cid:19) = (cid:18) e + l − l (cid:19) But we have X i ≥ (cid:18) n + i − i (cid:19) t i = 1(1 − t ) n EGULARITY, SINGULARITIES AND h -VECTOR OF GRADED ALGEBRAS 13 Since n = d + e , we have X i ≥ (cid:18) n + i − i (cid:19) t i ! (1 − t ) d = 1(1 − t ) e The l -th coefficient on both sides give exactly the equality we seek. (cid:3) From the equality above, the following is an immediate consequence of our mainTheorem 1.2. Corollary 5.2. Let R = k [ x , . . . , x n ] /I be a standard graded algebra over a field k with d = dim R and e = ht I . Let s i = dim k I i . Assume R satisfies ( S r ) and either:(1) char( k ) = 0 and X = Proj R is Du Bois.(2) char( k ) = p > and R is F -pure.Then for each l ≤ r , we have: l X j =0 ( − j s l − j (cid:18) dj (cid:19) ≤ (cid:18) e + l − l (cid:19) In particular, if I contains no elements of degree less than l , then s l ≤ (cid:0) e + l − l (cid:1) .Moreover, if equality happens for any such l , then R is Cohen-Macaulay. Remark 5.3. In [20] , it is established that for a monomial ideal I (not necessarilysquare-free), the deficiency modules of R = S/I satisfies the same regularity boundsas in Propositions 3.1 and 4.1, so our main results above applied for this situationas well. The next corollary fits the well-known but mysterious theme that “nice singular-ities of small codimension” should be Cohen-Macaulay. Corollary 5.4. Let R = k [ x , . . . , x n ] /I be a standard graded algebra over a field k of characteristic with e = ht I and d = dim R . Let d ≥ d ≥ . . . be thedegree sequence of a minimal set of generators for I . Assume that R is unmixed,equidimensional, Cohen-Macaulay in codimension l and X = Proj R has only MJ-log canonical singularities. If e + l ≥ d + · · · + d e , then R is Cohen-Macaulay.Proof. By [26, Proposition 2.7] (see also [16]), X is MJ-log canonical is equivalent tosaying that the pair ( P n − , e I e ) is log canonical. Let r = d + · · · + d e . Now applying[7, Corollary 5.1], we have H i m ( I ) j = 0 for all i > j ≥ r − n + 1.Since we have 0 → I → S → R → 0, the long exact sequence of local cohomologytells us that H i m ( R ) j = 0 for 0 < i < d and j ≥ r − n + 1. This is equivalent to sayingthat for each such i , [ K i ] ≤ n − r − = 0 with K i = Ext n − iS ( R, ω S ). However, since MJ-log canonical singularities are Du Bois by [6, Theorem 7.7], Proposition 3.1 implies We refer to [8] for detailed definition and properties of MJ-log canonical singularities. that reg K i ≤ dim K i ≤ d − l − 1. Our assumption says that d − l − ≤ n − r − K i = 0 for all i < d and hence R is Cohen-Macaulay. (cid:3) Corollary 5.5. Let R = k [ x , . . . , x n ] /I be a standard graded algebra over a field k of characteristic . Let e = ht I and d ≥ d ≥ . . . be the degree sequence of aminimal set of generators for I . Assume that Proj R is a smooth variety, n ≥ e + 3 and n > d + · · · + d e . Then R is Gorenstein. In particular, if e = 2 then R is acomplete intersection.Proof. By Corollary 5.4, R is Cohen-Macaulay. Let X = Proj R . The assumptionof smoothness and n ≥ e + 3 implies that Pic( X ) = Z , generated by O X (1), see[21, Theorem 11.4]. It follows that the class group of R is trivial, and therefore thecanonical module is free. (cid:3) Our methods also give streamlined proof and sometimes strengthen known results.Here is a statement on h that can be seen as a modest extension of [9, Theorem4.2]. Theorem 5.6. Let R = k [ x , . . . , x n ] /I be a standard graded algebra over an al-gebraically closed field k with e = ht I . Suppose that Proj R is connected in codi-mension one and the radical of I contains no linear forms. Then h ( R ) ≥ and e ( R ) ≥ e . If equality happens in either case, then R red is Cohen-Macaulay ofminimal multiplicity.Proof. Let J = √ I and T = R red = S/J . By assumption, J, I contains no linearforms, so h ( R ) = (cid:0) e +12 (cid:1) − dim k I and h ( T ) = (cid:0) e +12 (cid:1) − dim k J . It follows that h ( R ) ≥ h ( T ). One also easily see that e ( R ) ≥ e ( T ). So we may replace R by T and assume R is reduced. We may assume now that d = dim R ≥ l be a generic linear form, ¯ R = R/lR and R ′ = ¯ R/N with N = H m ( ¯ R ). BertiniTheorem tells us that one Proj R ′ is still connected in codimension one and reduced.Since R is connected and reduced, H m ( R ) ≤ = 0, so H m ( N ) ≤ = 0. It follows asin the proof of Proposition 2.7 that h i ( R ′ ) = h i ( R ) for i < h ( R ′ ) ≤ h ( R )and c ( R ) = c ( R ′ ). Thus by induction on dimension, we can assume dim R ′ = 2.But here H m ( ¯ R ) ≤ = H m ( ¯ R ) ≤ = 0 say precisely that ¯ R is ( M T ) (see Proposition2.4). Our Theorem 2.8 now applies to show the needed inequalities (note that e + 1 = h ( R ) + h ( R )). If any equality happens, then Theorem 2.9 asserts that¯ R is Cohen-Macaulay (necessarily of regularity 2). To finish we now claim that ifdim R ≥ R/lR is Cohen-Macaulay for a generic l , then so is R . Let N = H m ( R )and R ′ = R/N . We obtain an exact sequence 0 → N/lN → R/lR → R ′ /lR ′ → R/lR is Cohen-Macaulay of dimension at least 1, N/lN = 0, which forces N = 0, thus l is regular on R and we are done. (cid:3) Example 5.7. It is important that k is algebraically closed in the previous theorem.Take R = R [ s, t, is, it ] which is a domain. Then R ∼ = S/I with S = R [ a, b, c, d ] /I EGULARITY, SINGULARITIES AND h -VECTOR OF GRADED ALGEBRAS 15 and I = ( a + c , b + d , ad − bc, ab + cd ) and h ( R ) = − . Note that R ⊗ R C ∼ = C [ x, y, u, v ] / ( xu, xv, yu, yv ) . Next, we give a formula relating the h -vectors of the R and the (deficiency) Extmodules. Recall the notations at the beginning of section 2. Proposition 5.8. Let R = S/I be a standard graded algebra of dimension d . For i = 0 , . . . , d set K i = Ext n − iS ( R, ω S ) . Then we have the following relations betweenthe numerators of Hilbert series: p R (1 /t ) t d = d X i =0 ( − d − i p K i ( t )(1 − t ) d − dim K i Proof. Applying the main Theorem of [2] with M = R, N = ω S = S ( − n ), we obtainan equality of rational functions: X i ( − n − i H K i ( t ) = H R (1 /t ) H ω S ( t ) H S (1 /t )Since H K i ( t ) = p Ki ( t )(1 − t ) dim Ki , H ω S ( t ) = t n (1 − t ) n , H S (1 /t ) = t n ( t − n , the assertion follows. (cid:3) Corollary 5.9. Retain the notations of Proposition 5.8. Suppose that for i < d , wehave H im ( R ) > = 0 (for example, this holds if R is F -pure or if R is Du Bois [22,Theorem 4.4] ). Then h d ( R ) = X ≤ i ≤ d ( − d − i dim k H i m ( R ) . Proof. By (graded) local duality, our assumption implies that for each i < d ,[ K i ] < = 0, so the polynomial p K i ( t ) contains no negative powers of t . What weneed to prove follows immediately from Proposition 5.8. (cid:3) Next we point out that in certain situations, the negativity of h d ( R ) is guaranteedif R is not Cohen-Macaulay. This class of examples shows that one can not hope toimprove of our main theorem too much. Corollary 5.10. Let R = k [ x , . . . , x n ] /I be a standard graded algebra with dim R = d . Suppose that depth R = d − R ) = d − and H d m ( R ) − = 0 . Then h d ( R ) < .Proof. Since depth R = d − 1, reg R = d − H d − m ( R ) > = H d m ( R ) ≥ = 0. Since H d m ( R ) − = 0 and reg R = d − 1, we have H d − m ( R ) = 0. Nowby Corollary 5.9, h d ( R ) = − dim k H d − m ( R ) < (cid:3) Example 5.11. Let A be a standard graded Cohen-Macaulay k -algebra of dimensionat least with zero a -invariant and B = k [ x, y ] . Then the Segre product R = A♯B satisfies the assumptions of Corollary 5.10. It is even ( S d − ) . Some further results and open questions Our results and examples suggest the following natural question, which can beviewed as a vast generalization of everything discussed so far: Question 6.1. Assume that char( k ) = 0 , R satisfies ( S r ) and is Du Bois in codi-mension r − . If ( h , . . . , h s ) is the h -vector of R , is it true that h i ≥ whenever i = 0 , . . . , r ? If furthermore R has Castelnuovo-Mumford regularity less than r , isit true that R is Cohen-Macaulay? Remark 6.2. Question 6.1 is true if r = 2 : in fact we can assume k = ¯ k andthat I contains no linear forms. The condition that R is Du Bois in codimension0, together with ( S ) , says that I = √ I . Further, the condition that R satisfies ( S ) yields that Proj R is connected in codimension 1 by [12, Proposition 2.1, Theorem2.2] . So we can apply Theorem 5.6, in which we proved that R/lR is ( M T ) for ageneric linear form l . We can now conclude like in that proof. The first step to answer 6.1 would be to relax the condition X = Proj R is Du Bois(i.e., R is Du Bois in codimension ≤ d − 1) in key Proposition 3.1 to R being Du Boisin lower codimension. This amounts to strong Kodaira-vanishing type statements.At this point we can prove the following, which is a generalization of [15, Theorem3.15]. Proposition 6.3. Let < k ≤ d − . Suppose X is S d − k and is Du Bois incodimension d − k − . Then we have H j m ( R ) < = 0 for all j ≤ d − k , or equivalently, Ext n − jS ( R, ω S ) > = 0 for all j ≤ d − k .Proof. If k = d − 1, we know X is S and generically reduced and thus X is reduced.Then H m ( R ) < vanishes. For the case k < d − 1, notice that it is enough to show H d − k m ( R ) < = 0 (because the assumption remains valid for any k ′ ≥ k ). Hence itsuffices to prove H d − k − ( X, L − m ) = 0 for all m ≥ k < d − X is S d − k , the possible non-vanishing cohomology of ω • X are: h − ( d − ( ω • X ) , h − ( d − ( ω • X ) , . . . , h − ( d − k − ( ω • X ) , and we havedim h − ( d − ( ω • X ) ≤ k − , dim h − ( d − ( ω • X ) ≤ k − , . . . , dim h − ( d − k − ( ω • X ) ≤ . By Serre duality, H d − k − ( X, L − m ) is dual to H − d + k +1 ( X, ω • X ⊗ L m ). Using spectralsequence, the possible E -page contributions to H − d + k +1 ( X, ω • X ⊗ L m ) are: H k ( X, h − ( d − ( ω • X ) ⊗ L m ) , H k − ( X, h − ( d − ( ω • X ) ⊗ L m ) , . . . , H ( X, h − ( d − k − ( ω • X ) ⊗ L m ) . The latter ones H k − ( X, h − ( d − ( ω • X ) ⊗ L m ) , . . . , H ( X, h − ( d − k − ( ω • X ) ⊗ L m ) EGULARITY, SINGULARITIES AND h -VECTOR OF GRADED ALGEBRAS 17 all vanish because dimension reasons (for example, H k − ( X, h − ( d − ( ω • X ) ⊗ L m ) = 0since dim h − ( d − ( ω • X ) ≤ k − H k ( X, h − ( d − ( ω • X ) ⊗ L m ), note that by Theorem 3.2 we have a short exactsequence 0 → h − ( d − ( ω • X ) → h − ( d − ( ω • X ) → C → C ≤ k − X is Du Bois in codimension ( d − − k . Theinduced long exact sequence of sheaf cohomology:0 = H k ( X, h − ( d − ( ω • X ) ⊗ L m ) → H k ( X, h − ( d − ( ω • X ) ⊗ L m ) → H k ( X, C ⊗ L m ) = 0 , where the first 0 is by Theorem 3.3, shows that H k ( X, h − ( d − ( ω • X ) ⊗ L m ) = 0 for all m ≥ 1. Hence all E -page contributions to H − d + k +1 ( X, ω • X ⊗ L m ) are 0, it followsthat H − d + k +1 ( X, ω • X ⊗ L m ) = 0 for all m ≥ H d − k − ( X, L − m ) = 0 for all m ≥ (cid:3) References [1] F. Ambro, Quasi-log varieties , Tr. Mat. Inst. 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Department of Mathematics, University of Kansas, Lawrence, KS 66045-7523,USA E-mail address : [email protected] Department of Mathematics, Purdue University, West Lafayette, IN 47906, USA E-mail address : [email protected] Dipartimento di Matematica, Universit`a di Genova Via Dodecaneso, 35 16146 Gen-ova, Italy E-mail address ::