The Second Vanishing Theorem for Local Cohomology Modules
aa r X i v : . [ m a t h . A C ] F e b THE SECOND VANISHING THEOREM FOR LOCAL COHOMOLOGYMODULES
WENLIANG ZHANG
Abstract.
We prove the Second Vanishing Theorem for local cohomology modules of an un-ramified regular local ring in its full generality and provide a new proof of the Second VanishingTheorem in prime characteristic p . As an application of our vanishing theorem for unramifiedregular local rings, we extend our topological characterization of the highest Lyubeznik number ofan equi-characteristic local ring to the setting of mixed characteristic. An upper bound of localcohomological dimension in mixed characteristic is also obtained by partially extending Lyubeznik’svanishing theorem in prime characteristic p to mixed characteristic. Introduction
The study of vanishing of local cohomology modules has a long and rich history. In [Har67,p. 79], Grothendieck stated the following problem.
Problem . Let R be a commutative noetherian local ring, a be an ideal of R ,and n be an integer. Find conditions under which H i a ( M ) = 0 for all i > n and all R -modules M .Grothendieck proved that H i a ( M ) = 0 for all i > dim( R ) and all R -modules M ([Har67]), whichsolved Problem 1.1 for n = dim( R ). A solution to Problem 1.1 in the case when n = dim( R ) − n = dim( R ) −
2, we consider thefollowing definition:
Definition 1.2.
Let ( R, m ) be a noetherian local ring and let e R denote the completion of the stricthenselization of the completion of R . We say that the Second Vanishing Theorem holds for R if thefollowing conditions are equivalent:(1) H j a ( M ) = 0 for all j > d − R -modules M ;(2) dim( R/ a ) ≥ e R/ a e R is connected.If R is not regular, then the Second Vanishing Theorem may not hold for R , cf. [HZ18, 7.7].When R is regular, some positive results are known. When R is a polynomial ring over a fieldand a is a homogeneous ideal, then the Second Vanishing Theorem holds, as proved by Hartshorne([Har68, 7.5]) who also coined the name ‘Second Vanishing Theorem’ and proposed the problemto prove the Second Vanishing Theorem for all regular local rings ([Har68, p. 445]). Subsequently,Ogus ([Ogu73, Corollary 2.11]) proved that the Second Vanishing Theorem holds for regular localrings of equicharacteristic 0 and Peskine-Szpiro showed in [PS73, III 5.5] that the Second VanishingTheorem holds for regular local rings of equicharacteristic p . [HL90] provided a unified proof thatthe Second Vanishing Theorem holds for regular local rings of equicharacteristic. Extending theSecond Vanishing Theorem to rings that do not contain a field has been a major open problem inthe study of local cohomology. In §
2, we prove that:
Theorem 1.3. If R is an unramified regular local ring of mixed characteristic, then the SecondVanishing Theorem holds for R . The author is partially supported by NSF through DMS-1752081. special case of Theorem 1.3, when dim( R/ a ) ≥ and R/ a is equidimensional, can be found in[HNnBPW18].In prime characteristic p >
0, we produce a new proof of the Second Vanishing Theorem usingthe action of Frobenius and some equivalent formulations of the Second Vanishing Theorem incharacteristic p >
0, which can be found in § p isa vanishing theorem due to Lyubeznik [Lyu06b, Theorem 1.1]. In §
5, we present some extensions ofLyubeznik’s vanishing theorem to mixed characteristic. As a consequence of these extensions, weobtain the following result on local cohomological dimension. (Recall that the local cohomologicaldimension of an ideal I in a noetherian ring A , denoted by lcd A ( I ), is max { j | H jI ( A ) = 0 } .) Theorem 1.4.
Let ( R, m ) be an n -dimensional regular local ring of mixed characteristic (0 , p ) andlet a be an ideal that contains p . Then(1) lcd R ( a ) ≤ n − F -depth( R/ a ) , when R is unramified; and(2) lcd R ( a ) ≤ n + 1 − F -depth( R/ a ) , when R is ramified. For a noetherian local ring ( A, m ) of prime characteristic p , its local cohomology modules H i m ( A )are equipped with an action of Frobenius f : H i m ( A ) → H i m ( A ) induced by the Frobenius endomor-phism on A . The F -depth of A is defined as F -depth( A ) := min { j | H j m ( A ) is not nilpotent under f } . As an application of Theorem 1.3, we extend our results in [Zha07] to local rings of mixedcharacteristic. Before stating our extension, we recall the definition of the Hochster-Huneke graphof a local ring. Let A be a noetherian local ring. The Hochster-Huneke graph Γ A of A is defined asfollows. Its vertices are the top-dimensional minimal prime ideals of A , and two distinct vertices P and Q are joined by an edge if and only if ht B ( P + Q ) = 1.The main result in § Theorem 1.5.
Let ( A, m , k ) be a d -dimensional noetherian local ring. Assume that A = R/I where R is an n -dimensional unramified regular local ring ( R, m ) of mixed characteristic (0 , p ) .Then dim k (Hom R ( k, H d m H n − dI ( R ))) is the number of connected components of the Hochster-Hunekegraph Γ e A of e A , where e A is the completion of the strict Henselization of the completion of A . Proof of Theorem 1.3
To prove Theorem 1.3, we need the following result from [PS73].
Theorem 2.1 (Th´eor`eme III.5.1 in [PS73]) . Let ( R, m ) be a d -dimensional complete regular localring with a separably closed residue field and let a be an ideal of R . Assume that Spec( R/ a ) \{ m } is connected and dim( R/ a ) ≥ . Then the following are equivalent.(1) H i a ( R ) is artinian for all i ≥ d − ;(2) H i a ( R ) = 0 for all i ≥ d − .Remark . There are many characterizations of artinianness; one such characterization ( cf. [HK91, Remark 1.3]) asserts that, if ( A, m , k ) is complete local ring, then an A -module M isartinian if and only if that Supp( M ) = { m } and M has finite dimensional socle, i.e. Hom R ( k, M )is a finite dimensional k -space.We now prove Theorem 1.3. Proof of Theorem 1.3.
Since H j a ( M ) ⊗ R e R ∼ = H j a ( M ⊗ R e R ) by flat base change and e R is faithfullyflat over R , we may assume that R is complete with a separably closed residue field by replacing R with e R . ⇒ (2). If dim( R/ a ) ≤
1, then either H d − a ( R ) = 0 (when dim( R/ a ) = 1) or H d a ( R ) = 0 (whendim( R/ a ) = 0). Hence dim( R/ a ) ≥
2. If the punctured spectrum of R/ a were disconnected, thenthere would be two ideals I, J of height at most d − I ∩ J = a and √ I + J = m . TheMayer-Vietoris sequence says0 = H d − a ( R ) → H dI + J ( R ) = H d m ( R ) → H dI ( R ) ⊕ H dJ ( R ) . Since ht ( I ) , ht ( J ) ≤ d −
1, the Hartshorne-Lichtenbaum Vanishing Theorem ([Har68, Theorem3.1]) implies H di ( R ) = H dJ ( R ) = 0 which would imply that H d m ( R ) = 0, a contradiction. Hence thepunctured spectrum of R/ a must be connected.(2) ⇒ (1). It was observed in [Har68] that H jI ( M ) = 0 for all j > t and all R -modules M if andonly if H jI ( R ) = 0 for all j > t . Hence it suffices to show H d − a ( R = H d a ( R ) = 0 (since H >d a ( R ) = 0by Grothendieck Vanishing). Combining Theorem 2.1 and Remark 2.2, it suffices to show thatboth of H d a ( R ) and H d − a ( R ) are supported only at the maximal ideal and have finite dimensionalsocle. It follows from the Hartshorne-Lichtenbaum vanishing theorem ([Har68, Theorem 3.1]) that H d a ( R ) = 0 and Supp( H d − a ( R )) = { m } . On the other hand, [Lyu00, Theorem 1] (or [NnB13,Theorem 1.2]) shows that H d − a ( R ) has finite dimensional socle since R is an unramified completeregular local ring. This finishes the proof. (cid:3) Remark . The analogue of Theorem 1.3 in the ramified case remains open in general.It follows from our proof of Theorem 1.3 that: let ( R, m , k ) be an n -dimensional ramified completeregular local ring of mixed characteristic (0 , p ) and a be an ideal such that Spec( R/ a ) \{ m } isconnected and dim( R/ a ) ≥
2, if dim k Hom R ( k, H n − a ( R )) < ∞ , then H n − a ( R ) = 0.Note that dim k Hom R ( k, H n − a ( R )) is one of the Bass numbers of the local cohomology module H n − a ( R ). One ought to remark that the finiteness of Bass numbers of local cohomology modules ofa ramified regular local ring of mixed characteristic was first conjectured in [Hun92] and has beena long standing open problem. This is one of the reasons we consider a reduction and a new proofof the Second Vanishing Theorem in characteristic p in the subsequent sections.3. Reduction to dimension 2
In this section, we show that the Second Vanishing Theorem can be reduced to the case whenthe ideal a is a prime ideal of dimension 2, i.e. dim( R/ a ) = 2, and discuss a related approach toproving the Second Vanishing Theorem in general.Our main result is the following: Theorem 3.1.
Let ( R, m , k ) be a d -dimensional complete regular local ring with a separably closedresidue field. If H d p ( R ) = H d − p ( R ) = 0 for all prime ideals p with height d − , then H d a ( R ) = H d − a ( R ) = 0 for all ideals I such that Spec( R/ a ) \{ m } is connected and that dim( R/ a ) ≥ .Proof. We will use reverse induction on the height of a . Since the conclusion only depends only theradical of a , we may assume that a is radical. First, assume ht( a ) = d − a = ∩ ti =1 p i . SinceSpec( R/ a ) \{ m } is connected, each prime p i must have height d −
2. The Hartshorne-LichtenbaumVanishing Theorem implies that H d a ( R ) = 0 We will use induction on t to show that H d − a ( R ) = 0.When t = 1, this is precisely our assumption. Assume that t ≥
2. We consider the exact sequence · · · → H d − p ( R ) ⊕ H d − ∩ ti =2 p i ( R ) → H d − a ( R ) → H d p + ∩ ti =2 p i ( R ) . Since Spec( R/ a ) \{ m } is connected, p + ∩ ti =2 p i is not m -primary; consequently it follows fromHartshorne-Lichtenbaum Vanishing Theorem that H d p + ∩ ti =2 p i ( R ) = 0. The induction hypothesisasserts that H d − ∩ ti =2 p i ( R ) = 0. Therefore, H d − a ( R ) = 0. ow assume that ht( a ) ≤ d −
3. First, we treat the case when a is a prime ideal. In thiscase pick r ∈ m \ a , then by Faltings’ connectedness theorem Spec( R/ ( a , r )) \{ m } is connected sincethe number of generators of ( r ) in R/ a is at most dim( R/ a ) −
2. As ht(( a , r )) = ht( a ) + 1,induction hypotheses implies that H d − a ,r ) ( R ) = H d ( a ,r ) ( R ) = 0. Since Supp( H d − a ( R )) ⊆ { m } , wehave H d − a ( R ) r = 0. Now the exact sequence → H j − a ,r ) ( R ) → H j a ( R ) → H j a ( R ) r → shows that H d a ( R ) = H d − a ( R ) = 0. Next assume that all minimal prime ideals of a have he same height. Theninduction on the number of minimal primes of a shows that H d a ( R ) = H d − a ( R ) = 0. Finally, in thegeneral case, write a = I ∩ J where all minimal primes of I have height ht( a ) and ht( J ) ≥ ht( a ) + 1.Then induction hypothesis and the Mayer-Vietoris sequence → H j − I + J ( R ) → H j − I ( R ) ⊕ H j − j ( R ) → H j − a ( R ) → H jI + J ( R ) → prove that H d a ( R ) = H d − a ( R ) = 0. (cid:3) Based on Theorem 3.1, from now on we will assume that p is a height ( d −
2) prime ideal ina d -dimensional complete regular local ( R, m , k ). Let { I n } n ∈ N be a sequence ideals of R that iscofinite with { p n } n ∈ N . It is clear that, to prove the socle of H d − p ( R ) is finite, it suffices to show that H d − p ( R ) is finitely generated. According to [PS73, Proposition III.4.15], to show that H d − p ( R ) isfinitely generated, it suffices to show that the number of generators of Ext d − R ( R/I n , R ) is uniformlybounded ( i.e. bounded by a constant independent of n ). Proposition 3.2.
Let p be a height ( d − prime ideal in a d -dimensional complete regular local ( R, m , k ) . If there exist a sequence of p -primary ideals { I n } n ∈ N that is cofinite with { p n } n ∈ N anda constant C such that β d − ( R/I n ) ≤ C for all n (here β d − ( R/I n ) denotes the ( d − st Bettinumber of R/I n ), then the number of generators of Ext d − R ( R/I n , R ) is uniformly bounded (andconsequently H d − p ( R ) will be finitely generated in this case).Proof. We wish to show that there is a constant C such that dim k (Ext d − R ( R/I n , R ) ⊗ R k ) ≤ C forall n . Consider the following spectral sequence ([Dol62, Theorem 4.1]): E − i,j = Tor Ri (Ext jR ( R/I n , R ) , k ) ⇒ Ext j − iR ( R/I n , k ) . The maps coming into and going out of E ,d − r are E − r,d + r − r → E ,d − r → E r,d − rr . Since I n is p -primary, we have Ext dR ( R/I n , R ) = 0; and clearly Ext d + r − ( R/I n , R ) = 0 for r ≥
3. This shows that E − r,d + r − r = 0 for all r ≥
2. Also, clearly E r,d − rr = 0. Hence E ,d − = E ,d − ∞ . Therefore, to bounddim k (Ext d − R ( R/I n , R ) ⊗ R k ) uniformly, it suffices to bound dim k (Ext d − R ( R/I n , k )) uniformly.Since R is complete, we have Ext d − R ( R/I n , k ) = Ext d − R ( R/I n , k ∨ ) ∼ = Tor d − ( R/I n , k ) ∨ , where( − ) ∨ denotes the Matlis dual. This finishes the proof of our proposition. (cid:3) Inspired by Proposition 3.2, we would like to pose the following question.
Question . Let ( R, m , k ) be a d -dimensional complete regular local ring and let p be a height d − R . Let β i ( M ) denote the i -th Betti number of an R -module M .Do there exist(1) a sequence of p -primary ideals { I n } n ∈ N which is cofinal with { p n } n ∈ N ; and(2) a constant C independent of n such that β d − (cid:16) R/I n (cid:17) ≤ C for all n ?When char( R ) = p >
0, Question 3.3 has a positive answer. emark . Since R is regular and char( R ) = p >
0, the Frobenius powers p [ p n ] of p are p -primary.Let 0 → R ⊕ b d → R ⊕ b d − → · · · → R → R/ p → R/ p . Since R isregular, the Peskine-Szpiro functor F is exact. It is well-known that F n ( R/ p ) ∼ = p [ p n ] and that F n ( R ⊕ ℓ ) ∼ = R ⊕ ℓ for all n, ℓ ∈ N . Hence β d − ( R/ p [ p n ] ) = β d − ( R/ p ) for all n .4. Second Vanishing Theorem in prime characteristic p , Revisited As we note in Remark 2.3, to prove the Second Vanishing Theorem in the ramified case, oneapproach is to proving finiteness of Bass numbers of local cohomology modules in the ramifiedcase. However, the finiteness of Bass numbers of local cohomology modules in the ramified casehas been a long-standing open problem. Moreover, rings of mixed characteristic (0 , p ) and those ofcharacteristic p can be linked via a tilting correspondence ( cf. [Sch12]). This perspective has beensuccessfully exploited; for instance, vanishing of local cohomology of the absolute integral closurein mixed characteristic is proved in [Bha20] using the vanishing of corresponding local cohomologymodules in characteristic p via the tilting correspondence. Therefore, it is desirable to seek alternateapproaches, without using finiteness of Bass numbers of local cohomology modules, in characteristic p . In this section we provide a new proof of the Second Vanishing Theorem in characteristic p andsome equivalent formulations.We recall the following result due to Lyubeznik (Theorem 1.1 in [Lyu06b]). Theorem 4.1.
Let ( R, m ) be a regular local ring of dimension n of prime characteristic p and A be a homomorphic image of R . Let I be the kernel of R ։ A . Then H n − iI ( R ) = 0 if and only ifthe Frobenius action on H i m ( A ) is nilpotent.Remark . In the statement of the Second Vanishing Theorem, the residue field of e R is separablyclosed. By gonflement, one can find a flat local extension ( e R, e m ) → ( R ′ , m ′ ) such that such that m ′ = e m R ′ and R ′ / m ′ is an algebraic closure of e R/ e R . Since R ′ is faithfully flat over e R , one has H j a ( e R ) = 0 ⇔ H j a R ′ ( R ′ ) = 0. Furthermore, since e R/ e m is separably closed, the punctured spectrumof e R/ a is connected if and only if the same hold for R ′ / a R ′ . Hence when proving the SecondVanishing Theorem, one may assume that the residue field is algebraically closed.We are in position to present a new proof of the Second Vanishing Theorem in characteristic p . A new proof of Second Vanishing Theorem in characteristic p . Let ( R, m ) be an n -dimensionalnoetherian regular local ring of prime characteristic p and I be an ideal of height at most n − R is a complete noetherian regularlocal ring of prime characteristic p with an algebraically closed residual field and I is a height-( d − H nI ( R ) = H n − I ( R ) = 0 implies the connectednessof Spec( R/I ) \{ m } is the same as in the proof of Theorem 1.3 (the proof of this particular implicationis characteristic-free). We will focus on the other implication.Assume that Spec( R/I ) \{ m } is connected and we wish to show H nI ( R ) = H n − I ( R ) = 0.The vanishing H nI ( R ) = 0 follows from Hartshorne-Lichtenbaum vanishing. It remains to show H n − I ( R ) = 0. According to Theorem 4.1, this is equivalent to the niloptence of the Frobeniusaction on H m ( A ).Since A is a local integral domain, H m ( A ) is finitely generated and hence has finite length. Let f denote the Frobenius action on H m ( A ). Since H m ( A ) has finite length, every element in m H m ( A )is f -nilpotent. It follows that H m ( A ) is f -nilpotent if and only if ( H m ( A )) s := T t f t ( H m ( A )) is 0.Set U = Spec( R/I ) \{ m } . Then [HS77, 3.1] asserts that there is an exact sequence0 → H m ( A ) s → A s → H ( U, O U ) s → H m ( A ) s → . ince U is connected and A is a domain (whose residue field is algebraically closed), the map inthe middle A s → H ( U, O U ) s is the isomorphism k ∼ −→ k where k is the residue field of A . Hence H m ( A ) s = 0. This proves that H m ( A ) is f -nilpotent and hence H n − I ( R ) = 0. (cid:3) Next we will consider some equivalent formulations of the Second Vanishing Theorems in char-acteristic p . To this we recall some basic facts regarding S -ification from [HH94]. Remark . Let ( A, m ) be a complete local domain with a canonical module ω , then(1) Hom A ( ω, ω ) is a commutative complete local ring and the natural map A → Hom A ( ω, ω )is an injective module-finite ring homomorphism;(2) Hom A ( ω, ω ) satisfies Serre’s ( S )-condition as both an a -module and as a ring on its own. Theorem 4.4.
The following statements are equivalent:(1) The Second Vanishing Theorem holds for regular local rings of characteristic p .(2) Let ( A, m ) be a -dimensional complete local domain of prime characteristic p > with acanonical module ω . Assume A/ m is separably closed. For each element φ ∈ Hom A ( ω, ω ) there is an integer e such that φ p e ∈ A , i.e. there is an element a ∈ A such that φ p e is themultiplication by a on ω .(3) Let A be a 2-dimensional complete local domain of characteristic p with a separably closedresidue field. Then there exists a positive integer n such that, for all systems of parameters x, y , one has ( x : y ) [ p n ′ ] = ( x p n ′ ) for all n ′ ≥ n .Proof. First we prove that (1) ⇒ (2). Assume that the Second Vanishing Theorem holds in char-acteristic p . Let ( A, m ) be as in (2). Set S := Hom A ( ω, ω ). Consider the short exact sequence0 → A → S → C →
0. (2) is equivalent to proving that the Frobenius on C is nilpotent.The short exact sequence induces a long exact sequence on local cohomology0 = H m ( S ) → H m ( C ) → H m ( A ) → H m ( S ) = 0where H m ( S ) = H m ( S ) = 0 since S satisfies ( S )-condition. Write A = R/I where R is an n -dimensional complete regular local ring. By the Second Vanishing Theorem, H n − I ( R ) = 0. ByTheorem 4.1, the Frobenius on H m ( A ) must be nilpotent. Hence so is the Frobenius on H m ( C ).We claim that H m ( C ) = C and we reason as follows. It suffices to show that C is supported inthe maxmail ideal only. Let p be any non-maximal prime ideal. Since A p is Cohen-Macaulay (notethat dim( A ) = 2), we have S p ∼ = A p and hence C p = 0. This shows that C is supported in themaximal ideal only. It is clear that C is a finite A -module, thus H m ( C ) = C . So, the Frobenius on C is nilpotent and, equivalently, (2) holds.Next, we prove (2) ⇒ (1). Assume now (2) holds, and we wish to prove that Second VanishingTheorem. To this end, let ( R, m ) be an n -dimensional prime-characteristic complete regular localring with a separably closed residual field. By Proposition 3.1, it suffices to prove that H n − P ( R ) = 0for all prime ideals P of height n −
2. Set A = R/P . Then A satisfies that hypotheses in (2). Bythe argument in previous paragraph, we see that the Frobenius on H m ( A ) is nilpotent. Accordingto Theorem 4.1, we have H n − P ( R ) = 0. This completes the proof of (2) ⇒ (1) and hence (1) ⇔ (2).Next we prove that (1) ⇔ (3). We have seen that (1) is equivalent to H m ( A ) being f -nilpotentwhere ( A, m ) is a 2-dimensional complete local domain of characteristic p with a separably closedresidue field. Given an arbitrary system of parameters x, y in A , each element in H m ( A ) can bewritten as [ ax , by ] such that ay = bx . Since H m ( A ) is artinian, it is f -nilpotent if and only if thereis an integer n such that f n ( H m ( A )) = 0 (and consequently f n ′ ( H m ( A )) = 0 for all n ′ ≥ n ). Thisholds if and only if [ ax , by ] p n ′ = 0 for all a, b, x, y such that ay = bx . Note that [ ax , by ] p n ′ = 0 ifand only if a p n ′ ∈ ( x p n ′ ) and ay = bx if and only if a ∈ ( x : y ). This completes the proof of(1) ⇔ (3). (cid:3) emark . A direct proof of (2) or (3) will produce another proof of the Second VanishingTheorem in prime characteristic p .5. Extensions of Theorem 4.1 to mixed characteristic
Recall that Lyubeznik’s vanishing theorem links the vanishing of H n − i a ( R ) and the action ofFrobenius on H i m ( R/ a ) where ( R, m ) is a regular local ring of prime characteristic p and a is anideal of R . In this section, we consider some (partial) extensions to mixed characteristic. Theorem 5.1.
Let ( R, m ) be an n -dimensional unramified regular local ring of mixed characteristic (0 , p ) and a be an ideal of R . Assume that p ∈ a (hence R/ a contained a field of characteristic p ).If H i m ( R/ a ) is Frobenius nilpotent, then H n − i a ( R ) = 0 . Proof.
Set R = R/ ( p ) and a = a / ( p ). Then R/ a = R/ a . Since R is unramified, R is an ( n − p . By our assumption on H i m ( R/ a ) and Theorem4.1, we have H ( n − − i a ( R ) = 0. The short exact sequence 0 → R · p −→ R → R → H ( n − − i a ( R ) = H ( n − − i a ( R ) → H n − i a ( R ) · p −→ H n − i a ( R ) . Since p ∈ a , this forces H n − i a ( R ) = 0. (cid:3) We ought to remark that Theorem 5.1 only extends one implication in Theorem 4.1. As to theother implication, we ask:
Question . Does the converse to Theorem 5.1 hold?When i = 1, the answer is affirmative and it follows from the Second Vanishing Theorems incharacteristic p and in unramified mixed characteristic ( i.e. our Theorem 1.3). Theorem 5.3.
Let ( R, m ) be an n -dimensional unramified regular local ring of mixed characteristic (0 , p ) and a be an ideal of R . Assume that p ∈ a (hence R/ a contained a field of characteristic p ).Then H n − a ( R ) = 0 if and only if H m ( R/ a ) is Frobenius nilpotent.Proof. ⇐ is a special case of Theorem 5.1; it remains to prove ⇒ .Set R = R/ ( p ) and a = a / ( p ). Then R/ a = R/ a . Note that, since R is regular, H n a ( R ) = 0by the Hartshorne-Lichtenbaum vanishing theorem. By Theorem 1.3, the punctured spectrum of e R a e R is connected ( e R is the completion of the strict henselization of the completion of R ) anddim( R a ) ≥
2. Since p ∈ a , the same will hold for R and hence H n − a ( R ) = H n − a ( R ) = 0by the Second Vanishing Theorem in characteristic p . It now follows from Theorem 4.1 that H m ( R/ a ) = H m ( R/ a ) is Frobenius nilpotent. (cid:3) Question 5.2 concerns a specific cohomological degree. It turns out that one can draw a weakerconclusion if one considers all vanishing above a specific cohomological degree.
Proposition 5.4.
Let ( R, m ) be an n -dimensional unramified regular local ring of mixed character-istic (0 , p ) and a be an ideal of R . Assume that p ∈ a (hence R/ a contained a field of characteristic p ). If H n − j a ( R ) = 0 for all j ≤ t for a fixed integer t , then H j − m ( R/ a ) is Frobenius nilpotent for all j ≤ t .Proof. Set R = R/ ( p ) and a = a / ( p ). Then R/ a = R/ a . Consider the long exact sequence oflocal cohomology induced by 0 → R · p −→ R → R →
0. Since H n − j a ( R ) = 0 for all j ≤ t , one has H n − j a ( R ) = H n − j a ( R ) = 0 for all j ≤ t . Theorem 4.1 shows that H j − m ( R/ a ) = H j − m ( R/ a ) = H ( n − − ( n − j ) m ( R/ a ) s Frobenius nilpotent for j ≤ t . (cid:3) When R is a ramified, the situation seems to be much more mysterious since R/ ( p ) is no longera regular ring. We are only able to obtain a weaker version of Theorem 5.1 as follows. Theorem 5.5.
Let ( R, m ) be an n -dimensional ramified regular local ring of mixed characteristic (0 , p ) and a be an ideal of R . Assume that p ∈ a (hence R/ a contained a field of characteristic p ).Assume that H j m ( R/ a ) is Frobenius nilpotent for j ≤ t for a fixed integer t , then H n +1 − j a ( R ) = 0 for j ≤ t .Proof. Without loss of generality, we may assume that R is complete. By Cohen’s Structure The-orem, R ∼ = V [[ x , . . . , x n ]] / ( p − f ) where f ∈ m . Set A = ( V /pV )[[ x , . . . , x n ]] (an n -dimensionalregular local ring of characteristic p ). We will denote the image of f in A by f again. Then R/ ( p ) ∼ = A/ ( f ) . Set R = R/ ( p ) and a = a / ( p ). We may view a as an ideal in A/ ( f ). Let b be the ideal in A suchthat b / ( f ) = a . It is clear that R/ a ∼ = A/ b . And hence H j m ( A/ b ) is Frobenius nilpotent for j ≤ t . Theorem 4.1 asserts that H n − j b ( A ) = 0 for j ≤ t . The exact sequence of local cohomology induced by 0 → A · f −→ A → A/ ( f ) → H n − j b ( A/ ( f )) = 0for j ≤ t . (This is where we need to assume vanishing above a cohomological degree instead ofvanishing at a single degree.) Consequently H n − j a ( R/ ( p )) = H ( n − j a ( R/ ( p )) = 0 for j ≤ t .Consider the exact sequence of local cohomology induced by the exact sequence 0 → R · p −→ R → R → H n − j a ( R/ ( p )) → H n +1 − j a ( R ) · p −→ H n +1 − j a ( R )for j ≤ t . Since p ∈ a , this forces H n +1 − j a ( R ) = 0 for j ≤ t . (cid:3) We are in position to prove Theorem 1.4.
Proof of Theorem 1.4.
Since R is regular and p ∈ a , it follows from Theorem 5.1 (unramified case),for each t ≤ F -depth( R/ a ), H i a ( R ) = 0 for i ≥ n − t Therefore, lcd R ( a ) ≤ n − F -depth( R/ a ) . The ramified case follows similarly from Theorem 5.5. (cid:3) The highest Lyubeznik number of a local ring of mixed characteristic
Let ( A, m , k ) be a complete local ring of mixed characteristic. By Cohen’s structure theorem, A admits a surjection π : R ։ A from a complete unramified regular local ring ( R, m , k ). Let I bethe kernel of π and n denote dim( R ). We have the following. Proposition 6.1.
Let
A, R, I, n be as above. Then dim k Hom R ( k, H i m H n − jI ( R )) depends only on A, i, j , but not on the choices of R or π . roof. The proof follows the same line of ideas as in [Lyu93, 4.1] and [NnBW13, 3.4], hence we willprovide a sketch only.Let V be a coefficient ring of A (whose existence is guaranteed by Cohen’s structure theorem).Then one can reduce the proof to proving the following:(6.1.1) dim k Hom R ( k, H i m H n − jI ( R )) = dim k Hom R [[ x ]] ( k, H i ( m ,x ) H n +1 − j ( I,x ) ( R [[ x ]]))where x is an indeterminate over R . Set S = R [[ x ]]. For each R -module M , define G ( M ) := M ⊗ R H x ) ( S ) ( G was introduced in [Lyu93, Proof of 4.3] and further studied in [NnBW14, § G ( H i m H n − jI ( R )) = H i ( m ,x ) H n +1 − j ( I,x ) ( R [[ x ]]). Now (6.1.1) follows from[NnBW14, 3.12] which asserts that Hom R ( k, M ) = Hom S ( k, G ( M )) for all R -modules M . (cid:3) Definition 6.2.
Let ( A, m , k ) be a noetherian local ring of mixed characteristic and let ˆ A denoteits completion. Let π : R ։ ˆ A be a surjection from an n -dimensional complete unramified regularlocal ring ( R, m , k ) of mixed charaterisrtic. Define λ i,j ( A ) := dim k Hom R ( k, H i m H n − jI ( R )) . Remark . By Cohen’s Structure Theorem of complete local rings, if A is a complete local ringof mixed characteristic, then A admits a surjection π : R ։ ˆ A from a complete unramified regularlocal ring R .If A (not necessarily complete) itself admits a surjection R ′ ։ A from an n ′ -dimensional completeunramified regular local ring ( R ′ , m ′ , k ) with kernel I ′ , then one can check that λ i,j ( A ) = dim k Hom R ′ ( k, H i m ′ H n ′ − jI ′ ( R ′ )) . Remark . If ( A, m , k ) is a noetherian local ring containing a field and admits presentation A = R/I where ( R, m , k ) is an n -dimensional regular local ring containing the same field, then it followsfrom [Lyu93, 1.4, 4.1]) that(6.4.1) dim k Hom R ( k, H i m H n − jI ( R )) = dim k Ext iR ( k, H n − jI ( R )) . However, when A does not contain a field, (6.4.1) may no longer hold.Let R = Z [[ x , . . . , x ]] and let I be the ideal of R generated by the 10 monomials { x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x } . Let m denote (2 , x , . . . , x ) and set k = R/ m , A = R/I . Then it is proved in [DSZ19, 5.5] that H I ( R ) ∼ = H x ,...,x ) ( R/ (2)) . Since H x ,...,x ) ( R/ (2)) admits an injective resolution (as an R -module)0 → H x ,...,x ) ( R/ (2)) → E R ( R/ m ) · −→ E R ( R/ m ) → , one can check that dim k Hom R ( k, H m H I ( R )) = 0 = 1 = dim k Ext R ( k, H I ( R )) . Theorem 1.5 is one reason why we use dim k Hom R ( k, H i m H n − jI ( R )) as our definition of λ i,j ( A ).Next, we focus on λ d,d ( A ) where d = dim( A ) and prove our Theorem 1.5, which extends ourresults in [Zha07] to local rings of mixed characteristic. Proof of Theorem 1.5.
Since both completion and strict Henselization are faithfully flat, we mayassume that both A and R are complete with separably closed residue fields. Assume that Γ , . . . , Γ t are the connected components of Γ A . For 1 ≤ j ≤ t , let I j be the intersection of the minimal primes f R that are vertices of Γ j . Similar to the proof of [Lyu06a, Proposition 2.1], using the Mayer-Vietoris sequence of local cohomology, one can prove that H n − dI ( R ) = ⊕ tj =1 H n − dI j ( R ) . Hence dim k Hom R ( k, H d m H n − dI ( R )) = t X j =1 dim k Hom R ( k, H d m H n − dI j ( R )) . We are reduced to proving that dim k Hom R ( k, H d m H n − dI ( R )) = 1 when Γ A = Γ R/I is connected and A is equidimensional. The rest of the proof follows the same strategy as in [Zha07].We will use induction on dim( A ).First, assume that dim( A ) = dim( R/I ) = 2. Since Γ A is connected and A is equidimensional,Spec( A ) \{ m } is also connected. Our Theorem 1.3 now implies that H nI ( R ) = H n − I ( R ) = 0. Hencethe following is an injective resolution of H n − I ( R ):0 → H n − I ( R ) → M I ⊆ p ; ht( p )= n − E ( R/ p ) → M I ⊆ q ; ht( q )= n − E ( R/ q ) → E ( R/ m ) → . Therefore, we have H m H n − I ( R ) = E ( R/ m ) and H j m H n − I ( R ) = 0 for j = 2. This proves that casewhen dim( A ) = 2.Assume now dim( A ) ≥
3. We will pick an element r ∈ m as follows (same as in the proof of[Zha07, Theorem 1.4]). If Supp( H n − d +1 I ( R )) = { m } , then by prime avoidance we pick r that is notin any minimal prime of I nor in any minimal element of Supp( H n − d +1 I ( R )). If Supp( H n − d +1 I ( R )) = { m } , then we pick r ∈ m that is not in any minimal prime of I . Then dim( R/I + ( r )) = dim( A ) − R/I + ( r ) is also equidimensional. Our theorem now follows from the following statements:(1) H d m H n − dI ( R ) ∼ = H d − m H n − d +1 I +( r ) ( R ), and(2) Γ R/ √ I +( r ) is connected.These two statements appeared as Proposition 2.1 and Proposition 2.2, respectively, in [Zha07].The proofs of these two statements in [Zha07] do not require the ring to contain a field. Thiscompletes the proof our theorem. (cid:3) Corollary 6.5.
If a d -dimensional noetherian local ring A satisfies the Serre’s ( S ) condition, then λ d,d ( A ) = 1 .Proof. When A contains a field, this is known ( cf. [NnBWZ16, Theorem 4.6]). Assume A doesn’tcontain a field. According to our Theorem 1.5, it suffices to show that that Hochster-Huneke graphΓ e A of e A = d A sh is connected. Since A is S , so is e A . Then [Har62, Remark 2.4.1] implies that e A isequidimensional. Therefore, Γ e A must be connected by [HH94, Theorem 3.6]. (cid:3) Remark . Let ( A, m , k ) be a complete local ring of prime characteristic p . Then, by CohenStructure Theorem, one can write A = R/I where R = k [[ x , . . . , x n ]] is a formal pwer series ringover k . Denote the maximal ideal of R by n . One may consider(6.6.1) dim k Hom R ( k, H i n H n − jI ( R ))which agrees with the Lyubeznik number λ i,j ( A ) ( cf. [Lyu93, 1.4, 4.1]). On the other hand, A canbe also written as R ′ /I ′ where R ′ is a complete unramified regular local ring of mixed characteristic(0 , p ). Denote the maximal ideal of R ′ by n ′ and dim( R ′ ) by n ′ . Following Definition 6.2, one mayconsider(6.6.2) dim k Hom R ′ ( k, H i n ′ H n ′ − jI ′ ( R ′ )) . A natural question is that whether (6.6.1) agrees with (6.6.2) (for fixed i and j ). hen i = dim( A ) and j = dim( A ), it follows immediately from our Theorem 1.5 and the maintheorem in [Zha07] that the numbers (6.6.2) and (6.6.1) coincide, both of which agree with thenumber of the connected components of the Hochster-Huneke graph of A . Remark . The proof of Theorem 1.5 is an example of applying our Theorem 1.3 to extend results,previously only known in equicharacteristic, to mixed characteristic. One may also apply Theorem1.3 to extend other results (for instance, some results in [NnBSW19]) to mixed charateristic, whichwe will leave to another project.
Acknowledgement.
The author would like to thank Bhargav Bhatt and Gennady Lyubeznik forrelated conversations and Luis N´u˜nez-Betancourt for comments on a draft.
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Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago,Chicago, IL 60607
Email address : [email protected]@uic.edu