Bockstein homomorphisms in local cohomology
aa r X i v : . [ m a t h . A C ] J a n BOCKSTEIN HOMOMORPHISMS IN LOCALCOHOMOLOGY
ANURAG K. SINGH AND ULI WALTHER
Abstract.
Let R be a polynomial ring in finitely many variables overthe integers, and fix an ideal a of R . We prove that for all but finitelyprime integers p , the Bockstein homomorphisms on local cohomology, H k a ( R/pR ) −→ H k +1 a ( R/pR ), are zero. This provides strong evidencefor Lyubeznik’s conjecture which states that the modules H k a ( R ) have afinite number of associated prime ideals. Introduction
Let R be a polynomial ring in finitely many variables over Z , the ring ofintegers. Fix an ideal a of R . For each prime integer p , applying the localcohomology functor H • a ( − ) to0 −−−→ R/pR p −−−→ R/p R −−−→ R/pR −−−→ , one obtains a long exact sequence; the connecting homomorphisms in thissequence are the Bockstein homomorphisms for local cohomology, β kp : H k a ( R/pR ) −→ H k +1 a ( R/pR ) . We prove that for all but finitely many prime integers p , the Bocksteinhomomorphisms β kp are zero, Theorem 3.1.Our study here is motivated by Lyubeznik’s conjecture [Ly2, Remark 3.7]which states that for regular rings R , each local cohomology module H k a ( R )has finitely many associated prime ideals. This conjecture has been verifiedfor regular rings of positive characteristic by Huneke and Sharp [HS], andfor regular local rings of characteristic zero as well as unramified regularlocal rings of mixed characteristic by Lyubeznik [Ly2, Ly3]. It remainsunresolved for polynomial rings over Z , where it implies that for fixed a ⊆ R ,the Bockstein homomorphisms β kp are zero for almost all prime integers p ;Theorem 3.1 provides strong supporting evidence for Lyubeznik’s conjecture. Date : November 21, 2018.2000
Mathematics Subject Classification.
Primary 13D45; Secondary 13F20, 13F55.A.K.S. was supported by NSF grants DMS 0600819 and DMS 0608691.U.W. was supported by NSF grant DMS 0555319 and by NSA grant H98230-06-1-0012.
The situation is quite different when, instead of regular rings, one consid-ers hypersurfaces. In Example 4.2 we present a hypersurface R over Z , withideal a , such that the Bockstein homomorphism H a ( R/pR ) −→ H a ( R/pR )is nonzero for each prime integer p .Huneke [Hu, Problem 4] asked whether local cohomology modules of Noe-therian rings have finitely many associated prime ideals. The answer to thisis negative: in [Si1] the first author constructed an example where, for R a hypersurface, H a ( R ) has p -torsion elements for each prime integer p , andhence has infinitely many associated primes; see also Example 4.2. Theissue of p -torsion is central in studying Lyubeznik’s conjecture for finitelygenerated algebras over Z , and the Bockstein homomorphism is a first steptowards understanding p -torsion.For local or graded rings R , the first examples of local cohomology mod-ules H k a ( R ) with infinitely many associated primes were produced by Katz-man [Ka]; these are not integral domains. Subsequently, Singh and Swanson[SS] constructed families of graded hypersurfaces R over arbitrary fields, forwhich a local cohomology module H k a ( R ) has infinitely many associatedprimes; these hypersurfaces are unique factorization domains that have ra-tional singularities in the characteristic zero case, and are F -regular in thecase of positive characteristic.In Section 2 we establish some properties of Bockstein homomorphismsthat are used in Section 3 in the proof of the main result, Theorem 3.1.Section 4 contains various examples, and Section 5 is devoted to Stanley-Reisner rings: for ∆ a simplicial complex, we relate Bockstein homomor-phisms on reduced simplicial cohomology groups e H • (∆ , Z /p Z ) and Bock-stein homomorphisms on local cohomology modules H • a ( R/pR ), where a isthe Stanley-Reisner ideal of ∆. We use this to construct nonzero Bocksteinhomomorphisms on H • a ( R/pR ), for R a polynomial ring over Z .2. Bockstein homomorphisms
Definition 2.1.
Let R be a commutative Noetherian ring, and M an R -module. Let p be an element of R that is a nonzerodivisor on M .Let F • be an R -linear covariant δ -functor on the category of R -modules.The exact sequence0 −−−→ M p −−−→ M −−−→ M/pM −−−→ k ( M/pM ) δ kp −−−→ F k +1 ( M ) p −−−→ F k +1 ( M ) π k +1 p −−−→ F k +1 ( M/pM ) . OCKSTEIN HOMOMORPHISMS IN LOCAL COHOMOLOGY 3
The
Bockstein homomorphism β kp is the composition π k +1 p ◦ δ kp : F k ( M/pM ) −→ F k +1 ( M/pM ) . It is an elementary verification that β • p agrees with the connecting homo-morphisms in the cohomology exact sequence obtained by applying F • tothe exact sequence0 −−−→ M/pM p −−−→ M/p M −−−→ M/pM −−−→ . Let a be an ideal of R , generated by elements f , . . . , f t . The covariant δ -functors of interest to us are local cohomology H • a ( − ) and Koszul coho-mology H • ( f , . . . , f t ; − ) discussed next.Setting f e = f e , . . . , f et , the Koszul cohomology modules H • ( f e ; M ) arethe cohomology modules of the Koszul complex K • ( f e ; M ). For each e > K • ( f e ; M ) −→ K • ( f e +1 ; M ) , and thus a filtered direct system { K • ( f e ; M ) } e > . The direct limit of thissystem can be identified with the ˇCech complex ˇ C • ( f ; M ) displayed below:0 −→ M −→ M i M f i −→ M i Let F • and G • be covariant δ -functors on the category of R -modules, and let τ : F • −→ G • be a natural transformation. Since theBockstein homomorphism is defined as the composition of a connecting ho-momorphism and reduction mod p , one has a commutative diagramF k ( M/pM ) −−−→ F k +1 ( M/pM ) τ y y τ G k ( M/pM ) −−−→ G k +1 ( M/pM ) , where the horizontal maps are the respective Bockstein homomorphisms.The natural transformations of interest to us are H • ( f e ; − ) −→ H • ( f e +1 ; − )and H • ( f e ; − ) −→ H • a ( − ) , ANURAG K. SINGH AND ULI WALTHER where a = ( f , . . . , f t ).Let M be an R -module, let p be an element of R that is a nonzerodivisoron M , and let f = f , . . . , f t and g = g , . . . , g t be elements of R such that f i ≡ g i mod p for each i . One then has isomorphisms H • ( f ; M/pM ) ∼ = H • ( g ; M/pM ) , though the Bockstein homomorphisms on H • ( f ; M/pM ) and H • ( g ; M/pM )may not respect these isomorphisms; see Example 2.3. A key point in theproof of Theorem 3.1 is Lemma 2.4, which states that upon passing tothe direct limits lim −→ e H • ( f e ; M/pM ) and lim −→ e H • ( g e ; M/pM ), the Bocksteinhomomorphisms commute with the isomorphismslim −→ e H • ( f e ; M/pM ) ∼ = lim −→ e H • ( g e ; M/pM ) . Example 2.3. Let p be a nonzerodivisor on R . Let x be an element of R .The Bockstein homomorphism on Koszul cohomology H • ( x ; R/pR ) is(0 : R/pR x ) = H ( x ; R/pR ) −→ H ( x ; R/pR ) = R/ ( p, x ) Rr mod ( p ) rx/p mod ( p, x ) . Let y be an element of R with x ≡ y mod p . Comparing the Bocksteinhomomorphisms β, β ′ on H • ( x ; R/pR ) and H • ( y ; R/pR ) respectively, onesees that the diagram (cid:0) R/pR x (cid:1) β −−−→ R/ ( p, x ) R (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:0) R/pR y (cid:1) β ′ −−−→ R/ ( p, y ) R does not commute if rx/p and ry/p differ modulo the ideal ( p, y ) R ; foran explicit example, take R = Z [ w, x, z ] / ( wx − pz ) and y = x + p . Then β ( w ) = z , whereas β ′ ( w ) = z + w .Nonetheless, the diagram below does commute, hinting at Lemma 2.4. H ( x ; R/pR ) β −−−→ H ( x ; R/pR ) x −−−→ H ( x ; R/pR ) (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) H ( y ; R/pR ) β ′ −−−→ H ( y ; R/pR ) y −−−→ H ( y ; R/pR ) Lemma 2.4. Let M be an R -module, and let p be an element of R that is M -regular. Suppose a and b are ideals of R with rad( a + pR ) = rad( b + pR ) . OCKSTEIN HOMOMORPHISMS IN LOCAL COHOMOLOGY 5 Then there exists a commutative diagram · · · −−−→ H k a ( M/pM ) −−−→ H k +1 a ( M/pM ) −−−→ · · · y y · · · −−−→ H k b ( M/pM ) −−−→ H k +1 b ( M/pM ) −−−→ · · · where the horizontal maps are the respective Bockstein homomorphisms, andthe vertical maps are natural isomorphisms.Proof. It suffices to consider the case a = b + yR , where y ∈ rad( b + pR ).For each R -module N , one has an exact sequence −−−→ H k − b ( N ) y −−−→ H k a ( N ) −−−→ H k b ( N ) −−−→ H k b ( N ) y −−−→ which is functorial in N ; see for example [ILL, Exercise 14.4]. Using this for0 −−−→ M p −−−→ M −−−→ M/pM −−−→ , one obtains the commutative diagram below, with exact rows and columns. H k − b ( M/pM ) y −−−→ H k b ( M ) y p −−−→ H k b ( M ) y −−−→ H k b ( M/pM ) y y y y y H k a ( M/pM ) −−−→ H k +1 a ( M ) p −−−→ H k +1 a ( M ) −−−→ H k +1 a ( M/pM ) θ k y y y y θ k +1 H k b ( M/pM ) −−−→ H k +1 b ( M ) p −−−→ H k +1 b ( M ) −−−→ H k +1 b ( M/pM ) y y y y H k b ( M/pM ) y −−−→ H k +1 b ( M ) y p −−−→ H k +1 b ( M ) y −−−→ H k +1 b ( M/pM ) y Since H • b ( M/pM ) is y -torsion, it follows that H • b ( M/pM ) y = 0. Hence themaps θ • are isomorphisms, and the desired result follows. (cid:3) Main Theorem Theorem 3.1. Let R be a polynomial ring in finitely many variables overthe ring of integers. Let a = ( f , . . . , f t ) be an ideal of R .If a prime integer p is a nonzerodivisor on Koszul cohomology H k +1 ( f ; R ) ,then the Bockstein homomorphism H k a ( R/pR ) −→ H k +1 a ( R/pR ) is zero.In particular, the Bockstein homomorphisms on H • a ( R/pR ) are zero forall but finitely many prime integers p . We use the following notation in the proof, and also later in Section 5. ANURAG K. SINGH AND ULI WALTHER Notation 3.2. Let R be a ring with an endomorphism ϕ . Set R ϕ to be the R -bimodule with R as its underlying Abelian group, the usual action of R on the left, and the right R -action defined by r ′ r = ϕ ( r ) r ′ for r ∈ R and r ′ ∈ R ϕ . One thus obtains a functor M R ϕ ⊗ R M on the category of R -modules, where R ϕ ⊗ R M is viewed as an R -modulevia the left R -module structure on R ϕ . Proof of Theorem 3.1. The R -modules H k ( f ; R ) are finitely generated, so [ k Ass H k ( f ; R )is a finite set of prime ideals. These finitely many prime ideals containfinitely many prime integers, so the latter assertion follows from the former.Fix a prime p that is a nonzerodivisor on H k +1 ( f ; R ). Suppose R = Z [ x , . . . , x n ], set ψ to be the endomorphism of R with ψ ( x i ) = x pi for each i .For each positive integer e , consider R ψ e as in Notation 3.2. The module R ψ e is R -flat, so applying R ψ e ⊗ R ( − ) to the injective homomorphism H k +1 ( f ; R ) p −−−→ H k +1 ( f ; R )one obtains an injective homomorphism H k +1 ( ψ e ( f ); R ) p −−−→ H k +1 ( ψ e ( f ); R ) , where ψ e ( f ) = ψ e ( f ) , . . . , ψ e ( f t ). Thus, the connecting homomorphism inthe exact sequence H k ( ψ e ( f ); R/pR ) δ −−−→ H k +1 ( ψ e ( f ); R ) p −−−→ H k +1 ( ψ e ( f ); R )is zero, and hence so is the Bockstein homomorphism(3.2.1) H k ( ψ e ( f ); R/pR ) −−−→ H k +1 ( ψ e ( f ); R/pR ) . The families of ideals (cid:8)(cid:0) ψ e ( f ) (cid:1) R/pR (cid:9) e > and (cid:8) a e R/pR (cid:9) e > are cofinal, so H k a ( R/pR ) ∼ = lim −→ e H k ( ψ e ( f ); R/pR ) . Let η be an element of H k a ( R/pR ). There exists an integer e and an element e η ∈ H k ( ψ e ( f ); R/pR ) such that e η η . By Remark 2.2 and Lemma 2.4,one has a commutative diagram H k ( ψ e ( f ); R/pR ) −−−→ H k +1 ( ψ e ( f ); R/pR ) y y H k a ( R/pR ) −−−→ H k +1 a ( R/pR ) , OCKSTEIN HOMOMORPHISMS IN LOCAL COHOMOLOGY 7 where the map in the upper row is zero by (3.2.1). It follows that η mapsto zero in H k +1 a ( R/pR ). (cid:3) Examples Example 4.1 shows that the Bockstein β p : H a ( R/pR ) −→ H a ( R/pR )need not be zero for R a regular ring. In Example 4.2 we exhibit a hyper-surface R over Z , with ideal a , such that β p : H a ( R/pR ) −→ H a ( R/pR ) isnonzero for each prime integer p . Example 4.3 is based on elliptic curves,and includes an intriguing open question. Example 4.1. Let a = ( f , . . . , f t ) ⊆ R and let [ r ] ∈ H a ( R/pR ). Thereexists an integer n and a i ∈ R such that rf ni = pa i for each i . Using theˇCech complex on f to compute H • a ( R/pR ), one has β p ([ r ]) = (cid:2)(cid:0) a /f n , . . . , a t /f nt (cid:1)(cid:3) ∈ H a ( R/pR ) . For an example where β p is nonzero, take R = Z [ x, y ] / ( xy − p ) and a = xR .Then [ y ] ∈ H xR ( R/pR ), and β p ([ y ]) = [1 /x ] ∈ H xR ( R/pR ) . We remark that R is a regular ring: since R x = Z [ x, /x ] and R y = Z [ y, /y ]are regular, it suffices to observe that the local ring R ( x,y ) is also regular.Note, however, that R is ramified since R ( x,y ) is a ramified regular local ring. Example 4.2. We give an example where β p is nonzero for each prime in-teger p ; this is based on [Si1, Section 4] and [Si2]. Consider the hypersurface R = Z [ u, v, w, x, y, z ] / ( ux + vy + wz )and ideal a = ( x, y, z ) R . Let p be an arbitrary prime integer. Then theelement ( u/yz, − v/xz, w/xy ) ∈ R yz ⊕ R xz ⊕ R xy gives a cohomology class η = [( u/yz, − v/xz, w/xy )] ∈ H a ( R/pR ) . It is easily seen that β p ( η ) = 0; we verify below that β p ( F ( η )) is nonzero,where F denotes the Frobenius action on H a ( R/pR ). Indeed, if β (cid:0) F ( η ) (cid:1) = (cid:20) ( ux ) p + ( vy ) p + ( wz ) p p ( xyz ) p (cid:21) is zero, then there exists k ∈ N and elements c i ∈ R/pR such that(4.2.1) ( ux ) p + ( vy ) p + ( wz ) p p ( xyz ) k = c x p + k + c y p + k + c z p + k ANURAG K. SINGH AND ULI WALTHER in R/pR . Assign weights as follows: x : (1 , , , , u : ( − , , , ,y : (0 , , , , v : (0 , − , , ,z : (0 , , , , w : (0 , , − , . There is no loss of generality in taking the elements c i to be homogeneous, inwhich case deg c = ( − p, k, k, p ), so c must be a scalar multiple of u p y k z k .Similarly, c is a scalar multiple of v p z k x k and c of w p x k y k . Hence( ux ) p + ( vy ) p + ( wz ) p p ( xyz ) k ∈ ( xyz ) k (cid:0) u p x p , v p y p , w p z p (cid:1) R/pR . Canceling ( xyz ) k and specializing u , v , w 1, we have x p + y p + ( − x − y ) p p ∈ (cid:0) x p , y p (cid:1) Z /p Z [ x, y ] , which is easily seen to be false. Example 4.3. Let E ⊂ P Q be a smooth elliptic curve. Set R = Z [ x , . . . , x ]and let a ⊂ R be an ideal such that ( R/ a ) ⊗ Z Q is the homogeneous coor-dinate ring of E × P Q for the Segre embedding E × P Q ⊂ P Q . For all butfinitely many prime integers p , the reduction mod p of E is a smooth ellipticcurve E p , and R/ ( a + pR ) is a homogeneous coordinate ring for E p × P Z /p ;we restrict our attention to such primes.The elliptic curve E p is said to be ordinary if the Frobenius map H ( E p , O E p ) −→ H ( E p , O E p )is injective, and is supersingular otherwise. By well-known results on el-liptic curves [De, El], there exist infinitely many primes p for which E p isordinary, and infinitely many for which E p is supersingular. The local co-homology module H a ( R/pR ) is zero if E p is supersingular, and nonzero if E p is ordinary, see [HS, page 75], [Ly4, page 219], [SW1, Corollary 2.2], or[ILL, Section 22.1]. It follows that the multiplication by p map H a ( R ) p −−−→ H a ( R )is surjective for infinitely many primes p , and not surjective for infinitelymany p . Lyubeznik’s conjecture implies that this map is injective for almostall primes p , equivalently that the connecting homomorphism H a ( R/pR ) δ −−−→ H a ( R )is zero for almost all p . We do not know if this is true. However, Theorem 3.1implies that β p , i.e., the composition of the maps H a ( R/pR ) δ −−−→ H a ( R ) π −−−→ H a ( R/pR ) , OCKSTEIN HOMOMORPHISMS IN LOCAL COHOMOLOGY 9 is zero for almost all p . It is known that the ideal a R/pR is generated upto radical by four elements, [SW1, Theorem 1.1], and that it has height 3.Hence β kp = 0 for k = 3.5. Stanley-Reisner rings Bockstein homomorphisms originated in algebraic topology where theywere used, for example, to compute the cohomology rings of lens spaces. Inthis section, we work in the context of simplicial complexes and associatedStanley-Reisner ideals, and relate Bockstein homomorphisms on simplicialcohomology groups to those on local cohomology modules; see Theorem 5.8.We use this to construct nonzero Bockstein homomorphisms H k a ( R/pR ) −→ H k +1 a ( R/pR ), for R a polynomial ring over Z . Definition 5.1. Let ∆ be a simplicial complex with vertices 1 , . . . , n . Set R to be the polynomial ring Z [ x , . . . , x n ]. The Stanley-Reisner ideal of ∆ is a = ( x σ | σ ∆) R , i.e., a is the ideal generated by monomials x σ = Q ni =1 x σ i i such that σ is nota face of ∆. In particular, if { i } / ∈ ∆, then x i ∈ a .The ring R/ a is the Stanley-Reisner ring of ∆. Example 5.2. Consider the simplicial complex corresponding to a triangu-lation of the real projective plane RP depicted in Figure 1.PSfrag replacements 11 223 34 56 Figure 1. A triangulation of the real projective planeThe associated Stanley-Reisner ideal in Z [ x , . . . , x ] is generated by 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 . Remark 5.3. Let ∆ be a simplicial complex with vertex set { , . . . , n } . Let a be the associated Stanley-Reisner ideal in R = Z [ x , . . . , x n ], and set n tobe the ideal ( x , . . . , x n ). The ring R has a Z n -grading where deg x i is the i -th unit vector; this induces a grading on the ring R/ a , and also on theˇCech complex ˇ C • = ˇ C • ( x ; R/ a ). Note that a module( R/ a ) x i ··· x ik is nonzero precisely if x i · · · x i k / ∈ a , equivalently { i , . . . , i k } ∈ ∆. Hence[ ˇ C • ] , the (0 , . . . , C • , is the complex that com-putes the reduced simplicial cohomology e H • (∆; Z ), with the indices shiftedby one. This provides natural identifications(5.3.1) h H k n ( R/ a ) i = e H k − (∆; Z ) for k > . Similarly, for p a prime integer, one has h H k n ( R/ ( a + pR )) i = e H k − (∆; Z /p Z ) , and an identification of Bockstein homomorphisms e H k − (∆; Z /p Z ) β −−−→ e H k (∆; Z /p Z ) (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:2) H k n ( R/ ( a + pR )) (cid:3) β −−−→ (cid:2) H k +1 n ( R/ ( a + pR )) (cid:3) . Proposition 5.5 extends these natural identifications. Definition 5.4. Let ∆ be a simplicial complex, and let τ be a subset of itsvertex set. The link of τ in ∆ is the setlink ∆ ( τ ) = { σ ∈ ∆ | σ ∩ τ = ∅ and σ ∪ τ ∈ ∆ } . Proposition 5.5. Let ∆ be a simplicial complex with vertex set { , . . . , n } ,and let a in R = Z [ x , . . . , x n ] be the associated Stanley-Reisner ideal.Let G be an Abelian group. Given u ∈ Z n , set e u = { i | u i < } . Then H k n ( R/ a ⊗ Z G ) u = ( e H k − −| e u | (link ∆ ( e u ); G ) if u , if u j > for some j ,where n = ( x , . . . , x n ) . Moreover, for u , there is a natural identificationof Bockstein homomorphisms (cid:2) H k n ( R/ ( a + pR )) (cid:3) u β −−−→ (cid:2) H k +1 n ( R/ ( a + pR )) (cid:3) u (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)e H k − −| e u | (link ∆ ( e u ); Z /p Z ) β −−−→ e H k −| e u | (link ∆ ( e u ); Z /p Z ) . OCKSTEIN HOMOMORPHISMS IN LOCAL COHOMOLOGY 11 This essentially follows from Hochster [Ho], though we sketch a proofnext; see also [BH, Section 5.3]. Note that e = ∅ and link ∆ ( ∅ ) = ∆, so onerecovers (5.3.1) by setting u = and G = Z . Proof. We may assume G is nontrivial; set T = R/ a ⊗ Z G . As in Remark 5.3,we compute H k n ( T ) as the cohomology of the ˇCech complex ˇ C • = ˇ C • ( x ; T ).We first consider the case where u j > j . If the module h T x i ··· x ik i u is nonzero, then x j = 0 in T x i ··· x ik , and so { j, i , . . . , i k } ∈ ∆. Hence, if thecomplex [ ˇ C • ] u is nonzero, then it computes—up to index shift—the reducedsimplicial cohomology of a cone, with j the cone vertex. It follows that[ H k n ( T )] u = 0 for each k .Next, suppose u . Then the module [ T x i ··· x ik ] u is nonzero precisely if { i , . . . , i k } ∈ ∆ and e u ⊆ { i , . . . , i k } . Hence, after an index shift of | e u | + 1,the complex [ ˇ C • ] u agrees with a complex C • (link ∆ ( e u ); G ) that computesthe reduced simplicial cohomology groups e H • (link ∆ ( e u ); G ).The assertion about Bockstein maps now follows, since the complexes0 −→ [ ˇ C • ( x ; R/ a )] u p −→ [ ˇ C • ( x ; R/ a )] u −→ [ ˇ C • ( x ; R/ ( a + pR ))] u −→ −→ C • (link ∆ ( e u ); Z ) p −→ C • (link ∆ ( e u ); Z ) −→ C • (link ∆ ( e u ); Z /p ) −→ (cid:3) Thus far, we have related Bockstein homomorphisms on reduced simplicialcohomology groups to those on H • n ( R/ ( a + pR )). Our interest, however, isin the Bockstein homomorphisms on H • a ( R/pR ). Towards this, we need thefollowing duality result: Proposition 5.6. Let ( S, m ) be a Gorenstein local ring. Set d = dim S ,and let ( − ) ∨ denote the functor Hom S ( − , E ) , where E is the injective hullof S/ m . Suppose p ∈ S is a nonzerodivisor on S as well as a nonzerodivisoron a finitely generated S -module M . Then there are natural isomorphisms Ext k +1 S ( M, S/pS ) ∨ −→ Ext k +1 S ( M, S ) ∨ p −→ Ext k +1 S ( M, S ) ∨ −→ Ext kS ( M, S/pS ) ∨ y ∼ = y ∼ = y ∼ = y ∼ = H d − k − m ( M/pM ) −→ H d − k − m ( M ) p −→ H d − k − m ( M ) −→ H d − k − m ( M/pM ) , where the top row originates from applying Hom S ( M, − ) ∨ to the sequence −−−→ S p −−−→ S −−−→ S/pS −−−→ , and the bottom row from applying H m ( − ) to the sequence −−−→ M p −−−→ M −−−→ M/pM −−−→ . Proof. Let F • be a free resolution of M . The top row of the commutativediagram in the proposition is the homology exact sequence of ←−−− Hom S ( F • , S ) ∨ p ←−−− Hom S ( F • , S ) ∨ ←−−− Hom S ( F • , S/pS ) ∨ ←−−− (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ←−−− F • ⊗ S E p ←−−− F • ⊗ S E ←−−− F • ⊗ S E p ←−−− where E p = Hom S ( S/pS, E ).Let ˇ C • be the ˇCech complex on a system of parameters of S . Since S isGorenstein, ˇ C • is a flat resolution of H d ( ˇ C • ) = E , and therefore(5.6.1) H k ( F • ⊗ S E ) = Tor Sk ( M, E ) = H k ( M ⊗ S ˇ C • ) = H d − k m ( M ) . Since p is M -regular, the complex F • /pF • is a resolution of M/pM byfree S/pS -modules. Hence F • ⊗ S E p = F • ⊗ S ( S/pS ) ⊗ S/pS E p = ( F • /pF • ) ⊗ S/pS E p . Repeating the proof of (5.6.1) over the Gorenstein ring S/pS , which hasdimension d − 1, we see that H k ( F • ⊗ S E p ) = H d − − k m ( M/pM ) . (cid:3) Remark 5.7. Let R = Z [ x , . . . , x n ] be a polynomial ring. Fix an integer t > 2, and set ϕ to be the endomorphism of R with ϕ ( x i ) = x ti for each i ;note that ϕ is flat. Consider R ϕ as in Notation 3.2; the functor Φ withΦ( M ) = R ϕ ⊗ R M is an exact functor Φ on the category of R -modules. There is an isomorphismΦ( R ) ∼ = R given by r ′ ⊗ r ϕ ( r ) r ′ . More generally, for M a free R -module,one has Φ( M ) ∼ = M . For a map α of free modules given by a matrix ( α ij ),the map Φ( α ) is given by the matrix ( ϕ ( α ij )). Since Φ takes finite freeresolutions to finite free resolutions, it follows that for R -modules M and N , one has natural isomorphismsΦ (cid:0) Ext kR ( M, N ) (cid:1) ∼ = Ext kR (cid:0) Φ( M ) , Φ( N ) (cid:1) , see [Ly4, § 2] or [SW2, Remark 2.6].Let a be an ideal generated by square-free monomials. Since ϕ ( a ) ⊆ a ,there is an induced endomorphism ϕ of R/ a . The image of ϕ is spanned bythose monomials in x t , . . . , x td that are not in a . Using the map that is theidentity on these monomials, and kills the rest, one obtains a splitting of ϕ .It follows that the endomorphism ϕ : R/ a −→ R/ a is pure.Since the family of ideals { ϕ e ( a ) R } is cofinal with the family { a e } , themodule H k a ( R ) is the direct limit of the systemExt kR ( R/ a , R ) −→ Φ (cid:0) Ext kR ( R/ a , R ) (cid:1) −→ Φ (cid:0) Ext kR ( R/ a , R ) (cid:1) −→ · · · . OCKSTEIN HOMOMORPHISMS IN LOCAL COHOMOLOGY 13 The maps above are injective; see [Ly1, Theorem 1], [Mu, Theorem 1.1], or[SW2, Theorem 1.3]. Similarly, one has injective maps in the systemlim −→ e Φ e (cid:0) Ext kR ( R/ a , R/pR ) (cid:1) ∼ = H k a ( R/pR ) , and hence a commutative diagram with injective rows and exact columns: Ext kR ( R/ a , R/pR ) −−→ Φ (cid:0) Ext kR ( R/ a , R/pR ) (cid:1) −−→ · · · −−→ H k a ( R/pR ) y y y Ext k +1 R ( R/ a , R ) −−→ Φ (cid:0) Ext k +1 R ( R/ a , R ) (cid:1) −−→ · · · −−→ H k +1 a ( R ) p y p y y p Ext k +1 R ( R/ a , R ) −−→ Φ (cid:0) Ext k +1 R ( R/ a , R ) (cid:1) −−→ · · · −−→ H k +1 a ( R ) y y y Ext k +1 R ( R/ a , R/pR ) −−→ Φ (cid:0) Ext k +1 R ( R/ a , R/pR ) (cid:1) −−→ · · · −−→ H k +1 a ( R/pR ) It follows that the vanishing of the Bockstein homomorphism H k a ( R/pR ) −→ H k +1 a ( R/pR )is equivalent to the vanishing of the Bockstein homomorphismExt kR ( R/ a , R/pR ) −→ Ext k +1 R ( R/ a , R/pR ) . Theorem 5.8. Let ∆ be a simplicial complex with vertices , . . . , n . Set R = Z [ x , . . . , x n ] , and let a ⊆ R be the Stanley-Reisner ideal of ∆ . Foreach prime integer p , the following are equivalent: (1) the Bockstein H k a ( R/pR ) −→ H k +1 a ( R/pR ) is zero; (2) the Bockstein homomorphism e H n − k − −| e u | (link ∆ ( e u ); Z /p Z ) −→ e H n − k − −| e u | (link ∆ ( e u ); Z /p Z ) is zero for each u ∈ Z n with u . Setting u = immediately yields: Corollary 5.9. If the Bockstein homomorphism e H j (∆; Z /p Z ) −→ e H j +1 (∆; Z /p Z ) is nonzero, then so is the Bockstein homomorphism H n − j − a ( R/pR ) −→ H n − j − a ( R/pR ) . Proof of Theorem 5.8. By Remark 5.7, condition (1) is equivalent to thevanishing of the Bockstein homomorphismExt kR ( R/ a , R/pR ) −→ Ext k +1 R ( R/ a , R/pR ) . Set m = ( p, x , . . . , x n ). Using Proposition 5.6 for the Gorenstein local ring R m , this is equivalent to the vanishing of the Bockstein homomorphism H n − k − m ( R/ ( a + pR )) −→ H n − k m ( R/ ( a + pR )) , which, by Lemma 2.4, is equivalent to the vanishing of the Bockstein H n − k − n ( R/ ( a + pR )) −→ H n − k n ( R/ ( a + pR )) , where n = ( x , . . . , x n ). Proposition 5.5 now completes the proof. (cid:3) Example 5.10. Let ∆ be the triangulation of the real projective plane RP from Example 5.2, and a the corresponding Stanley-Reisner ideal. Let p bea prime integer. We claim that the Bockstein homomorphism(5.10.1) H a ( R/pR ) −→ H a ( R/pR )is nonzero if and only if p = 2.For the case p = 2, first note that the cohomology groups in question are e H ( RP ; Z ) = 0 , e H ( RP ; Z ) = 0 , e H ( RP ; Z ) = Z / , e H ( RP ; Z / 2) = 0 , e H ( RP ; Z / 2) = Z / , e H ( RP ; Z / 2) = Z / , so 0 −→ Z −→ Z −→ Z / −→ −→ e H ( RP ; Z / δ −→ e H ( RP ; Z ) −→ e H ( RP ; Z ) π −→ e H ( RP ; Z / −→ (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) −→ Z / δ −→ Z / −→ Z / π −→ Z / −→ Since δ and π are isomorphisms, so is the Bockstein homomorphism e H ( RP ; Z / −→ e H ( RP ; Z / p = 2.If p is an odd prime, then R/ ( a + pR ) is Cohen-Macaulay: this may beobtained from Proposition 5.5, or see [Ho, page 180]. Hence H k m ( R/ ( a + pR )) = 0 for each k = 3 and p > . By [Ly4, Theorem 1.1] it follows that H − k a ( R/pR ) = 0 for each k = 3 and p > , so the Bockstein homomorphism (5.10.1) must be zero for p an odd prime.We mention that the arithmetic rank of the ideal a R/pR in R/pR is 4,independent of the prime characteristic p ; see [Ya, Example 2]. Example 5.11. Let Λ m be the m -fold dunce cap , i.e., the quotient of theunit disk obtained by identifying each point on the boundary circle with itstranslates under rotation by 2 π/m ; specifically, for each θ , the points e i ( θ +2 πr/m ) for r = 0 , . . . , m − , OCKSTEIN HOMOMORPHISMS IN LOCAL COHOMOLOGY 15 are identified. The 2-fold dunce cap Λ is homeomorphic to the real projec-tive plane from Examples 5.2 and 5.10.The complex 0 −→ Z m −→ Z −→ 0, supported in homological degrees 1 , m . Let ℓ > ℓ need not be prime. The reduced simplicial cohomology groups of Λ m withcoefficients in Z and Z /ℓ are e H (Λ m ; Z ) = 0 , e H (Λ m ; Z ) = 0 , e H (Λ m ; Z ) = Z /m , e H (Λ m ; Z /ℓ ) = 0 , e H (Λ m ; Z /ℓ ) = Z /g , e H (Λ m ; Z /ℓ ) = Z /g , where g = gcd( ℓ, m ). Consequently, 0 −→ Z ℓ −→ Z −→ Z /ℓ −→ −→ e H (Λ m ; Z /ℓ ) δ −→ e H (Λ m ; Z ) ℓ −→ e H (Λ m ; Z ) π −→ e H (Λ m ; Z /ℓ ) −→ (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) −→ Z /g δ −→ Z /m ℓ −→ Z /m π −→ Z /g −→ δ is the cyclic subgroup of Z /m generated by the image of m/g . Consequently the Bockstein homomorphism e H (Λ m ; Z /ℓ ) −→ e H (Λ m ; Z /ℓ )is nonzero if and only if g does not divide m/g , equivalently, g does notdivide m .Suppose m is the product of distinct primes p , . . . , p r . By the abovediscussion, the Bockstein homomorphisms e H (Λ m ; Z /p i ) −→ e H (Λ m ; Z /p i )are nonzero. Let ∆ be a simplicial complex corresponding to a triangulationof Λ m , and let a in R = Z [ x , . . . , x n ] be the corresponding Stanley-Reisnerideal. Corollary 5.9 implies that the Bockstein homomorphism H n − a ( R/p i R ) −→ H n − a ( R/p i R )is nonzero for each p i . It follows that the local cohomology module H n − a ( R )has a p i torsion element for each i = 1 , . . . , r . Example 5.12. We record an example where the Bockstein homomorphism H k a ( R/pR ) −→ H k +1 a ( R/pR ) is zero though H k +1 a ( R ) has p -torsion; thistorsion is detected by “higher” Bockstein homomorphisms H k a ( R/p e R ) −→ H k +1 a ( R/p e R ) , i.e., those induced by 0 −→ Z p e −→ Z −→ Z /p e −→ . It follows from Example 5.11 that theBockstein homomorphism Z / e H (Λ ; Z / −→ e H (Λ ; Z / 2) = Z / is zero, whereas the Bockstein homomorphism Z / e H (Λ ; Z / −→ e H (Λ ; Z / 4) = Z / a in R = Z [ x , . . . , x ] be the Stanley-Reisner ideal corre-sponding to the triangulation of Λ depicted in Figure 2. While we haverestricted to p -Bockstein homomorphisms, corresponding results may be de-rived for p e -Bockstein homomorphisms; it then follows that the Bocksteinhomomorphism H a ( R/ R ) −→ H a ( R/ R ) is zero, whereas the Bockstein H a ( R/ R ) −→ H a ( R/ R ) is nonzero.PSfrag replacements 11 11 22 223 3 334 56789 Figure 2. A triangulation of the 4-fold dunce capGiven finitely many prime integers p , . . . , p r , Example 5.11, provides apolynomial ring R = Z [ x , . . . , x n ] with monomial ideal a ⊆ R such that, forsome k , the Bockstein homomorphism H k − a ( R/p i R ) −→ H k a ( R/p i R )is nonzero for each p i , in particular, H k a ( R ) has nonzero p i -torsion elements.The following theorem shows that for a a monomial ideal, each H k a ( R ) hasnonzero p -torsion elements for at most finitely many primes p . Theorem 5.13. Let R = Z [ x , . . . , x n ] be a polynomial ring, and a an idealthat is generated by monomials. Then each local cohomology module H k a ( R ) has at most finitely many associated prime ideals. In particular, H k a ( R ) hasnonzero p -torsion elements for at most finitely many prime integers p .Proof. Consider the N n –grading on R where deg x i is the i -th unit vec-tor. This induces an N n -grading on H k a ( R ), and it follows that each associ-ated prime of H k a ( R ) must be N n -graded, hence of the form ( x i , . . . , x i k ) or OCKSTEIN HOMOMORPHISMS IN LOCAL COHOMOLOGY 17 ( p, x i , . . . , x i k ) for p a prime integer. Thus, it suffices to prove that H k a ( R )has nonzero p -torsion elements for at most finitely many primes p .After replacing a by its radical, assume a is generated by square-freemonomials. Fix an integer t > ϕ be theendomorphism of R with ϕ ( x i ) = x ti for each i . Then H k a ( R/pR ) ∼ = lim −→ e Φ e (cid:0) Ext kR ( R/ a , R ) (cid:1) , where the maps in the direct system are injective. It suffices to verify that M has nonzero p -torsion if and only if Φ( M ) has nonzero p -torsion; this isindeed the case since Φ is an exact functor. (cid:3) References [BH] W. Bruns and J. Herzog, Cohen-Macaulay rings , revised edition, Cambridge Stud-ies in Advanced Mathematics , Cambridge University Press, Cambridge, 1998.[De] M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenk¨orper ,Abh. Math. Sem. Hansischen Univ. (1941), 197–272.[El] N. D. Elkies, The existence of infinitely many supersingular primes for every ellipticcurve over Q , Invent. Math. (1987), 561–567.[HS] R. Hartshorne and R. Speiser, Local cohomological dimension in characteristic p ,Ann. of Math. (2) (1977), 45–79.[Ho] M. Hochster, Cohen-Macaulay rings, combinatorics, and simplicial complexes , in:Ring theory II (Oklahoma, 1975), 171–223, Lecture Notes in Pure and Appl.Math. , Dekker, New York, 1977.[Hu] C. Huneke, Problems on local cohomology , in: Free resolutions in commutative al-gebra and algebraic geometry (Sundance, Utah, 1990), 93–108, Res. Notes Math. ,Jones and Bartlett, Boston, MA, 1992.[HS] C. Huneke and R. Sharp, Bass numbers of local cohomology modules , Trans. Amer.Math. Soc. (1993), 765–779.[ILL] S. B. Iyengar, G. J. Leuschke, A. Leykin, C. Miller, E. Miller, A. K. Singh, andU. Walther, Twenty-four hours of local cohomology , Grad. Stud. Math. , Amer-ican Mathematical Society, Providence, RI, 2007.[Ka] M. Katzman, An example of an infinite set of associated primes of a local coho-mology module , J. Algebra (2002), 161–166.[Ly1] G. Lyubeznik, On the local cohomology modules H i a ( R ) for ideals a generatedby monomials in an R -sequence , in: Complete Intersections, Lecture Notes inMath. , Springer, 1984, pp. 214–220.[Ly2] G. Lyubeznik, Finiteness properties of local cohomology modules (an application of D -modules to commutative algebra) , Invent. Math. (1993), 41–55.[Ly3] G. Lyubeznik, Finiteness properties of local cohomology modules for regular localrings of mixed characteristic: the unramified case , Comm. Alg. (2000), 5867–5882.[Ly4] G. Lyubeznik, On the vanishing of local cohomology in characteristic p > 0, Com-pos. Math. (2006), 207–221.[Mu] M. Mustat¸ˇa, Local cohomology at monomial ideals , J. Symbolic Comput. (2000),709–720. [Si1] A. K. Singh, p -torsion elements in local cohomology modules , Math. Res. Lett. (2000), 165–176.[Si2] A. K. Singh, p -torsion elements in local cohomology modules II , in: Local cohomol-ogy and its applications (Guanajuato, 1999), 155–167, Lecture Notes in Pure andAppl. Math. , Dekker, New York, 2002.[SS] A. K. Singh and I. Swanson, Associated primes of local cohomology modules and ofFrobenius powers , Int. Math. Res. Not. (2004), 1703–1733.[SW1] A. K. Singh and U. Walther, On the arithmetic rank of certain Segre products ,Contemp. Math. (2005) 147–155.[SW2] A. K. Singh and U. Walther, Local cohomology and pure morphisms , Illinois J.Math. (2007), 287–298.[Ya] Z. Yan, An ´etale analog of the Goresky-MacPherson formula for subspace arrange-ments , J. Pure Appl. Algebra (2000), 305–318. Department of Mathematics, University of Utah, 155 South 1400 East, SaltLake City, UT 84112, USA E-mail address : [email protected] Department of Mathematics, Purdue University, 150 N. University Street,West Lafayette, IN 47907, USA E-mail address ::