Let $(R,m)$ be a local Noetherian ring, let $I\subset R$ be any ideal and let $M$ be a finitely generated $R$-module. In 1990 Craig Huneke conjectured that the local cohomology modules $H^i_I(M)$ have finitely many associated primes for all $i$. In this paper I settle this conjecture by constructing a local cohomology module of a local $k$-algebra with an infinite set of associated primes, and I do this for any field $k$

Brodmann

Huneke

Katzman

Lyubeznik

Mordechai Katzman

Singh

English

arXiv

Katzman, Mordechai

Elsevier - Publisher Connector 

An example of an infinite set of associated primes of a local cohomology module 

Katzman, M.

White Rose Research Online

promoting access to White Rose research papers    White Rose Research Online eprints@whiterose.ac.uk    Universities of Leeds, Sheffield and York http://eprints.whiterose.ac.uk/    This is an author produced version of a paper published in Journal of Algebra.    White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/10159     Published paper  Katzman, M. (2002) An example of an infinite set of associated primes of a local cohomology module, Journal of Algebra, 252 (1), pp. 161-166 http://dx.doi.org/doi:10.1016/S0021-8693(02)00032-7    arXiv:math/0209353v1  [math.AC]  25 Sep 2002AN EXAMPLE OF AN INFINITE SET OF ASSOCIATED PRIMES OF ALOCAL COHOMOLOGY MODULEMORDECHAI KATZMAN0. IntroductionLet (R, m) be a local Noetherian ring, let I ⊂ R be any ideal and let M be a finitely generatedR-module. It has been long conjectured that the local cohomology modules HiI(M) have finitelymany associated primes for all i (see Conjecture 5.1 in [H] and [L].)If R is not required to be local these sets of associated primes may be infinite, as shown byAnurag Singh in [S], where he constructed an example of a local cohomology module of a finitelygenerated module over a finitely generated Z-algebra with infinitely many associated primes. Thislocal cohomology module has p-torsion for all primes p ∈ Z.However, the question of the finiteness of the set of associated primes of local cohomology modulesdefined over local rings and over k-algebras (where k is a field) has remained open until now. In thispaper I settle this question by constructing a local cohomology module of a local finitely generatedk-algebra with an infinite set of associated primes, and I do this for any field k.1. The exampleLet k be any field, let R0 = k[x, y, s, t] and let S = R0[u, v]. Define a grading on S by declaringdeg(x) = deg(y) = deg(s) = deg(t) = 0 and deg(u) = deg(v) = 1. Let f = sx2v2−(t+s)xyuv+ty2u2and let R = S/fS. Notice that f is homogeneous and hence R is graded. Let S+ be the ideal of Sgenerated by u and v and let R+ be the ideal of R generated by the images of u and v.Consider the local cohomology module H2R+(R): it is homogeneously isomorphic to H2S+(S/fS)and we can use the exact sequenceH2S+(S)(−2)f−→ H2S+(S) −→ H2S+(S/fS) −→ 0of graded R-modules and homogeneous homomorphisms (induced from the exact sequence0 −→ S(−2)f−→ S −→ S/fS −→ 0)Date: February 1, 2008.1991 Mathematics Subject Classification. Primary 13D45, 13E05, 13A02, 13P99.Key words and phrases. Graded commutative Noetherian ring, graded local cohomology module, infinite set ofassociated primes.12 MORDECHAI KATZMANto study H2R+(R). Furthermore, we can realize H2S+(S) as the module R0[u−, v−] of inverse polyno-mials described in [BS, 12.4.1]: this graded S-module vanishes beyond degree −2, and, for each d ≥ 2,its (−d)-th component is a free R0-module of rank d−1 with base(u−αv−β)α,β>0, α+β=−d. We willstudy the graded components of H2S+(S/fS) by considering the cokernels of the R0-homomorphismsf−d : R0[u−, v−]−d−2 −→ R0[u−, v−]−d (d ≥ 2)given by multiplication by f . In order to represent these R0-homomorphisms between free R0-modules by matrices, we specify an ordering for each of the above-mentioned bases by declaringthatuα1vβ1 < uα2vβ2(where α1, β1, α2, β2 < 0 and α1 + β1 = α2 + β2) precisely when α1 > α2. If we use this ordering forboth the source and target of each fd, we can see that each fd (d ≥ 2) is given by multiplication onthe left by the tridiagonal d − 1 by d + 1 matrixAd−1 :=sx2 −xy(t + s) ty2 0 . . . 00 sx2 −xy(t + s) ty2 0 . . . 00 0 sx2 −xy(t + s) ty2 . . . 0. . .0 . . . sx2 −xy(t + s) ty2.We also defineAd−1 :=s −(t + s) t 0 . . . 00 s −(t + s) t 0 . . . 00 0 s −(t + s) t . . . 0. . .0 . . . s −(t + s) tobtained by substituting x = y = 1 in Ad−1.Let also τi = (−1)i(ti + sti−1 + · · · + si−1t + si).1.1. Lemma.(i) Let Bi be the submatrix of Ai obtained by deleting its first and last columns. Then detBi =τi for all i ≥ 1.(ii) Let S be an infinite set of positive integers. Suppose that either k has characteristic zeroor that k has prime characteristic p and S contains infinitely many integers of the formpm − 2. The (k[s, t]-)irreducible factors of {τi}i∈S form an infinite set.AN EXAMPLE OF AN INFINITE SET OF ASSOCIATED PRIMES OF A LOCAL COHOMOLOGY MODULE 3Proof. We prove the first statement by induction on i. SincedetB1 = det (−t − s) = −t − s and detB2 = det−t − s ts −t − s = t2 + st + s2,the lemma holds for i = 1 and i = 2. Assume now that i ≥ 3. Expanding the determinant of Bi byits first row and applying the induction hypothesis we obtaindetBi = (−t − s) detBi−1 − st detBi−2= (−1)i−1(−t − s)(ti−1 + · · · + si−2t + si−1) − (−1)i−2st(ti−2 + · · · + si−3t + si−2)= (−1)i[(ti + · · · + si−2t2 + si−1t) + (sti−1 + · · · + si−1t + si) − (sti−1 + · · · + si−2t2 + si−1t)]= (−1)i(ti + sti−1 + · · · + si−1t + si).We now prove the second statement. Define σi = ti + ti−1 + · · ·+ t+1 and notice that it is enoughto show that the set of irreducible factors of {σi}i∈S is infinite. Let I be the set of irreducible factorsof {σi}i∈S . If k has characteristic zero consider Q[I] ⊇ Q, the splitting field of this set of irreduciblefactors. If I is finite, Q[I] ⊇ Q is finite extension which contains all ith roots of unity for all i ∈ S,which is impossible.Assume now that k has prime characteristic p. Let F be the algebraic closure of the prime fieldof k. For any positive integer mddtt(tpm−1 − 1) = −1so σpm−2 = (tpm−1 − 1)/(t − 1) has pm − 2 distinct roots in F and, therefore, the roots of {σs}s∈Sform an infinite set. 1.2. Theorem. For every d ≥ 2 the R0-module H2R+(R)−d has τd−1-torsion. Hence H2R+(R) hasinfinitely many associated primes.Proof. For the purpose of this proof we introduce a bigrading in R0 by declaring deg(x) = (1, 0),deg(y) = (1, 1) and deg(t) = deg(s) = (0, 0).We also introduce a bigrading on the free R0-modules Rn0 by declaring deg(xαyβsatbej) = (α +β, β + j) for all non-negative integers α, β, a, b and all 1 ≤ j ≤ n. Notice that Rn0 is a bigradedR0-module when R0 is equipped with the bigrading mentioned above.Consider the R0-module CokerAd−1; the columns of Ad−1 are bihomogeneous of bidegrees(2, 1), (2, 2), . . . , (2, d + 1).We can now consider CokerAd−1 as a k[s, t] module generated by the natural images of xαyβejfor all non-negative integers α, β and all 1 ≤ j ≤ d − 1. The k[s, t]-module of relations among4 MORDECHAI KATZMANthese generators is generated by k[x, y]-linear combinations of the columns of Ad−1, and since thesecolumns are bigraded, the k[s, t]-module of relations will be bihomogeneous and we can writeCokerAd−1 =⊕0≤D, 1≤j(CokerAd−1)(D,j) .Consider the k[s, t]-module (CokerAd−1)(d,d), the bihomogeneous component of CokerAd−1 of bide-gree (d, d). It is generated by the images ofxyd−1e1, x2yd−2e2, . . . , xd−2y2ed−2, xd−1yed−1and the relations among these generators are given by k[s, t]-linear combinations ofyd−2c2, xyd−3c3, . . . , xd−3ycd−1, xd−2cdwhere c1, . . . , cd+1 are the columns of Ad−1. So we have(CokerAd−1)(d,d) = CokerBd−1where Bd−1 is viewed as a k[s, t]-homomorphism k[s, t]d−1 → k[s, t]d−1.Using Lemma 1.1(i) we deduce that for all d ≥ 2 the direct summand (CokerAd−1)(d,d) ofCokerAd−1 has τd−1 torsion, and so does CokerAd−1 itself.Lemma 1.1(ii) applied with S = N now shows that there exist infinitely many irreducible homo-geneous polynomials {pi ∈ k[s, t] : i ≥ 1} each one of them contained in some associated prime ofthe R0-module ⊕d≥2 CokerAd−1. Clearly, if i 6= j then any prime ideal P ⊂ R0 which contains bothpi and pj must contain both s and t.Since the localisation of (CokerAd−1)(d,d) at s does not vanish, there exist Pi, Pj ∈ AssR0 CokerAd−1which do not contain s and such that pi ⊂ Pi, pj ⊂ Pj , and the previous paragraph shows thatPj 6= Pj .The second statement now follows from the fact that H2R+(R) is R0-isomorphic to ⊕d≥2 CokerAd−1.1.3. Corollary. Let T be the localisation of R at the irrelevant maximal ideal m = 〈s, t, x, y, u, v〉.Then H2(u,v)T (T ) has infinitely many associated primes.Proof. Since τi ∈ m for all i ≥ 1, H2(u,v)T (T )∼= (H2(u,v)R(R))m has τi-torsion for all i ≥ 1. 2. A connection with associated primes of Frobenius powersIn this section we apply a technique similar to the one used in section 1 to give a proof of a slightlymore general statement of Theorem 12 in [K]. The new proof is simpler, open to generalisations andAN EXAMPLE OF AN INFINITE SET OF ASSOCIATED PRIMES OF A LOCAL COHOMOLOGY MODULE 5it gives a connection between associated primes of Frobenius powers of ideals and of local cohomologymodules, at least on a purely formal level.Let k be any field, let S = k[x, y, s, t], let F = xy(x− y)(sx− ty) = sx3y− (t + s)x2y2 + txy3 andlet R = S/FS.2.1. Theorem. Let S be an infinite set positive integers and suppose that either k has characteristiczero or that k has characteristic p and that S contains infinitely many powers of p. The set⋃n∈SAssR(R〈xn, yn〉)is infinite.Proof. We introduce a grading in S by setting deg(x) = deg(y) = 1 and deg(s) = deg(t) = 0. SinceF is homogeneous, R is also graded.Fix some n > 0 and consider the graded R-module T = R/〈xn, yn〉. For each d > 4 considerTd, the degree d homogeneous component of T , as a k[s, t]-module. If d < n, Td is generated bythe images of yd, xyd−1, . . . , xd−1y, xd and the relations among these generators are obtained fromyd−4F, xyd−5F, . . . , xd−5yF, xd−4F . Using these generators and relations, in the given order, wewrite Td = CokerMd whereMd =0 0 . . . 0t−t − s ts −t − ss. . .t−t − s ts −t − ss0 0 . . . 0.When d = n, Td is isomorphic to the cokernel of the submatrix of Md obtained by deleting thefirst and last rows which correspond to the generators yn, xn of Tn.When d = n + 1, Td is isomorphic to the cokernel of the submatrix of Md obtained by deletingthe first two rows and and last two rows which correspond to the generators yn+1, xyn, xny, xn+1 ofTn+1, and the resulting submatrix is Bn−2 defined in Lemma 1.1; the result now follows from thatlemma. 6 MORDECHAI KATZMANThis technique for finding associated primes of non-finitely generated graded modules and ofsequences of graded modules has been applied in [BKS] and [KS] to yield further new and surprisingproperties of top local cohomology modules.AcknowledgmentI would like to thank Rodney Sharp, Anurag Singh and Gennady Lyubeznik for reading a firstdraft of this paper and for their helpful suggestions.References[BKS] M. Brodmann, M. Katzman and R. Y. Sharp. Associated primes of graded components of local cohomologymodules, Transactions of the American Mathematical Society, to appear.[BS] M. P. Brodmann and R. Y. Sharp. Local cohomology: an algebraic introduction with geometric applications,Cambridge University Press, 1998.[H] C. Huneke. Problems on local cohomology, in Free resolutions in commutative algebra and algebraic geometry(Sundance UT, 1990), 93–108, Research Notes in Mathematics 2. Jones and Bartlett, Boston, MA, 1992.[K] M. Katzman. Finiteness of ∪eAssF e(M) and its connections to tight closure. Illinois Journal of Mathematics40 (1996) pp. 330-337.[KS] M. Katzman and R. Y. Sharp. Some properties of top graded local cohomology modules, preprint.[L] G. Lyubeznik, A partial survey of local cohomology, in Local Cohomology and Its Applications, ed. G.Lyubeznik, Marcel Dekker (2001).[S] A. K. Singh. p-torsion elements in local cohomology modules, Math. Research Letters 7 (2000) 165–176.Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UnitedKingdom, Fax number: 0044-114-222-3769E-mail address: M.Katzman@sheffield.ac.uk

An example of an infinite set of associated primes of a local cohomology module

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An example of an infinite set of associated primes of a local cohomology module

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