Let $R_0$ be any domain, let $R=R_0[U_1, ..., U_s]/I$, where $U_1, ..., U_s$ are indeterminates of some positive degrees, and $I\subset R_0[U_1, ..., U_s]$ is a homogeneous ideal. The main theorem in this paper is states that all the associated primes of $H:=H^s_{R_+}(R)$ contain a certain non-zero ideal $c(I)$ of $R_0$ called the

``content'' of $I$. It follows that the support of $H$ is simply $V(\content(I)R + R_+)$ (Corollary 1.8) and, in particular, $H$ vanishes if and only if $c(I)$ is the unit ideal. These results raise the question of whether local cohomology modules have finitely many minimal associated primes-- this paper provides further evidence in favour of such a result. Finally, we give a very short proof of a weak version of the monomial conjecture based on these results

Katzman, M.

English

arXiv

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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/    White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/10161     Katzman, M. (2003) The support of top graded local cohomology modules   arXiv.org   http://arxiv.org/abs/math/0312333v1     arXiv:math/0312333v1  [math.AC]  17 Dec 2003The support of top graded local cohomol-ogy modulesMordechai Katzman, Department of Pure Mathematics, University of Sheffield,United Kingdom. E-mail: M.Katzman@sheffield.ac.uk1 IntroductionLet R0 be any domain, let R = R0[U1, . . . , Us]/I, where U1, . . . , Us are inde-terminates of positive degrees d1, . . . , ds, and I ⊂ R0[U1, . . . , Us] is a homo-geneous ideal.The main theorem in this paper is Theorem 2.6, a generalization of Theo-rem 1.5 in [KS], which states that all the associated primes of H := HsR+(R)contain a certain non-zero ideal c(I) of R0 called the “content” of I (seeDefinition 2.4.) It follows that the support of H is simply V(c(I)R + R+)(Corollary 1.8) and, in particular, H vanishes if and only if c(I) is the unitideal.These results raise the question of whether local cohomology moduleshave finitely many minimal associated primes– this paper provides furtherevidence in favour of such a result (Theorem 2.10 and Remark 2.12.)Finally, we give a very short proof of a weak version of the monomialconjecture based on Theorem 2.6.2 The vanishing of top local cohomology modulesThroughout this section R0 will denote an arbitrary commutative Noetheriandomain. We set S = R0[U1, . . . , Us] where U1, . . . , Us are indeterminates ofdegrees d1, . . . , ds, and R = S/I where I ⊂ R0[U1, . . . , Us] is an homogeneousideal. We define ∆ = d1 + · · ·+ ds and denote with D the sub-semi-group ofN generated by d1, . . . , ds.For t ∈ Z, we shall denote by (•)(t) the t-th shift functor (on the categoryof graded R-modules and homogeneous homomorphisms).For any multi-index λ = (λ(1), . . . , λ(s)) ∈ Zs we shall write Uλ forUλ(1)1 . . . Uλ(s)s and we shall set |λ| = λ(1) + · · · + λ(s).Lemma 2.1 Let I be generated by homogeneous elements f1, . . . , fr ∈ S.Then there is an exact sequence of graded S-modules and homogeneous ho-momorphismsr⊕i=1HsS+(S)(− deg fi)(f1,...,fr)−−−−−−→ HsS+(S) −→ HsR+(R) −→ 0.Proof: The functor HsS+(•) is right exact and the natural equivalence be-tween HsS+(•) and ( • )⊗S HsS+(S) (see [BS, 6.1.8 & 6.1.9]) actually yields ahomogeneous S-isomorphismHsS+(S)/(f1, . . . , fr)HsS+(S) ∼= HsS+(R).To complete the proof, just note that there is an isomorphism of gradedS-modules HsS+(R)∼= HsR+(R), by the Graded Independence Theorem [BS,13.1.6].We can realize HsS+(S) as the module R0[U−1 , . . . , U−s ] of inverse poly-nomials described in [BS, 12.4.1]: this graded R-module vanishes beyonddegree −∆. More generally R0[U−1 , . . . , U−s ]−d 6= 0 if and only if d ∈ D.For each d ∈ D, R0[U−1 , . . . , U−s ]−d is a free R0-module with base B(d) :=(Uλ)−λ∈Ns,|λ|=−d. We combine this realisation with the previous lemma tofind a presentation of each homogeneous component of HsR+(R) as the cok-ernel of a matrix with entries in R0.Assume first that I is generated by one homogeneous element f of degreeδ. For any d ∈ D we have, in view of Lemma 2.1, a graded exact sequenceR0[U−1 , . . . , U−s ]−d−δφd−→ R0[U−1 , . . . , U−s ]−d −→ HsR+(R)−d −→ 0.The map of free R0-modules φd is given by multiplication on the left by a#B(d) × #B(d + δ) matrix which we shall denote later by M(f ; d).In the general case, where I is generated by homogeneous elementsf1, . . . , fr ∈ S, it follows from Lemma 2.1 that the R0-module HsR+(R)−dis the cokernel of a matrix M(f1, . . . , fr; d) whose columns consist of all thecolumns of M(f1, d), . . . ,M(fr, d).Consider a homogeneous f ∈ S of degree δ. We shall now describe thematrix M(f ; d) in more detail and to do so we start by ordering the basesof the source and target of φd as follows. For any λ, µ ∈ Zs with negativeentries we declare that Uλ < Uµ if and only if U−λ <Lex U−µ where “<Lex”is the lexicographical term ordering in S with U1 > · · · > Us. We orderthe bases B(d), and by doing so also the columns and rows of M(f ; d), inascending order. We notice that the entry in M(f ; d) in the Uα row and Uβcolumn is now the coefficient of Uα in fUβ.Lemma 2.2 Let ν ∈ Zs have negative entries and let λ1, λ2 ∈ Ns. IfUλ1 <Lex Uλ2 and UνUλ1 , UνUλ2 ∈ R0[U−1 , . . . , U−s ] do not vanish thenUνUλ1 > UνUλ2 .Proof: Let j be the first coordinate in which λ1 and λ2 differ. Then λ(j)1 <λ(j)2 and so also −ν(j) − λ(j)1 > −ν(j) − λ(j)2 ; this implies that U−ν−λ1 >LexU−ν−λ2 and Uν+λ1 > Uν+λ2 .Lemma 2.3 Let f 6= 0 be a homogeneous element in S. Then, for all d ∈ D,the matrix M(f ; d) has maximal rank.Proof: We prove the lemma by producing a non-zero maximal minor ofM(f ; d). Write f =∑λ∈Λ aλUλ where aλ ∈ R0 \ {0} for all λ ∈ Λ and letλ0 be such that Uλ0 is the minimal member of{Uλ : λ ∈ Λ}with respect tothe lexicographical term order in S.Let δ be the degree of f . Each column of M(f ; d) corresponds to amonomial Uλ ∈ B(d + δ); its ρ-th entry is the coefficient of Uρ in fUλ ∈R0[U−1 , . . . , U−s ]−d.Fix any Uν ∈ B(d) and consider the column cν corresponding to Uν−λ0 ∈B(d + δ). The ν-th entry of cν is obviously aλ0 .By the previous lemma all entries in cν below the νth row vanish. Con-sider the square submatrix of M(f ; d) whose columns are the cν (ν ∈ B(d));its determinant is clearly a power of aλ0 and hence is non-zero.Definition 2.4 For any f ∈ R0[U1, . . . , Us] write f =∑λ∈Λ aλUλ whereaλ ∈ R0 for all λ ∈ Λ. For such an f ∈ R0[U1, . . . , Us] we define the contentc(f) of f to be the ideal 〈aλ : λ ∈ Λ〉 of R0 generated by all the coefficientsof f . If J ⊂ R0[U1, . . . , Us] is an ideal, we define its content c(J) to be theideal of R0 generated by the contents of all the elements of J . It is easy tosee that if J is generated by f1, . . . , fr, then c(J) = c(f1) + · · · + c(fr).Lemma 2.5 Suppose that I is generated by homogeneous elementsf1, . . . , fr ∈ S. Fix any d ∈ D. Let t := rankM(f1, . . . , fr; d) andlet Id be the ideal generated by all t × t minors of M(f1, . . . , fr; d). Thenc(I) ⊆√Id.Proof: It is enough to prove the lemma when r = 1; let f = f1. Writef =∑λ∈Λ aλUλ where aλ ∈ R0 \ {0} for all λ ∈ Λ. Assume that c(I) 6⊆√Idand pick λ0 so that Uλ0 is the minimal element in{Uλ : λ ∈ Λ}(with respectto the lexicographical term order in S) for which aλ /∈√Id. Notice thatthe proof of Lemma 2.3 shows that Uλ0 cannot be the minimal element of{Uλ : λ ∈ Λ}.Fix any Uν ∈ B(d) and consider the column cν corresponding to Uν−λ0 ∈B(d + δ). The ν-th entry of cν is obviously aλ0 . Lemma 2.2 shows that, forany other λ1 ∈ Λ with Uλ1 >Lex Uλ0, either ν − λ0 + λ1 has a non-negativeentry, in which case the corresponding term of fUν−λ0 ∈ R0[U−1 , . . . , U−s ]−dis zero, or Uν > Uν−λ0+λ1 .Similarly, for any other λ1 ∈ Λ with Uλ1 <Lex Uλ0 , either ν − λ0 + λ1has a non-negative entry, in which case the corresponding term of fUν−λ0 ∈R0[U−1 , . . . , U−s ]−d is zero, or Uν < Uν−λ0+λ1 .We have shown that all the entries below the ν-th row of cν are in√Id.Consider the matrix M whose columns are cν (ν ∈ B(d)) and let : R0 →R0/√Id denote the quotient map. We have0 = det(M) = det(M) = aλ0(d−1s−1)and, therefore, aλ0 ∈√Id, a contradiction.Theorem 2.6 Suppose that I is generated by homogeneous elements f1, . . . , fr ∈S. Fix any d ∈ D. Then each associated prime of HsR+(R)−d contains c(I).In particular HsR+(R)−d = 0 if and only if c(I) = R0.Proof: Recall that for any p, q ∈ N with p ≤ q and any p × q matrixM of maximal rank with entries in any domain, Coker M = 0 if and onlyif the ideal generated by the maximal minors of M is the unit ideal. LetM = M(f1, . . . , fr; d), so that HsR+(R)−d ∼= Coker M .In view of Lemmas 2.3 and 2.5, the ideal c(I) is contained in the radicalof the ideal generated by the maximal minors of M ; therefore, for eachx ∈ c(I), the localization of Coker M at x is zero; we deduce that c(I) iscontained in all associated primes of Coker M .To prove the second statement, assume first that c(I) is not the unitideal. Since all minors of M are contained in c(I), these cannot generate theunit ideal and Coker M 6= 0. If, on the other hand, c(I) = R0 then Coker Mhas no associated prime and Coker M = 0.Corollary 2.7 Let the situation be as in 2.6. The following statementsare equivalent:1. c(I) = R0;2. HsR+(R)−d = 0 for some d ∈ D;3. HsR+(R)−d = 0 for all d ∈ D.Consequently, HsR+(R) is asymptotically gap-free in the sense of [BH, (4.1)].Corollary 2.8 The R-module HsR+(R) has finitely many minimal associ-ated primes, and these are just the minimal primes of the ideal c(I)R + R+.Proof: Let r ∈ c(I). By Theorem 2.6, the localization of HsR+(R) at ris zero. Hence each associated prime of HsR+(R) contains c(I)R. Such anassociated prime must contain R+, since HsR+(R) is R+-torsion.On the other hand, HsR+(R)−∆∼= R0/ c(I) and HsR+(R)i = 0 for alli > −∆; therefore there is an element of the (−∆)-th component of HsR+(R)that has annihilator (over R) equal to c(I)R+R+. All the claims now followfrom these observations.Remark 2.9 In [Hu, Conjecture 5.1], Craig Huneke conjectured that everylocal cohomology module (with respect to any ideal) of a finitely generatedmodule over a local Noetherian ring has only finitely many associated primes.This conjecture was shown to be false (cf. [K, Corollary 1.3]) but Corollary2.8 provides some evidence in support of the weaker conjecture that everylocal cohomology module (with respect to any ideal) of a finitely generatedmodule over a local Noetherian ring has only finitely many minimal associ-ated primes.The following theorem due to Gennady Lyubeznik ([L]) gives furthersupport for this conjecture:Theorem 2.10 Let R be any Noetherian ring of prime characteristic p andlet I ⊂ R be any ideal generated by f1, . . . , fs ∈ R. The support of HsI (R) isZariski closed.Proof: We first notice that the localization of HsI (R) at a prime P ⊂ Rvanishes if and only if there exist positive integers α and β such that(f1 · · · · · fs)α ∈ 〈fα+β1 , . . . , fα+βs 〉in the localization RP . This is because if we can find such α and β we canthen take q := pe powers and obtain(f1 · · · · · fs)qα ∈ 〈f qα+qβ1 , . . . , f qα+qβs 〉for all such q. This shows that all elements in the direct limit sequenceR/〈f1, . . . fs〉f1·...·fs−−−−→ R/〈f21 , . . . f2s 〉f1·...·fs−−−−→ . . .map to 0 in the direct limit and hence HsI (R) = 0.But if(f1 · · · · · fs)α ∈ 〈fα+β1 , . . . , fα+βs 〉in RP , we may clear denominators and deduce that this occurs on a Zariskiopen subset containing P .Thus the complement of the support is a Zariski open subset.It may be reasonable to expect that non-top local cohomology modulesmight also have finitely many minimal associated primes; the only examplesknown to me of non-top local cohomology modules with infinitely manyassociated primes are the following: Let k be any field, let R0 = k[x, y, s, t]and let S be the localisation of R0[u, v, a1, . . . , an] at the maximal ideal mgenerated by x, y, s, t, u, v, a1, . . . , an. Let f = sx2v2−(t+s)xyuv+ty2u2 ∈ Sand let R = S/fS. Denote by I the ideal of S generated by u, v and by Athe ideal of S generated by a1, . . . , an.Theorem 2.11 Assume that n ≥ 2. The local cohomology module H2I∩A(R)has infinitely many associated primes and Hn+1I∩A(R) 6= 0.Proof: Consider the following segment of the Mayer-Vietoris sequence· · · → H2I+A(R) → H2I(R) ⊕ H2A(R) → H2I∩A(R) → . . .Notice that a1, . . . , an, u form a regular sequence on R so depthI+A R ≥n+1 ≥ 3 and the leftmost module vanishes. Thus H2I(R) injects into H2I∩A(R)and Corollary 1.3 in [K] shows that H2I∩A(R) has infinitely many associatedprimes.Consider now the following segment of the Mayer-Vietoris sequence· · · → Hn+1I∩A(R) → Hn+2I+A(R) → Hn+2I (R) ⊕ Hn+2A (R) → . . .The direct summands in the rightmost module vanish since both I and A canbe generated by less than n+2 elements, so Hn+1I∩A(R) surjects onto Hn+2I+A(R).Now c(f) is the ideal of R0 generated by sx2,−(t + s)xy and ty2 soc(f) ⊂ 〈x, y〉 6= R0. Corollary 2.7 now shows that Hn+2I+A(R) does not vanishand, therefore, nor does Hn+1I∩A(R).Remark 2.12 When n ≥ 3, H3I+A(R) = 0 and the argument above showsthat H2I(R) ⊕ H2A(R) ∼= H2I∩A(R). Corollary 2.8 implies that H2I(R) hasfinitely many minimal primes and since the only associated prime of H2A(R)is A, H2I∩A(R) has finitely many minimal primes.When n = 2 we obtain a short exact sequence0 → H2I(R) ⊕ H2A(R) → H2I∩A(R) → H3I+A(R) → 0.The short exact sequence0 → S f→ S → R → 0implies that H3I+A(R) injects into the local cohomology module H4I+A(S)whose only associated prime is I + A, so again we see that H2I∩A(R) hasfinitely many minimal associated primes.3 An application: a weak form of the MonomialConjecture.In [Ho] Mel Hochster suggested reducing the Monomial Conjecture to theproblem of showing the vanishing of certain local cohomology modules whichwe now describe.Let C be either Z or a field of characteristic p > 0, let R0 = C[A1, . . . , As]where A1, . . . , As are indeterminates, S = R0[Us, . . . , Us] where U1, . . . , Usare indeterminates and R = S/Fs,tS whereFs,t = (U1 · . . . · Us)t −s∑i=1AiUt+1i .Suppose thatHs,t := Hs〈U1,...,Us〉(R)vanishes with C = Z. If for some local ring T we can find a system ofparameters x1, . . . , xs so that (x1 · . . . · xs)t ∈ 〈xt+11 , . . . , xt+1s 〉, i.e., if thereexist a1, . . . , as ∈ T so that (x1 · . . . · xs)t =∑ti=1 aixt+1i we can define anhomomorphism R → T by mapping Ai to ai and Ui to xi. We can view Tas an R-module and we have an isomorphism of T -modulesHs〈x1,...,xs〉(T )∼= Hs〈U1,...,Us〉(R) ⊗R Tand we deduce thatHs〈x1,...,xs〉(T ) = 0but this cannot happen since x1, . . . , xs form a system of parameters in T .We have just shown that the vanishing of Hs,t for all t ≥ 1 implies theMonomial Conjecture in dimension s. In [Ho] Mel Hochster proved that theselocal cohomology modules vanish when s = 2 or when C has characteristicp > 0, but in [R] Paul Roberts showed that, when C = Z, H3,2 6= 0,showing that Hochster’s approach cannot be used for proving the MonomialConjecture in dimension 3. This can be generalized further:Proposition 3.1 When C = Z, Hs,2 6= 0 for all s ≥ 3.Proof: We proceed by induction on s; the case s = 3 is proved in [R].Assume that for some s ≥ 1, α ≥ 0 and δ > α the monomial xα1 . . . xαs+1is in the ideal of C[x1, . . . , xs+1, a1, . . . , as+1] generated by xα+β1 , . . . , xα+βs+1and Fs+1,t.Define Gs+1,2 to be the result of substituting as+1 = 0 in Fs+1,2, i.e.,Gs+1,2 = (x1 . . . xs+1)2 −s∑i=1aix3i .Ifxα1 . . . xαs+1 ∈ 〈xα+β1 , . . . , xα+βs+1 , Fs+1,2〉 (1)then by setting as+1 = 0 we see thatxα1 . . . xαs+1 ∈ 〈xα+β1 , . . . , xα+βs+1 , Gs+1,2〉.If we assign degree 0 to x1, . . . , xs, degree 1 to xs+1 and degree 2 to a1, . . . , asthen the ideal 〈xα+β1 , . . . , xα+βs+1 , Gs+1,2〉 is homogeneous and we must havexα1 . . . xαs+1 ∈ 〈xα+β1 , . . . , xα+βs , Gs+1,2〉.If we now set xs+1 = 1 we obtainxα1 . . . xαs ∈ 〈xα+β1 , . . . , xα+βs , Fs,2〉. (2)Now Hs+1,2 = 0 if and only if for each β ≥ 1 we can find an α ≥ 0 so thatequation (1) holds and this implies that for each β ≥ 1 we can find an α ≥ 0so that equation (2) holds which is equivalent to Hs,2 = 0. The inductionhypothesis implies that Hs,2 6= 0 and so Hs+1,2 6= 0.The local cohomology modules Hs,t are a good illustration for the failureof the methods of the previous section in the non-graded case. For example,one cannot decide whether Hs,t is zero just by looking at Fs,t: the vanishingof Hs,t depends on the characteristic of C! Compare this situation to thefollowing graded problem.Theorem 3.2 (A Weaker Monomial Conjecture) Let T be a local ringwith system of parameters x1, . . . , xs. For all t ≥ 0 we have(x1 · . . . · xs)t /∈ 〈xst1 , . . . , xsts 〉.Proof: Let S = Z[A1, . . . , As][X1, . . . ,Xs] where deg Ai = 0 and deg Xi = 1for all 1 ≤ i ≤ s. Following Hochster’s argument we reduce to the problemof showing thatHs〈X1,...,Xs〉(S/fS) = 0wheref = (X1 · . . . · Xs)t −s∑i=1AiXsti .Since f is homogeneous and c(f) is the unit ideal, the result follows fromTheorem 2.6.References[BH] M. Brodmann and M. Hellus, Cohomological patterns of coherentsheaves over projective schemes, J. Pure Appl. Algebra 172 (2002)2-3,pp. 165–182.[BS] M. P. Brodmann and R. Y. Sharp, Local cohomology: an algebraicintroduction with geometric applications, Cambridge University Press,1998.[Ho] M. Hochster, Canonical elements in local cohomology modules and thedirect summand conjecture, J. Algebra 84 (2) (1983), pp. 503–553.[Hu] C. Huneke, Problems on local cohomology, in Free resolutions in com-mutative algebra and algebraic geometry, Sundance 90, ed. D. Eisenbudand C. Huneke, Research Notes in Mathematics 2, Jones and BartlettPublishers, Boston, 1992, pp. 93–108.[K] M. Katzman, An example of an infinite set of associated primes of alocal cohomology module, J. Algebra 252(1) (2002), pp. 161–166.[KS] M. Katzman and R. Y. Sharp, Some properties of top graded localcohomology modules, J. Algebra 259(2) (2003) , pp. 599–612.[L] G. Lyubeznik, private communication.[R] P. Roberts, A computation of local cohomology, in Commutative alge-bra: syzygies, multiplicities, and birational algebra (South Hadley, MA,1992), Contemp. Math., 159, Amer. Math. Soc., Providence, RI, 1994,pp. 351–356.

The support of top graded local cohomology modules

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