Surjectivity of Some Local Cohomology Map and the Second Vanishing Theorem 3

Home , Local cohomology

arXiv:2107.09041v1 [math.AC] 19 Jul 2021 hoe 1.2. Theorem h hoe,w hwta eti oa ooooympindu map cohomology local certain a in element that regular extra show case the ramified we and add theorem, unramified we the with setting, deal our and In modules cohomology modules. cohomology local on olwn ttmnsaeequivalent. are statements following > p narglrlclrn.Teeulcaatrsi aei wel Spei is and case Hartshorne characteristic 0, equal characteristic The in ring. theorem vanishing local regular a on H hoe.Ti scle h e eodvnsigtermi [ in theorem vanishing 1.1. second Theorem new regarde the i is called which and result, is rings following This local the prove theorem. regular to ramified is for paper this it unramifie e proved for the theorem partially in vanishing [1] treatment second uniform the proved a gave [16] [8] Zhang Lyubeznik and Huneke and oa ooooymodules cohomology local da with ideal i n[]adi eeoe xesvl n[2.Teoiia defi original The [12]. in extensively developed is and [7] in ihsprbycoe eiu field residue closed separably with ≥ m 3 n[] h bv hoe setbihdi h naie cas unramified the in established is theorem above the [5], In h eodvnsigtermi nasrino h vanishin the of assertion an is theorem vanishing second The e od n phrases. and words Key 2020 h dat rv hstermi oiae yteter fsu of theory the by motivated is theorem this prove to idea The ( 2 h ucue spectrum punctured The (2) (1) 0 R/pR 0 hntemlilcto map multiplication the Then . i nideal an Fix . UJCIIYO OELCLCHMLG A N THE AND MAP COHOMOLOGY LOCAL SOME OF SURJECTIVITY ahmtc ujc Classification Subject Mathematics rv h eodvnsigtermi h aie ae u i Our case. ramified the in theorem is rings elements. vanishing local regu second regular complete the ramified of prove complete vanishing of the case on the assertion However, an is and modules Abstract. H I d − + dim( 1 ( q R ) R/I 0 = ) aefiielnt as length finite have Let ( hoe 3.4 Theorem h eodvnsigtermhsaln itr ntetheory the in history long a has theorem vanishing second The R ) ( . R, I ssretv,wihi ttda follows. as stated is which surjective, is ≥ ⊂ rbnu a,lclchmlg,scn aihn theorem vanishing second cohomology, local map, Frobenius m 3 HNOUEIHR N AUASHIMOMOTO KAZUMA AND ISHIRO SHINNOSUKE k , m H n h oa ooooymodules cohomology local the and EODVNSIGTHEOREM VANISHING SECOND ) m d n eua element regular a and − ea be ) i ( Let R/I Spec d k dmninlcmlt eua oa igo ie characte mixed of ring local regular complete -dimensional ( H ) R, 33,1D5 13H10. 13D45, 13A35, : ◦ uhthat such R and I ( i +1 R/I m mdlsfreeymnmlpieideal prime minimal every for -modules 1. k , ( R H Introduction ) ) ) m d scnetdi h aik topology. Zariski the in connected is eaddmninlrglrlclrn fcharacteristic of ring local regular d-dimensional a be − −→ x i − R/pR H 1 1 ( I i R/I +1 x ( ∈ R is ) ) r ffiielnt as length finite of are I F ups that Suppose . ssurjective. is sa nlgeo h eodvanishing second the of analogue an as d neovd nti ae,w partially we paper, this In unresolved. pr.Asm that Assume -pure. e 9 rvdi ncharacteristic in it proved [9] ser sursle ngnrl h i of aim The general. in unresolved is t iini ie for given is nition a oa ig ne oecondition. some under rings local lar -nw.Ou 1]poe h second the proved [15] Ogus l-known. fcranlclchmlg modules cohomology local certain of g eua oa ig.Bhattacharyyat rings. local regular d H 5]. e ssteter fsurjective of theory the uses dea m 1 iutnosy nodrt prove to order In simultaneously. s odto ntefiieeso local of finiteness the on condition ihu h ntns condition finiteness the without e ulcaatrsi ae Recently, case. characteristic qual ( jcieeeet hc sdefined is which elements rjective e yamlilcto ysome by multiplication a by ced R/pR + R/xR flclcohomology local of ujcieelement. surjective , q ) , H R H I mdlsfrsome for -modules m is i m 2 q ( ⊂ ( R R/pR of F o all for ) R pr n the and -pure I hnthe Then . saproper a is + q i ) > p ristic ≥ and 0, 0 2 S.ISHIRO AND K.SHIMOMOTO

i where m is the maximal ideal of a local ring R. In the present article, we work with HI(R), where I is an arbitrary proper ideal of R for some i with some extra conditions. We remark that the hypothesis in Theorem 1.1 applies to any unramified regular local ring and any ramified regular local ring of the form R = C(k)[[x ,...,x ]]/(p f), where f is a square-free monomial in the 1 d − variables x1,...,xd, because R/pR is known to be F -pure by Fedder’s criterion [2]. We emphasize that this paper is guided by the following general question and we will see that the main theorem gives one such realization: Let R be a ring of mixed characteristic p> 0. Then R/pR is a ring of prime characteristic. • Can one find a ring S of prime characteristic with a regular element t S such that ∈ S/tS ∼= R/pR and S inherits certain good properties from R? We construct a nontrivial example which satisfies the assumption of Theorem 1.1 in Example 4.5.

1.1. Notations and conventions. All rings in this paper are assumed to be commutative noetherian with unity. • For a local ring (R, m, k), the punctured spectrum is the set Spec(R) m denoted by • Spec◦(R). \ { } −1 For a ring R and an element f R, Rf = S R is the ring of fractions of A with respect • to S = f n . ∈ { }n≥0

2. Preliminaries; local cohomology modules We refer the reader to [14] for local cohomology modules.

2.1. Deﬁnition of local cohomology modules and long exact sequences. Let R be a Noe- therian ring and let I be an ideal generated by f1,...,fr of R. For an R-module M, we have the Cechˇ complex

• (2.1) C (I; M) : 0 M Y Mfj Y Mfj fj Mf1···fr 0. → → → 1 2 →···→ → 1≤j≤r 1≤j1

i Then its i-th cohomology is called the i-th local cohomology module and denoted by HI(M). We often deal with the long exact sequences of three diﬀerent types. First for a short exact sequence 0 N M L 0, we obtain the long exact sequence → → → → (2.2) Hi (N) Hi(M) Hi(L) Hi+1(N) . ···→ I → I → I → I →··· Take an element f R, an ideal I R and an R-module M. Then we obtain the long exact sequence ∈ ⊂ (2.3) Hi (M) Hi(M) Hi(M ) Hi+1 (M) ···→ I+f → I → I f → I+f →··· which is induced from the short exact sequence of complexes 0 C•(I + f; M) C•(I; M) • → → → C (I; Mf ) 0. Finally take→ ideals I, J R and an R-module M, we obtain the long exact sequence called the Mayer-Vietoris long exact⊂ sequence (2.4) Hi (M) Hi(M) Hi (M) Hi (M) Hi+1 (M) . ···→ I+J → I ⊕ J → I∩J → I+J →··· SURJECTIVITY OF SOME LOCAL COHOMOLOGY MAP AND THE SECOND VANISHING THEOREM 3

2.2. Connectedness of punctured spectra. The punctured spectrum Spec◦(R) of a local ring (R, m, k) is the set of all primes p = m with the topology induced by the Zariski topology on Spec R. Let I be an ideal of R. The6 punctured spectrum Spec◦(R/I) is connected if the following property holds; For any ideals a and b of R with √a b = √I and √a + b = m, √a or √b equals ∩ m. Equivalently, √a or √b equals √I. Remark 2.1. If R is a local domain, then it is easy to see that the punctured spectrum Spec◦(R) is connected.

3. Some surjective maps of local cohomology modules In this section, we will investigate the surjectivity of certain local cohomology maps. This is motivated by the notion of surjective elements. In the articles [7], [12], surjective elements are considered only for local cohomology modules supported by the maximal ideal. We apply its idea for local cohomology modules supported by an arbitrary ideal. Recall that a ring R of characteristic p> 0 is F-pure (resp. F-split), if the Frobenius endomor- phism F : R R is pure (resp. split). → Definition 3.1. Let R be a local ring of characteristic p > 0 and let M be an R-module with a Frobenius action F : M M. Then M is antinilpotent, if for any submodule N M which satisfies F (N) N, the induced→ Frobenius action on M/N is injective. ⊆ ⊆ i Lemma 3.2. Let R be an F -split local ring and let I be an ideal of R. Then if HI(R) is artinian for some fixed i 0, it is antinilpotent. ≥ Proof. In the proof of [11, Theorem 3.7], the required conditions that we need to prove is that local cohomology modules supported at the maximal ideal are artinian. Thus the proof in [11, Theorem 3.7] can directly be applied without any changes. Lemma 3.3. Let (R, m, k) be a local ring of characteristic p > 0 and let M be an R-module. If F e (e) M is antinilpotent, then R(M) := R R M M is surjective for some (equivalently, every) e 1. ⊗ → ≥ Proof. This is [12, Lemma 2.1]. Theorem 3.4. Let (R, m, k) be a d-dimensional regular local ring of characteristic p> 0. Fix an ideal I m and a regular element x I. Suppose that R/xR is F -pure and the local cohomology ⊂ d−i d−i−1 ∈ modules Hm (R/I) and Hm (R/I) are of finite length as R-modules for some i 0. Then the x ≥ multiplication map Hi+1(R) Hi+1(R) is surjective. I −→ I Proof. There is a short exact sequence 0 R x R R/xR 0 together with its associated long exact sequence: → −→ → → x (3.1) Hi (R) Hi(R) Hi(R/xR) Hi+1(R) . ···→ I −→ I → I → I →··· d−i d−i−1 Since Hm (R/I) and Hm (R/I) are finite length R-modules, it follows from [10, Corollary i i+1 3.3] and [10, Corollary 3.4] that HI(R) and HI (R) are artinian R-modules. This implies that i HI(R/xR) is artinian via (3.1), since the class of artinian modules is stable under extension. Consider the composition map

α βe F e (Hi (R/xR)) e R/xR F e (Hi (R/xR)) = F e (Hi(R/xR)) Hi(R/xR) R I −→ ⊗R R I ∼ R/xR I −→ I for every e 1. Remark that R/xR is F -split, since a complete F -pure ring is known to be F - ≥ split by [2, Lemma 1.2]. Then applying Lemma 3.2 and Lemma 3.3, βe is surjective. Moreover, 4 S.ISHIRO AND K.SHIMOMOTO

F e i i αe is always surjective, so we conclude that the composite map R(HI (R/xR)) HI (R/xR) is i pe i → surjective. Since this composite map factors through HI(R/x R) HI (R/xR), it follows that i pe i pe →։ HI(R/x R) HI (R/xR) is surjective. Finally note that R/x R R/xR factors through the → h ։ e i pe surjective map R/x R R/xR for any 1 h p . Then the induced map HI(R/x R) i ≤ i ≤ h i → HI(R/xR) factors through the surjective map HI (R/x R) HI (R/xR). Next we show the following claim. →

j−l Claim 3.5. For each l> 0 and j l, the multiplication map R/xlR x R/xj induces an injective j−l ≥ −−−→ map Hi+1(R/xlR) x Hi+1(R/xjR). I −−−→ I Proof of Claim 3.5. By induction, it suﬃce to show the case j = l + 1. The short exact sequence x 0 R/xlR R/xl+1R R/xR 0 → −→ → → induces the long exact sequence of local cohomology modules φ δ Hi (R/xl+1R) i Hi(R/xR) i Hi+1(R/xlR) Hi+1(R/xl+1R) . ···→ I −→ I −→ I → I →··· Then since φ is surjective, δ is a zero map. So Hi+1(R/xlR) Hi+1(R/xl+1R) is injective. i i I → I x The short exact sequence 0 R R R/xR 0 induces the long exact sequence of local cohomology modules → −→ → →

x πi+1 δ Hi+1(R) Hi+1(R) Hi+1(R/xR) Hi+2(R) . ···→ I −→ I −−−→ I −→ I →··· By Claim 3.5, we obtain an injection Hi+1(R/xlR) x Hi+1(R/xl+1R) for any l> 0. This implies I −→ I that Hi+1(R/xkR) forms an injective direct system. Then applying the discussion of [12, { I }k≥0 i+1 δ i+2 Proposition 3.3], we can show that HI (R/xR) HI (R) is injective. Therefore we obtain x −→ Im π = Ker δ = 0. This implies that Hi+1(R) Hi+1(R) is surjective. i+1 I −→ I We do not know that the characterization of surjective elements holds for local cohomology modules supported by an arbitrary ideal without the ﬁniteness condition on the length. So we suggest the following question. Question 1. Do the analogues of [12, Proposition 3.2] and [12, Proposition 3.3] hold for a local cohomology module supported by an arbitrary ideal?

4. Application to vanishing of local cohomology modules First we deﬁne two graphs. These are important to prove the main theorem. Deﬁnition 4.1. Let (R, m, k) be a local ring.

(1) Let p1,..., pt be the set of minimal primes of R. Then the graph ΘR has vertices labeled 1,...,t and there is an edge between two (distinct) vertices i and j, precisely when pi + pj is not m-primary. (2) ([6, Deﬁnition 3.4]) Let p1,..., pr be the set of minimal primes of R such that dim(R/pi)= dim(R). Then the Hochster-Huneke graph of R, which is denoted by ΓR, has vertices 1, . . . , r and there is an edge between two (distinct) verticies i and j, precisely if pi + pj has height one.

Huneke and Lyubeznik pointed out the importance of the graph ΘR when we investigate the connectedness of punctured spectra. SURJECTIVITY OF SOME LOCAL COHOMOLOGY MAP AND THE SECOND VANISHING THEOREM 5

Lemma 4.2. (Huneke-Lyubeznik) Let (R, m, k) be a local ring. Then Spec◦(R) is connected if and only if The graph ΘR is connected. The following lemma is important to prove Lemma 4.4. Lemma 4.3. ([5, Proposition 3.5]) Let (R, m, k) be a complete local domain and let x m be a ∈ non-zero element. Then the Hochster-Huneke graph ΓR/xR of R/xR is connected. We prove the following lemma which is similar to [5, Lemma 3.7], but the proof is different in nature. Lemma 4.4. Let (R, m, k) be a d-dimensional complete regular local ring of mixed characteristic with separably closed residue field k such that R/pR is F -pure. Let q be a prime ideal of R such that 1 2 dim(R/q) 3. Put J := pR + q. Assume that the local cohomology modules Hm(R/J), Hm(R/J) 3 ≥ d−1 and Hm(R/J) are of finite length as R-modules. Then Hq (R) = 0. Proof. Since R is a complete regular local ring of mixed characteristic, it is isomorphic to C(k)[ x ,...,x ]/(p f) | 1 d| − 2 where f is an element in m mC(k)[|x1,...,x |] or f = x1, and C(k) is a complete discrete C(k)[|x1,...,xd|] \ d valuation ring such that C(k)/pC(k) ∼= k. First consider the case p q. Put S := k[ x ,...,x ]. Then we have the isomorphism R/pR = ∈ | 1 d| ∼ S/fS where f is the image of C(k)[ x ,...,x ] ։ k[ x ,...,x ]. Thus we obtain the short sequence | 1 d| | 1 d| f (4.1) 0 S S R/pR 0. → −→ → → Applying the long exact sequence for (4.1), we get

f (4.2) Hi(S) Hi(S) Hi(R/pR) Hi+1(S) , ···→ p −→ p → q → p →··· where p S is the inverse image of the ideal q(R/pR) under S ։ R/pR. Let us show the vanishing d−2 ⊆ Hq (R/pR) = 0. Since p is contained in q and q is a prime ideal, the punctured spectrum of (R/pR)/(q/pR) = R/q is connected. In addition, we have the inequality dim R/q 2. So we obtain d−1 ∼ ≥ Hp (S) = 0 by the second vanishing theorem [8, Theorem 2.9]. Moreover since S/fS ∼= R/pR is F -pure and f S is a regular element by assumption, it follows from Theorem 3.4 that the ∈ f multiplication map Hd−2(S) Hd−2(S) in (4.2) is surjective. To summarize the above, we obtain p −→ p f (4.3) Hd−2(S) Hd−2(S) Hd−2(R/pR) Hd−1(S) = 0. ···→ p −→ p → q → p d−2 d−2 d−2 f d−2 Remarking that Hp (S) Hq (R/pR) is trivial by the surjectivity of Hp (S) Hp (S), we d−2 →d−1 d−2 −→ see that Hq (R/pR) Hp (S) = 0 is injective. This implies that Hq (R/pR) = 0. Finally, → p attached to the short exact sequence: 0 R R R/pR 0, we get the exact sequence → −→ → → p (4.4) 0 Hd−1(R) Hd−1(R). → q −→ q Assume that Hd−1(R) = 0. Since Hd−1(R) is q-torsion and p q, every element of Hd−1(R) is q 6 q ∈ q n d−1 p annihilated by p for some n > 0. This gives a contradiction to the injectivity of Hq (R) d−1 d−1 −→ Hq (R). Thus we obtain the vanishing Hq (R) = 0. Next we consider the case p / q. For distinct minimal primes p and p of Spec(R/J), if ∈ 1 2 htR/J (p1 + p2) = 1, then p1 + p2 can not be primary to the maximal ideal of R/J, since we have the inequality dim(R/J) 2. This implies that the Hochster-Huneke graph Γ is a subgraph ≥ R/J 6 S.ISHIRO AND K.SHIMOMOTO of ΘR/J . Moreover the two graphs ΓR/J and ΘR/J have the same vertices. Indeed, since R/q is a complete local domain which is catenary, R/J is equidimensional. Since R/q is a complete local domain, the Hochster-Huneke graph ΓR/J of R/J is connected by Lemma 4.3 by taking x = p. Since ΓR/J and ΘR/J have the same vertices, ΘR/J is also connected. ◦ This implies that Spec (R/J) is connected by Lemma 4.2. Remark that R/J ∼= S/J, where J = J/pR and dim(R/J) 2. Then we obtain Hd−1(S) = 0 by the second vanishing theorem [8, J ≥ d−1 Theorem 2.9]. Applying the above discussion replacing p for J, we obtain HJ (R) = 0. Finally we have the long exact sequence Hd−1(R) Hd−1(R) Hd−1(R ) . ···→ J → q → q p →··· d−1 d−1 We already proved HJ (R) = 0. So it suffices to show that Hq (Rp) = 0. Since p is in the maximal ideal m, we obtain dim Rp = d 1 and dim(Rp/qRp) 2. It then suffices to show that the d−1 − ≥ localization of H (R ) at every prime p Spec R vanishes, where p / p, q p and dim Rp = d 1. q p ∈ ∈ ⊂ − Notice that (Rp)p ∼= Rp is a regular local ring of dimension d 1 and dim(Rp/qRp) 2. Thus d−1 d−1 − ≥ we obtain (H (R ))p = H (Rp) = 0 by the Hartshorne-Lichtenbaum vanishing theorem. Thus q p ∼ qRp d−1 Hq (Rp) = 0. Now we prove Theorem 1.1. The proof is based on [5, Theorem 3.8]. d−1 ◦ Proof of Theorem 1.1. Suppose that HI (R) = 0. If Spec (R/I) is not connected, there exist ideals J , J of R such that √J + J = m and √J J = √I, but √J and √J are not equal to 1 2 1 2 1 ∩ 2 1 2 both m and √I. By the Mayer-Vietoris exact sequence, we get Hd−1(R) Hd (R) Hd (R) Hd (R) . ···→ I → m → J1 ⊕ J2 →··· d−1 d d Then we have HI (R) = 0 by assumption and HJ1 (R) = HJ2 (R) = 0 by the Hartshorne- d Lichtenbaum vanishing theorem. This is a contradiction for Hm(R) = 0 by the Grothendieck’s vanishing theorem. 6 Conversely, suppose that Spec◦(R/I) is connected. We proceed by induction on the number of minimal primes t of I. The case that t = 1 was established in Lemma 4.4. Assume that the implication holds for all ideals a of R with t 1 0 minimal primes, for which dim(R/p) 3 for some minimal primes p of a. − ≥ ≥ Fix an ideal I with t minimal primes such that dim(R/p) 3 for any minimal prime p of I, and ◦ ≥ for which Spec (R/I) is connected. Then ΘR/I is also connected by Lemma 4.2. Thus, there is an ordering q1, q2,..., qt of the minimal primes of I such that the induced subgraph of ΘR/I on indices q1, q2,..., qi is connected for all 1 i t. This means that given 1 i t, if Ji = q1 qi, ≤ ≤ ◦ ≤ ≤ ∩···∩ then the graph ΘR/Ji is connected. So we deduce that Spec (R/Ji) is connected. Consider the Mayer-Vietoris exact sequence in local cohomology associated to Jt−1 and qt Hd−1 (R) Hd−1(R) Hd−1(R) Hd (R) 0. ···→ Jt−1 ⊕ qt → I → Jt−1+qt → By the inductive hypothesis, Hd−1 (R) = 0. In addition, Hd−1(R) = 0 by Lemma 4.4. Moreover, Jt−1 qt q m d since the punctured spectrum of R/I is connected, √Jt−1 + t ( , so that HJt−1+qt (R)=0 by d−1 the Hartshorne-Lichtenbaum vanishing theorem, which proves HI (R) = 0. Example 4.5. We construct a nontrivial example fitting into the setting of Theorem 1.1. Let k be an algebraically closed residue field of characteristic p > 0 and let W (k) be the Witt ring of k. First, take A = k[[X , X , X , X ,Y ,Y ,Y ,Y ]]/(X , X , X , X ) (Y ,Y ,Y ,Y ). 1 2 3 4 1 2 3 4 1 2 3 4 ∩ 1 2 3 4 SURJECTIVITY OF SOME LOCAL COHOMOLOGY MAP AND THE SECOND VANISHING THEOREM 7

1 i This is a 4-dimensional F -pure and Buchsbaum local ring and HmA (A)= k, HmA (A)=0for i = 1, 4 (see [3, Example 2.4]). In particular, we have depth(A) = 1. 6 Put R := R = W (k)[[X , X , X , X ,Y ,Y ,Y ,Y ]]/(p X Y ) i,j 1 2 3 4 1 2 3 4 − i j with 1 i, j 4 and I = (X , X , X , X ) (Y ,Y ,Y ,Y ). Fix any pair (i, j). Then R is a ≤ ≤ 1 2 3 4 ∩ 1 2 3 4 8-dimensional ramiﬁed regular local ring. Then R/(pR + I) ∼= R/I ∼= A. There are 2 minimal primes of I, which are

p = (X1, X2, X3, X4) and q = (Y1,Y2,Y3,Y4).

Then R/(pR + p) and R/(pR + q) are regular local rings of dimension 4. The graph ΘA is not ◦ 7 connected. So by Lemma 4.2, Spec (A) is not connected and hence HI (R) = 0. Note that the topological structure of Spec◦(A) cannot be detected by [4, Proposition 2.1], because6 depth(A) = 1. References [1] R. Bhattacharyya, A note on the second vanishing theorem, https://arxiv.org/abs/2004.02075v3. [2] R. Fedder, F -purity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), 461–480. [3] S. Goto and T. Ogawa, A note on rings with finite local cohomology, Tokyo J. Math. 6 (1983), 403–411. [4] R. Hartshorne, Complete intersections and connectedness, Amer. J. Math. 84 (1962), 497–508. [5] D. J. Hernández, L. Núnez-Betancourt, J. F. Pérez, and E. Witt, Cohomological dimension, Lyubeznik numbers, and connectedness in mixed characteristic, J. Algebra 514 (2018), 442–467. [6] M. Hochster and C. Huneke, Indecomposable canonical modules and connectedness, Commutative algebra:syzygies, multiplicities, and birational algebra (South Hadley,MA,1992), Comtemp. Math. 159 (1994) 197– 208, Amer. Math. Soc., Providence, RI. [7] J. Horiuchi, L. E. Miller and K. Shimomoto, Deformation of F -injectivity and local cohomology, Indiana Univ. Math. J. 63 (2014), 1139–1157. [8] C. Huneke and G. Lyubeznik, On the vanishing of local cohomology modules, Invent. Math. 102 (1990), 73–93. [9] R. Hartshorne and R. Speiser, Local cohomological dimension in characteristic p, Annals of Mathematics (1977), 45–79. [10] G. Lyubeznik, On the vanishing of local cohomology in characteristic p > 0, Composition Mathematica 142 (2006), 207–221. [11] L. Ma, Finiteness properties of local cohomology for F -pure local rings, International Mathematics Research Notices 20 (2014), 5489–5509. [12] L. Ma and P. H. Quy, Frobenius actions on local cohomology modules and deformation, Nagoya Math. J. 232 (2018), 55–75. [13] H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge (1986). [14] S. B. Iyengar, G. J. Leuschke, A. Leykin, C. Miller, E. Miller, A. K. Singh, and U. Walther, Twenty-Four Hours of Local Cohomology, Graduate Studies in Mathematics 87, American Mathematical Society, Providence (2007). [15] A. Ogus, Local cohomological dimension of algebraic varieties, Annals of Mathematics (1973), 327–365. [16] W. Zhang, The second vanishing theorem for local cohomology modules, https://arxiv.org/abs/2102.12545.

Email address: [email protected] Email address: [email protected]