Zero-dimensional Non-Artinian local cohomology modules
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Archiv der Mathematik
Zero-dimensional Non-Artinian local cohomology modules Farzaneh Vahdanipour, Kamal Bahmanpour , and Ghader Ghasemi
Abstract. Let (R, m, k) be a Noetherian local ring of dimension d ≥ 4. Assume that 2 ≤ i ≤ d − 2 is an integer and x1 , . . . , xi is a part of a system of parameters for R. Let Υi denote the set of all prime ideals i (R/p) = {m}, and p of R such that dim R/p = i + 1, Supp H(x 1 ,...,xi )R i dimk SocR H(x1 ,...,xi )R (R/p) = ∞. In this paper, it is shown that Υi is an infinite set. Mathematics Subject Classification. Primary 13D45, Secondary 14B15, 13E05. Keywords. Artinian module, Cohomological dimension, Local cohomology, Noetherian ring, Symbolic power.
1. Introduction. Throughout this paper, let R denote a commutative Noetherian ring and I be an ideal of R. In the sequel, let the symbol V (I) denote the set Supp R/I = {p ∈ Spec R : p ⊇ I}. The local cohomology modules HIi (M ), i = 0, 1, 2, . . . , of an R-module M with respect to I were introduced by Grothendieck, [9]. They arise as the derived functors of the left exact functor ΓI (−), where for an R-module M , of M consisting of all elements annihilated by some ΓI (M ) is the submodule ∞ power of I, i.e., n=1 (0 :M I n ). There is a natural isomorphism: HIi (M ) ∼ ExtiR (R/I n , M ). = lim −→ n≥1
The geometric roots of local cohomology come back to some works of J.P. Serre in algebraic geometry. For example, the concept of local cohomology plays an important role in the study of sheaf cohomologies. Nowadays, local cohomology is one of the basic and important tools in modern algebraic geometry and local algebra. We refer the reader to [9] or [4] for more details about algebraic and geometric aspects of local cohomology.
F. Vahdanipour et al.
Arch. Math.
Recall that, if (R, m, k) is a local ring, then for any R-module L, the socle of L, denoted by SocR L, is defined as SocR L = (0 :L m) HomR (k, L), which is a k-vector space. It is a well known result that if (R, m, k) is a Noetherian local ring, then for i (M ) is each finitely generated R-module M and each i ∈ N0 , the R-module Hm i Artinian, hence the R-module HomR (k, Hm (M )) is finitely generated. Taking this fact, Grothendieck (see [8, Expos´e XIII, Conjecture 1.1]) conjectured the following: Conjecture. For each ideal I of a Noetherian ring R and each finitely generated R-module M , the R-modules HomR (R/I, HIi (M )) are finitely generated for all i ∈ N0 . Hartshorne, by constructing a counterexample, showed that this conjecture is false even when R is regular. But, this valuable example contains some new information about the local cohomology modules. In fact, this example is the first evidence which shows that some of the finitely generated modules over Noetherian local rings have examples of local cohomology modules with infinite dimensional socles. Hartshorne’s example ([10, §3]). Let k be a field, R = k[[u, v]][x, y], I = 2 (RP /f RP ) is (x, y)R, P = (u, v, x, y)R, and f = ux + vy. Then SocRP HIR P infinite dimensional. In 2004, a similar family of such info
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