scholarly journals Cohomologically Full Rings

2019 ◽  
Author(s):  
Hailong Dao ◽  
Alessandro De Stefani ◽  
Linquan Ma
Keyword(s):  
Local Cohomology ◽  
Projective Dimension ◽  
Base Change ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Local Cohomology Modules ◽  
Flat Base ◽  
Du Bois Singularities ◽  
Cohomology Modules ◽  
Algebra Over A Field

Abstract Inspired by a question raised by Eisenbud–Mustaţă–Stillman regarding the injectivity of maps from ${\operatorname{Ext}}$ modules to local cohomology modules and the work by the third author with Pham, we introduce a class of rings, which we call cohomologically full rings. This class of rings includes many well-known singularities: Cohen–Macaulay rings, Stanley–Reisner rings, F-pure rings in positive characteristics, and Du Bois singularities in characteristics $0$. We prove many basic properties of cohomologically full rings, including their behavior under flat base change. We show that ideals defining these rings satisfy many desirable properties, in particular they have small cohomological and projective dimension. When $R$ is a standard graded algebra over a field of characteristic $0$, we show under certain conditions that being cohomologically full is equivalent to the intermediate local cohomology modules being generated in degree $0$. Furthermore, we obtain Kodaira-type vanishing and strong bounds on the regularity of cohomologically full graded algebras.

scholarly journals Local cohomology of Du Bois singularities and applications to families

2017 ◽  
Vol 153 (10) ◽  
pp. 2147-2170 ◽  
Author(s):  
Linquan Ma ◽  
Karl Schwede ◽  
Kazuma Shimomoto
Keyword(s):  
Commutative Algebra ◽  
Local Cohomology ◽  
American Mathematical Society ◽  
Vanishing Theorem ◽  
Du Bois ◽  
Local Cohomology Modules ◽  
Du Bois Singularities ◽  
Cohomology Modules ◽  
Kodaira Vanishing ◽  
Koszul Cohomology

In this paper we study the local cohomology modules of Du Bois singularities. Let $(R,\mathfrak{m})$ be a local ring; we prove that if $R_{\text{red}}$ is Du Bois, then $H_{\mathfrak{m}}^{i}(R)\rightarrow H_{\mathfrak{m}}^{i}(R_{\text{red}})$ is surjective for every $i$. We find many applications of this result. For example, we answer a question of Kovács and Schwede [Inversion of adjunction for rational and Du Bois pairs, Algebra Number Theory 10 (2016), 969–1000; MR 3531359] on the Cohen–Macaulay property of Du Bois singularities. We obtain results on the injectivity of $\operatorname{Ext}$ that provide substantial partial answers to questions in Eisenbud et al. [Cohomology on toric varieties and local cohomology with monomial supports, J. Symbolic Comput. 29 (2000), 583–600] in characteristic $0$. These results can also be viewed as generalizations of the Kodaira vanishing theorem for Cohen–Macaulay Du Bois varieties. We prove results on the set-theoretic Cohen–Macaulayness of the defining ideal of Du Bois singularities, which are characteristic-$0$ analogs and generalizations of results of Singh–Walther and answer some of their questions in Singh and Walther [On the arithmetic rank of certain Segre products, in Commutative algebra and algebraic geometry, Contemporary Mathematics, vol. 390 (American Mathematical Society, Providence, RI, 2005), 147–155]. We extend results on the relation between Koszul cohomology and local cohomology for $F$-injective and Du Bois singularities first shown in Hochster and Roberts [The purity of the Frobenius and local cohomology, Adv. Math. 21 (1976), 117–172; MR 0417172 (54 #5230)]. We also prove that singularities of dense $F$-injective type deform.

Matlis Duality and Finiteness Properties of Generalized Local Cohomology Modules

2010 ◽  
Vol 17 (04) ◽  
pp. 637-646 ◽  
Author(s):  
Hero Saremi
Keyword(s):  
Local Cohomology ◽  
Projective Dimension ◽  
Regular Sequence ◽  
Local Cohomology Module ◽  
Local Cohomology Modules ◽  
Matlis Duality ◽  
Noetherian Dimension ◽  
Generalized Local Cohomology Modules ◽  
Finiteness Properties ◽  
Cohomology Modules

Let [Formula: see text] be an ideal of a commutative Noetherian local ring [Formula: see text] and M, N be two finitely generated R-modules such that M is of finite projective dimension n. Let t be a positive integer. We show that if there exists a regular sequence [Formula: see text] with [Formula: see text] and the i-th local cohomology module [Formula: see text] of N with respect to [Formula: see text] is zero for all i > t, then [Formula: see text], where D(-):= Hom R(-,E). Also, we prove that if N is a Cohen-Macaulay R-module of dimension d, then the generalized local cohomology module [Formula: see text] is co-Cohen-Macaulay of Noetherian dimension d. Finally, with an elementary proof, we show that [Formula: see text] is finite.

FINITENESS DIMENSION AND BASS NUMBERS OF GENERALIZED LOCAL COHOMOLOGY MODULES

2013 ◽  
Vol 12 (07) ◽  
pp. 1350036 ◽  
Author(s):  
HERO SAREMI ◽  
AMIR MAFI
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Betti Numbers ◽  
Projective Dimension ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Bass Numbers ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M, N two nonzero finitely generated R-modules. Let t be a non-negative integer. It is shown that dim Supp [Formula: see text] for all i < t if and only if there exists an ideal 𝔟 of R such that dim R/𝔟 ≤ 1 and [Formula: see text] for all i < t. As a consequence all Bass numbers and all Betti numbers of generalized local cohomology modules [Formula: see text] are finite for all i < t, provided that the projective dimension pd (M) is finite.

scholarly journals Artinianness of local cohomology modules

2014 ◽  
Vol 52 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Moharram Aghapournahr ◽  
Leif Melkersson
Keyword(s):  
Local Cohomology ◽  
Local Cohomology Modules ◽  
Cohomology Modules

On the associated primes of local cohomology modules

1999 ◽  
Vol 27 (12) ◽  
pp. 6191-6198 ◽  
Author(s):  
K. Khashyarmanesh ◽  
Sh Salarian
Keyword(s):  
Local Cohomology ◽  
Associated Primes ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Some properties of local homology and local cohomology modules

2013 ◽  
Vol 50 (1) ◽  
pp. 129-141
Author(s):  
Tran Nam
Keyword(s):  
Local Cohomology ◽  
Local Homology ◽  
Local Cohomology Modules ◽  
Cohomology Modules ◽  
Compact Module

We study some properties of representable or I-stable local homology modules HiI (M) where M is a linearly compact module. By duality, we get some properties of good or at local cohomology modules HIi (M) of A. Grothendieck.

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