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2022•Journal of Algebra and Its ApplicationsRequires access

Cohomological dimension of DG-modules

Yanping Rao, Zhongkui Liu, Xiaoyan Yang

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Abstract

Let [Formula: see text] be a commutative noetherian non-positive DG-ring, [Formula: see text] an ideal of [Formula: see text], and [Formula: see text]. In this paper, we introduce the notion of cohomological dimension of DG-modules and investigate the interplay between [Formula: see text] and [Formula: see text]. It is shown that [Formula: see text] for any [Formula: see text]. As an application, we recover a DG-version of Grothendieck’s vanishing and non-vanishing theorems for local cohomology. We also study the cohomological dimension of Koszul DG-modules and get some interesting results.

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What this paper is about

Let [Formula: see text] be a commutative noetherian non-positive DG-ring, [Formula: see text] an ideal of [Formula: see text], and [Formula: see text]. In this paper, we introduce the notion of cohomological dimension of DG-modules and investigate the interplay between [Formula: see text] and [Formula: see text]. It is shown that [Formula: see text] for any [Formula: see text]. As an application, we recover a DG-version of Grothendieck’s vanishing and non-vanishing theorems for local cohomology. We also study the cohomological dimension of Koszul DG-modules and get some interesting results.

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Available abstract

Let [Formula: see text] be a commutative noetherian non-positive DG-ring, [Formula: see text] an ideal of [Formula: see text], and [Formula: see text]. In this paper, we introduce the notion of cohomological dimension of DG-modules and investigate the interplay between [Formula: see text] and [Formula: see text]. It is shown that [Formula: see text] for any [Formula: see text]. As an application, we recover a DG-version of Grothendieck’s vanishing and non-vanishing theorems for local cohomology. We also study the cohomological dimension of Koszul DG-modules and get some interesting results.

Key concepts: Cohomological dimension, Mathematics, Local cohomology, Noetherian ring, Global dimension, Dimension (graph theory), Commutative property, Pure mathematics

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