# Boundedness of cohomolgy

Brodmann, M; Jahangiri, M; Linh, C H (2010). Boundedness of cohomolgy. Journal of Algebra, 323(2):458-472.

## Abstract

Let View the MathML source and let View the MathML source denote the class of all pairs (R,M) in which View the MathML source is a Noetherian homogeneous ring with Artinian base ring R0 and such that M is a finitely generated graded R-module of dimension less-than-or-equals, slantd. For such a pair (R,M) let View the MathML source denote the (finite) R0-length of the n-th graded component of the i-th R+-transform module View the MathML source.

The cohomology table of a pair View the MathML source is defined as the family of non-negative integers View the MathML source. We say that a subclass View the MathML source of View the MathML source is of finite cohomology if the set View the MathML source is finite. A set View the MathML source is said to bound cohomology, if for each family View the MathML source of non-negative integers, the class View the MathML source is of finite cohomology. Our main result says that this is the case if and only if View the MathML source contains a quasi diagonal, that is a set of the form {(i,ni)|i=0,…,d−1} with integers n0>n1>cdots, three dots, centered>nd−1.

We draw a number of conclusions of this boundedness criterion.

Let View the MathML source and let View the MathML source denote the class of all pairs (R,M) in which View the MathML source is a Noetherian homogeneous ring with Artinian base ring R0 and such that M is a finitely generated graded R-module of dimension less-than-or-equals, slantd. For such a pair (R,M) let View the MathML source denote the (finite) R0-length of the n-th graded component of the i-th R+-transform module View the MathML source.

The cohomology table of a pair View the MathML source is defined as the family of non-negative integers View the MathML source. We say that a subclass View the MathML source of View the MathML source is of finite cohomology if the set View the MathML source is finite. A set View the MathML source is said to bound cohomology, if for each family View the MathML source of non-negative integers, the class View the MathML source is of finite cohomology. Our main result says that this is the case if and only if View the MathML source contains a quasi diagonal, that is a set of the form {(i,ni)|i=0,…,d−1} with integers n0>n1>cdots, three dots, centered>nd−1.

We draw a number of conclusions of this boundedness criterion.

## Citations

1 citation in Web of Science®
1 citation in Scopus®

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 15 January 2010 15 Nov 2010 16:17 05 Apr 2016 14:16 Elsevier 0021-8693 https://doi.org/10.1016/j.jalgebra.2009.07.032 http://arxiv.org/abs/0905.2471
Permanent URL: https://doi.org/10.5167/uzh-36134

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