# Boundedness of cohomology

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

## Abstract

Let d∈N and let Dd denote the class of all pairs (R,M) in which R=⊕n∈N0Rn is a Noetherian homogeneous ring with Artinian base ring R0 and such that M is a finitely generated graded R-module of dimension ⩽d. 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 (R,M)∈Dd is defined as the family of non-negative integers View the MathML source. We say that a subclass C of Dd is of finite cohomology if the set {dM|(R,M)∈C} is finite. A set S⊆{0,…,d−1}×Z is said to bound cohomology, if for each family (hσ)σ∈S 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 S contains a quasi diagonal, that is a set of the form {(i,ni)|i=0,…,d−1} with integers n0>n1>⋯>nd−1.

We draw a number of conclusions of this boundedness criterion.

## Abstract

Let d∈N and let Dd denote the class of all pairs (R,M) in which R=⊕n∈N0Rn is a Noetherian homogeneous ring with Artinian base ring R0 and such that M is a finitely generated graded R-module of dimension ⩽d. 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 (R,M)∈Dd is defined as the family of non-negative integers View the MathML source. We say that a subclass C of Dd is of finite cohomology if the set {dM|(R,M)∈C} is finite. A set S⊆{0,…,d−1}×Z is said to bound cohomology, if for each family (hσ)σ∈S 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 S contains a quasi diagonal, that is a set of the form {(i,ni)|i=0,…,d−1} with integers n0>n1>⋯>nd−1.

We draw a number of conclusions of this boundedness criterion.

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Physical Sciences > Algebra and Number Theory English 2010 08 Jan 2012 20:58 03 Aug 2021 15:11 Elsevier 0021-8693 Green https://doi.org/10.1016/j.jalgebra.2009.07.032 http://arxiv.org/abs/0905.2471