 # Boundedness of cohomology

Brodmann-Maeder, Monika; Jahangiri, M; Linh, C H (2010). Boundedness of cohomology. 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.

## 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.

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