# Boundedness of Cohomology - Mathematics > Commutative Algebra

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Abstract: Let $d \in \N$ and let $\D^d$ denote the class of all pairs $R,M$ in which$R = \bigoplus {n \in \N 0} R n$ is a Noetherian homogeneous ring with Artinianbase ring $R 0$ and such that $M$ is a finitely generated graded $R$-module ofdimension $\leq d$.The cohomology table of a pair $R,M \in \D^d$ is defined as the family ofnon-negative integers $d M:= d^i Mn {i,n \in \N \times \Z}$. We say thata subclass $\mathcal{C}$ of $\D^d$ is of finite cohomology if the set $\{d M\mid R,M \in \C\}$ is finite. A set $\mathbb{S} \subseteq \{0,

.,d-1\}\times \Z$ is said to bound cohomology, if for each family$h^\sigma {\sigma \in \mathbb{S}}$ of non-negative integers, the class$\{R,M \in \D^d\mid d^i Mn \leq h^{i,n} {for all} i,n \in \mathbb{S}\}$is of finite cohomology. Our main result says that this is the case if and onlyif $\mathbb{S}$ contains a quasi diagonal, that is a set of the form$\{i,n i| i=0,

., d-1\}$ with integers $n 0> n 1 >

. > n {d-1}$. We draw anumber of conclusions of this boundedness criterion.

Author: ** Markus Brodmann, Maryam Jahangiri, Cao Huy Linh**

Source: https://arxiv.org/