Boundedness of Cohomology

Abstract

Let $d\in \mathbb{N}$ and let $\D^d$ denote the class of all pairs $(R,M)$ in which $R={⨁}_{n\in {\mathbb{N}}_0}{R}_n$ is a Noetherian homogeneous ring with Artinian base ring $R_0$ and such that $M$ is a finitely generated graded $R$-module of dimension $\le d$. The cohomology table of a pair $(R,M) \in \D^d$ is defined as the family of non-negative integers $d_M:=(d_M^i(n){)}_{(i,n)\in \mathbb{N}\times \mathbb{Z}}$. We say that a subclass $\mathcal{C}$ of $\D^d$ is of finite cohomology if the set $\{d_M \mid (R,M) \in \C\}$ is finite. A set $\mathbb{S}\subseteq \{0,...,d-1\}\times \mathbb{Z}$ is said to bound cohomology, if for each family $(h^{\sigma }{)}_{\sigma \in \mathbb{S}}$ of non-negative integers, the class $\{(R,M) \in \D^d\mid d^i_M(n) \leq h^{(i,n)} {for all} (i,n) \in \mathbb{S}\}$ is of finite cohomology. Our main result says that this is the case if and only if $\mathbb{S}$ 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 a number of conclusions of this boundedness criterion.

18 pages