Boundedness of Cohomology

TL;DR

This paper characterizes when the cohomology of graded modules over Noetherian rings is uniformly bounded, showing it depends on the presence of a specific quasi diagonal subset in the cohomology table.

Contribution

It provides a necessary and sufficient condition for boundedness of cohomology in terms of the existence of a quasi diagonal subset, advancing understanding of cohomological finiteness.

Findings

Bounded cohomology occurs iff a quasi diagonal exists in the cohomology table.

The main theorem characterizes boundedness via the presence of a quasi diagonal.

Several consequences of the boundedness criterion are discussed.

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 Artinian base ring $R_0$ and such that $M$ is a finitely generated graded $R$-module of dimension $\leq d$. The cohomology table of a pair $(R,M) \in \D^d$ is defined as the family of non-negative integers $d_M:= (d^i_M(n))_{(i,n) \in \N \times \Z}$. 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 $\mathbb{S} \subseteq \{0,... ,d-1\}\times \Z$ 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.