Boundedness of Cohomology

Let $d\in \mathbb{N}$ and let ${\text{D}}^d$ 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 $(R,M)\in {\text{D}}^d$ 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 ${\text{D}}^d$ is of finite cohomology if the set ${d_M \mid (R,M) \in \C}$isfinite\mathrm{.}Aset$$\mathbb{S}$ \subseteq {0,… ,d-1}\times \Z$issaidtoboundcohomology,ifforeachfamily$(h^\sigma)_{\sigma \in $\mathbb{S}$}$ofnon-negativeintegers,theclass${(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 $\{(i,n_i)\mathrm{\mid }i=0,\mathrm{.}\mathrm{.}\mathrm{.},d-1\}$ with integers $n_0>n_1>\mathrm{.}\mathrm{.}\mathrm{.}>n_{d-1}$. We draw a number of conclusions of this boundedness criterion.