  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.

Commutative Algebra;Algebraic Geometry