Boundedness of cohomology

Let d∈N and let Dd denote the class of all pairs (R,M) in which R=⊕n∈N0Rn is a Noetherian homogeneous ring with Artinian base ring R0 and such that M is a finitely generated graded R-module of dimension ⩽d. For such a pair (R,M) let denote the (finite) R0-length of the n-th graded component of the i-th R+-transform module .The cohomology table of a pair (R,M)∈Dd is defined as the family of non-negative integers . We say that a subclass C of Dd is of finite cohomology if the set {dM|(R,M)∈C} is finite. A set S⊆{0,…,d−1}×Z is said to bound cohomology, if for each family (hσ)σ∈S of non-negative integers, the class is of finite cohomology. Our main result says that this is the case if and only if S contains a quasi diagonal, that is a set of the form {(i,ni)|i=0,…,d−1} with integers n0>n1>⋯>nd−1.We draw a number of conclusions of this boundedness criterion.