Boundedness of Cohomology
Markus Brodmann, Maryam Jahangiri, Cao Huy Linh

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 and let denote the class of all pairs in which is a Noetherian homogeneous ring with Artinian base ring and such that is a finitely generated graded -module of dimension . The cohomology table of a pair is defined as the family of non-negative integers . We say that a subclass of is of finite cohomology if the set is finite. A set is said to bound cohomology, if for each family of non-negative integers, the class is of finite cohomology. Our main result says that this is the case if and only if contains a quasi diagonal,…
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Taxonomy
TopicsCommutative Algebra and Its Applications · Advanced Combinatorial Mathematics · Algebraic Geometry and Number Theory
