Boundedness of Cohomology

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dc.creatorBrodmann, Markus
dc.creatorJahangiri, Maryam
dc.creatorLinh, Cao Huy
dc.date2009-05-15
dc.date.accessioned2026-07-07T13:15:34Z
dc.date.available2026-07-07T13:15:34Z
dc.descriptionLet $d \in \N$ 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 $(h^σ)_{σ\in \mathbb{S}}$ 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.
dc.description18 pages
dc.identifierhttps://arxiv.org/abs/0905.2471
dc.identifierhttp://arxiv.org/abs/0905.2471
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/230510
dc.subjectCommutative Algebra
dc.subjectAlgebraic Geometry
dc.subject13D45, 14B15
dc.titleBoundedness of Cohomology
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