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Spaces with Noetherian cohomology. (English) Zbl 1263.55005
Let \(Y\) be a connected space with a finite fundamental group \(\pi_1 Y\). The authors consider the following questions. Is the cohomology \(H^*(Y;{\mathbb{F}}_p)\) over the finite field \({\mathbb{F}}_p\) a Noetherian graded algebra? Is the cohomology \(H^*(Y;{\mathbb{Z}}^\wedge_p)\) over the \(p\)-adic integers \({\mathbb{Z}}^\wedge_p\) a Noetherian graded algebra? If \(M\) is a \({\mathbb{Z}}^\wedge_p[\pi_1 Y]\)-module finitely generated over \({\mathbb{Z}}^\wedge_p\), is \(H^*(Y;M)\) a Noetherian module over \(H^*(Y;{\mathbb{Z}}^\wedge_p)\)? They show that the answers to these questions are related as follows: first, \(H^*(Y;{\mathbb{Z}}^\wedge_p)\) is Noetherian if and only if \(H^*(Y;{\mathbb{F}}_p)\) is Noetherian and \(H^*(Y;{\mathbb{Z}}^\wedge_p)\) has bounded torsion; second, if \(\pi_1 Y\) is a \(p\)-group and \(H^*(Y;{\mathbb{Z}}^\wedge_p)\) is Noetherian then \(H^*(Y;M)\) is Noetherian. As applications they show that \(p\)-compact groups and \(p\)-local finite groups have Noetherian cohomology.

MSC:
55U20 Universal coefficient theorems, Bockstein operator
13E05 Commutative Noetherian rings and modules
55N25 Homology with local coefficients, equivariant cohomology
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
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