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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.

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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