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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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