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Noetherian loop spaces. (English) Zbl 1229.55011
This paper studies properties of loop spaces having Noetherian mod $$p$$ cohomology, paying attention to differences between such loop spaces and those with finite mod $$p$$ cohomology; here $$p$$ is any prime number.
More precisely, a $$p$$-Noetherian group is a triple $$(X,BX,e)$$ such that the cohomology $$H^*(X; \mathbb F_p)$$ is a Noetherian algebra (it is finitely generated as an algebra), $$BX$$ is a $$p$$-complete space, and $$e: X\rightarrow \Omega BX$$ (where $$\Omega BX$$ is the space of loops on $$BX$$) is a weak homotopy equivalence. In this setting, $$X$$ is called the Noetherian loop space, and $$BX$$ is referred to as the classifying space of $$X$$. In particular, $$p$$-compact groups in the sense of W. Dwyer and C. Wilkerson [Ann. Math. (2) 139, No. 2, 395–442 (1994; Zbl 0801.55007)] are $$p$$-Noetherian groups.
The authors study the cohomology of the classifying space $$BX$$ and show that it is as small as expected. More precisely, they prove that for any $$p$$-Noetherian group $$(X,BX,e)$$, $$H^*(BX; \mathbb F_p)$$ is finitely generated as an algebra over the Steenrod algebra and, in addition, the module of indecomposable elements, $$QH^*(BX; \mathbb F_p)=\tilde H^*(BX; \mathbb F_p)/\tilde H^*(BX; \mathbb F_p)\cdot \tilde H^*(BX; \mathbb F_p)$$, belongs to $$\mathcal U_1$$ of the Krull filtration, $$\mathcal U_1\subset \mathcal U_2\subset \cdots$$, of the category $$\mathcal U$$ of unstable modules over the Steenrod algebra ($$\mathcal U_0$$ consists of the locally finite unstable modules and contains $$QH^*(X; \mathbb F_p)$$; that is, $$QH^*(BX; \mathbb F_p)$$ lies only one stage higher than where $$QH^*(X; \mathbb F_p)$$ lies).
##### MSC:
 55R35 Classifying spaces of groups and $$H$$-spaces in algebraic topology 13A02 Graded rings 13E05 Commutative Noetherian rings and modules 55P20 Eilenberg-Mac Lane spaces 55P60 Localization and completion in homotopy theory 55S10 Steenrod algebra 55T10 Serre spectral sequences 57T10 Homology and cohomology of Lie groups
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