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Nilpotence and descent in equivariant stable homotopy theory. (English) Zbl 1420.55024
This is a deep paper that unifies and extends many strands of previous research.
There are three parts. The first one develops the concepts of nilpotence, torsion and completeness in the general setting of a stable monoidal $$\infty$$-category. The following is the key notion. Suppose $$A$$ is an algebra object in a monoidal $$\infty$$-category $$\mathcal{C}$$. An object $$M$$ of $$\mathcal{C}$$ is $$A$$-nilpotent if it belongs to the thick $$\otimes$$-ideal of $$\mathcal{C}$$ generated by $$A$$. Roughly speaking, this means that $$M$$ is equivalent to a finite homotopy colimit of free $$A$$-modules, or is a retract of such a colimit. Several characterizations of nilpotent objects are given. There is a corresponding discussion of torsion objects and complete objects.
In the second part, the authors specialize to an important example in equivariant stable homotopy. Let $$G$$ be a finite group, and $$\mathcal{F}$$ be a family of subgroups of $$G$$ (recall that a family is a set of subgroups closed under conjugation and passing to smaller subgroups). Let $$S^0_G$$ be the $$G$$-equivariant sphere spectrum. The following product is an algebra object in the category of $$G$$-spectra: $\prod_{H\in\mathcal{F}}F(G/H_+,S^0_G).$ The authors say that a $$G$$-spectrum $$M$$ is $$\mathcal{F}$$-nilpotent if it is nilpotent over this ring spectrum. They explore general properties of this notion. One of their main theorems says that a $$G$$-equivariant ring spectrum $$\mathcal{R}$$ is $$\mathcal{F}$$-nilpotent if and only if the geometric fixed points $$\Phi^H\mathcal{R}$$ are contractible for all $$H\in \mathcal{F}$$. The authors focus mostly on general theory, but this notion of nilpotence has many applications, some of which are developed in the sequel papers [A. Mathew et al., Geom. Topol. 23, No. 2, 541–636 (2019; Zbl 1422.19001)] and [D. Clausen et al., “Descent in algebraic $$K$$-theory and a conjecture of Ausoni-Rognes”, Preprint, arXiv:1606.03328].
The third and final part concerns the structure of categories of modules over certain equivariant ring spectra for compact Lie groups, with special focus on connected Lie groups. Numerous applications are given, for example to nilpotence and to equivariant $$K$$-theory. To give a flavour of this part, we quote one sample result. Let $$U(n)$$ be the unitary group.
Theorem: Let $$\mathcal{R}$$ be an even-periodic $$E_\infty$$-ring. There is an equivalence of symmetric monoidal $$\infty$$-categories between $$U(n)$$-equivariant $$\mathcal{R}$$-modules and complete modules over the non-equivariant spectrum $$F(BU(n)_+,\mathcal{R})$$. Here completion is taken with respect to the augmentation ideal.
One can interpret this result as a kind of topological Koszul duality between the ring spectra $$\mathcal{R}\wedge U(n)_+$$ ($$\mathcal{R}$$-chains on $$U(n)$$) and $$F(BU(n)_+,\mathcal{R})$$ ($$\mathcal{R}$$-cochains on $$BU(n)$$). It generalizes results of J. P. C. Greenlees and B. E. Shipley on rational equivariant homotopy theory [Math. Z. 269, No. 1–2, 373–400 (2011; Zbl 1230.55008)].

MSC:
 55P91 Equivariant homotopy theory in algebraic topology 55P42 Stable homotopy theory, spectra
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