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Excellent rings in transchromatic homotopy theory. (English) Zbl 1387.55008
The Lubin-Tate spectra $$E_n$$ are fundamental examples of $$K(n)$$-local spectra, but their study sometimes benefits from understanding $$L_{K(t)}E_n$$ for $$t<n$$ first. While $$\pi_0E_n$$ is a power series ring, $$\pi_0L_{K(t)}E_n$$ is a completion of a localization of it and thus more complicated. Recent work by O. Gabber, resp. K. Kurano and K. Shimomoto [“Ideal-adic completion of quasi-excellent rings (after Gabber)”, Preprint, arXiv:1609.09246] in commutative algebra implies that this ring is still an excellent normal domain. The authors also generalize this to iterated localizations of $$E_n$$.
The main applications lie in transchromatic character theory that aims to understand $$\hat{C}_{t,k}\otimes_{E_n^0}(E_n)^0(BG)$$ for a certain ring $$\hat{C}_{t,k}$$ (with suitable $$k$$) in terms of $$L_{K(t)}E_n$$; see [N. Stapleton Algebr. Geom. Topol. 13, No. 1, 171–203 (2013; Zbl 1300.55011)]. The name character theory derives from the case $$n=1$$, where $$E_1$$ is $$p$$-adically complete $$K$$-theory and thus $$(E_1)^0(BG)$$ is a completion of the representation ring $$R(G)$$. The authors show that $$\hat{C}_{t,k}$$ is an excellent normal domain as well and hence that the colimit $$\mathrm{colim}_k \hat{C}_{t,k}$$ is at least normal.
##### MSC:
 55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology 13F40 Excellent rings
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