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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.
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
13F40 Excellent rings
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