Invertible modules for commutative -algebras with residue fields |
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Authors: | Andrew Baker Birgit Richter |
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Affiliation: | (1) Mathematics Department, University of Glasgow, Glasgow, G12 8QW, Scotland;(2) Fachbereich Mathematik der Universität Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany |
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Abstract: | The aim of this note is to understand under which conditions invertible modules over a commutative -algebra in the sense of Elmendorf, Kriz, Mandell & May give rise to elements in the algebraic Picard group of invertible graded modules over the coefficient ring by taking homotopy groups. If a connective commutative -algebra R has coherent localizations for every maximal ideal , then for every invertible R-module U, U*=π*U is an invertible graded R*-module. In some non-connective cases we can carry the result over under the additional assumption that the commutative -algebra has ‘residue fields’ for all maximal ideals if the global dimension of R* is small or if R is 2-periodic with underlying Noetherian complete local regular ring R0. We apply these results to finite abelian Galois extensions of Lubin-Tate spectra. |
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Keywords: | Commutative S-algebra invertible module Picard group 55P15 55P42 55P60 |
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