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Invertible modules for commutative -algebras with residue fields

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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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Authors and Affiliations

  1. Mathematics Department, University of Glasgow, Glasgow, G12 8QW, Scotland

    Andrew Baker

  2. Fachbereich Mathematik der Universität Hamburg, Bundesstrasse 55, 20146, Hamburg, Germany

    Birgit Richter

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  1. Andrew Baker
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Baker, A., Richter, B. Invertible modules for commutative -algebras with residue fields. manuscripta math. 118, 99–119 (2005). https://doi.org/10.1007/s00229-005-0582-1

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