manuscripta mathematica

, Volume 118, Issue 1, pp 99–119

Invertible modules for commutative Open image in new window-algebras with residue fields

Article

DOI: 10.1007/s00229-005-0582-1

Cite this article as:
Baker, A. & Richter, B. manuscripta math. (2005) 118: 99. doi:10.1007/s00229-005-0582-1

Abstract

The aim of this note is to understand under which conditions invertible modules over a commutative Open image in new window-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 Open image in new window-algebra R has coherent localizations Open image in new window for every maximal ideal Open image in new window, 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 Open image in new window-algebra has ‘residue fields’ for all maximal ideals Open image in new window 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.

Keywords

Commutative S-algebra invertible module Picard group 55P15 55P42 55P60 

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of GlasgowGlasgowScotland
  2. 2.Fachbereich Mathematik der Universität HamburgHamburgGermany