Invertible modules for commutative -algebras with residue fields

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.