Invertible modules for commutative $\mathbb{S}$-algebras with residue fields

Details Invertible modules for commutative $\mathbb{S}$-algebras with residue fields Invertible modules for commutative $\mathbb{S}$-algebras with residue fields

2004-05-27

arxiv.org/abs/math/0405525v4

Mathematics

Algebraic Topology

Revised version. One serious flaw corrected

Scientific paper

The aim of this note is to understand under which conditions invertible modules over a commutative S-algebra in the sense of Elmendorf, Kriz, Mandell and 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 S-algebra R has coherent localizations (R_*)_m for every maximal ideal m in R_*, then for every invertible R-module U, 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 S-algebra has `residue fields' for all maximal ideals m in R_* if the global dimension of R_* is small or if R is 2-periodic with underlying Noetherian complete local regular ring R_0.

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