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

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.Comment: Revised version. One serious flaw correcte

On the Computation of Ainfinity-Maps

Starting from a chain contraction (a special chain homotopy equivalence) connecting a differential graded algebra A with a differential graded module M, the so-called homological perturbation technique “tensor trick” [8] provides a family of maps, {mi}i≥1, describing an A∞- algebra structure on M derived from the one of algebra on A. In this paper, taking advantage of some annihilation properties of the component morphisms of the chain contraction, we obtain a simplified version of the existing formulas of the mentioned A∞-maps, reducing the computational cost of computing mn from O(n!2) to O(n!)