Derivation of Exact Triples and Leray-Koszul Spectral Sequences

Abstract

We introduce the new concept of ‘exact triple’ and show how is attached, a Leray-Koszul spectral sequence generalizing those of Massey exact couples. We generalize also a result of Browder-Eckmann-Hilton, computing the limit of Bockstein spectral sequences. The letter A means a unitary associative algebra over a commutative ring K. Modules, without any supplementary precisions, are left A-modules and morphisms are modules morphisms. Differential modules and differential morphisms deal with the algebra A[d]/(d ² ). For us, a Leray-Koszul spectral sequence is a sequence of differential modules(E ⁿ, d ⁿ _n ) such that E ⁿ⁺¹ = H(E ⁿ, d ⁿ ) (homology), n ≥ 0. A limit module, usually denoted byE ^∞, is defined; if the spectral sequence is stationary (∃n, ∀N ≥ n,d ^N = 0), then: E ⁿ = E ⁿ⁺¹ = ... = E ^∞.