Higher Extension Modules and the Yoneda Product

Abstract. A chain of c submodules E =: E0 ≥ E1 ≥ · · · ≥ Ec ≥ Ec+1 := 0 gives rise to c composable 1-cocycles in Ext1(Ei−1/Ei, Ei/Ei+1), i = 1, . . . , c. In this paper we follow the converse question: When are c composable 1-cocycles induced by a module E together with a chain of submodules as above? We call such modules c-extension modules. The case c = 1 is the classical correspondence between 1-extensions and 1cocycles. For c = 2 we prove an existence theorem stating that a 2-extension module exists for two composable 1-cocycles η L ∈ Ext(M,L) and η N ∈ Ext(L,N), if and only if their Yoneda product η L ◦ η N ∈ Ext(M,N) vanishes. We further prove a modelling theorem for c = 2: In case the set of all such 2-extension modules is non-empty it is an affine space modelled over the abelian group that we call the first extension group of 1-cocycles, Ext(η L , η N ) := Ext(M,N)/(Ext(M,L) ◦ η N + η L ◦ Ext(L,N)).