Sweedler’s two-cocycles and generalizations of theorems on Amitsur cohomology

Abstract:

For any (not necessarily commutative) algebra C over a commutative ring k Sweedler defined a cohomology set, denoted here by ${\mathcal {H}^2}(C/k)$, which generalizes Amitsur’s second cohomology group ${H^2}(C/k)$. In this paper, if I is a nilpotent ideal of C and $\bar C \equiv C/I$ is K-projective, a natural bijection ${\mathcal {H}^2}(C/k)\tilde \to {\mathcal {H}^2}(\bar C{\text {/}}k)$ is established. Also, when $k \subset B$ are fields and C is a commutative B-algebra, the sequence $\{ 1\} \to {H^2}(B{\text {/}}k)\xrightarrow {{{l^{\ast }}}}{H^2}(C/k)\xrightarrow {r}{H^2}(C/B)$ is shown to be exact if the natural map $C{ \otimes _k}C \to C{ \otimes _B}C$ induces a surjection on units, ${l^ {\ast } }$ is induced by the inclusion, and r is the “restriction” map.