We define abelian extensions of algebras in congruence-modular varieties. The theory is sufficiently general that it includes, in a natural way, extensions of R-modules for a ring R. We also define a cohomology theory, which we call clone cohomology, such that the cohomology group in dimension one is the group of equivalence classes of extensions.

We prove a vanishing theorem for the p-adic cohomology of exponential sums on An. In particular, we obtain new classes of exponential sums on An that have a single nonvanishing p-adic cohomology group. The dimension of this cohomology group equals a sum of Milnor numbers.

We focus on the rational cohomology of Cornalba’s moduli space of spin curves of genus 1 with n marked points. In particular, we show that both its first and its third cohomology group vanish and the second cohomology group is generated by boundary classes.

A coring approach to non-Abelian descent cohomology of [P. Nuss & M. Wambst, Non-Abelian Hopf cohomology, Preprint arXiv:math.KT/0511712, (2005)] is described and a definition of a Galois cohomology for partial group actions is proposed.