The third cohomology group of a ring and the commutative cohomology theory

A Cohomology Theory for Commutative Algebras. Ii1

1. Introduction. In the first paper of this series [l], henceforth referred to as I, we define a cohomology theory for a commutative algebra P with coefficients in an P-module M for which H2(R, M) is the group of singular extensions of P by M. In this paper we generalize to a cohomology 22(5, <b, M) where (S, <p) is a singular extension of P and M is an P-module. The second group is again a gro...

A Cohomology Theory for Commutative Monoids

Extending Eilenberg–Mac Lane’s cohomology of abelian groups, a cohomology theory is introduced for commutative monoids. The cohomology groups in this theory agree with the pre-existing ones by Grillet in low dimensions, but they differ beyond dimension two. A natural interpretation is given for the three-cohomology classes in terms of braided monoidal groupoids.

A Cohomology Theory for Commutative Algebras. I1

1. Introduction. D. K. Harrison has recently developed a co-homology theory for commutative algebras over a field [2]. A few key theorems are proved and the results applied to the theory of local rings and eventually to algebraic geometry. The main problem is that both his definitions and proofs require involved calculations. In this paper we define a cohomology theory which (a) relies on more ...

Discrete Morse Theory and the Cohomology Ring

In Morse Theory for Cell Complexes, we presented a discrete Morse theory that can be applied to general cell complexes. In particular, we defined the notion of a discrete Morse function, along with its associated set of critical cells. We also constructed a discrete Morse cocomplex, built from the critical cells and the gradient paths between them, which has the same cohomology as the underlyin...

The Cohomology of a Coxeter Group with Group Ring Coefficients