The \(\mathsf {D}\) -Cohomology of Monoids

In this chapter we present the \(\mathsf {D}\) -cohomology of a monoid, also referred to as Leech cohomology in some of the literature. Section 2.1 constructs a functor \(\mathsf {D}\) that assigns to each monoid S a small category \(\mathsf {D}(S)\) , which represents the division structure of S. The \(\mathsf {D}\) -cohomology of S uses \(\mathsf {D}(S)\) -modules as its source of coefficients to calculate the cohomology groups of S. This is introduced in Sect. 2.2 where we construct a free resolution of the constant \(\mathsf {D}(S)\) -module \(\mathbb {Z}\) . This in turn is used to create cochain complexes from which the cohomology groups are computed. Various aspects of this cohomology are discussed, with the section ending with the long exact cohomology sequence induced by a surjective homomorphism of monoids. In the third section the focus is on \(H^1(S, \mathcal {A})\) , which is the result of factoring out a group of inner derivations from the full group of derivations, much as in the familiar case of group cohomology. In the fourth section, we show that the second cohomology groups of S classify certain coextensions of S. Of particular interest are, of course, \(\mathcal {H}\) -coextensions. \(\mathsf {D}(S)\) -modules and their 2-cocycles (i.e., factor systems) are used to create such structures. Section 2.5 is dedicated to the cohomological \(\mathsf {D}\) -dimension of monoids, and the chapter ends in Sect. 2.6 with the calculation of the cohomology groups of cyclic monoids.