The Cohomology of Algebras

Abstract

We have reached another stage of machinery building in our development of the theory of associative algebras. This time the formalism of the cohomology of algebras is introduced. The reader is warned that the ratio of definitions to theorems in the first five sections of the chapter is very high. However, the cohomology of associative algebras plays an important part in the study of central simple algebras, as we will see in Chapter 14. In this chapter the cohomology theory is used to give a streamlined proof of the Wedderbum—Malcev Principal Theorem, one of the landmarks in the theory of associative algebras. The chapter ends with a discussion of the Principal Theorem in the general theory of associative algebras. The results on extensions enables us to formulate the work of Chapter 8 in a more natural way.KeywordsExact SequenceAssociative AlgebraCohomology TheoryFollow Diagram CommuteHochschild CohomologyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.