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Hochschild Cohomology

Sarah Witherspoon
2020 Notices of the American Mathematical Society  
Introduction At the end of the nineteenth century, Poincaré created invariants to distinguish different topological spaces and their features, foreshadowing the homology and cohomology groups that would appear later. Towards the middle of the twentieth century, these notions were imported from topology to algebra: The subject of group homology and cohomology was founded by Eilenberg and Mac Lane, and the subject of Hochschild homology and cohomology by Hochschild. The uses of homological
more » ... homological techniques continued to grow and spread, spilling out of algebra and topology into many other fields. In this article we will focus on Hochschild cohomology, which now appears in the settings of algebraic geometry, category theory, functional analysis, topology, and beyond. There are strong connections to cyclic homology and K-theory. Many mathematicians use Hochschild cohomology in their research, and many continue to develop theoretical and computational techniques for better understanding. Hochschild cohomology is a broad and growing field, with connections to diverse parts of mathematics. Our scope here is exclusively Hochschild cohomology for algebras, including Hochschild's original design [3] and just a few of the many important uses and recent developments in algebra. Some details and further references may be found in [5, 12, 13 ]. We will begin this story by setting the scene: We are interested here in a ring that is also a vector space over a field such as ℝ or ℂ. We require the multiplication map on to be bilinear; that is, the map × → given by ( , ) ↦ for , in is bilinear (over ). A ring with this additional structure is called an algebra over . Some examples are polynomial rings = [ 1 , ... , ], which are commutative, and matrix rings = ( ), which are noncommutative when > 1. Our focus will be on multilinear maps × ⋯ × → and what they can tell us about the structure and the representations, that is, modules, of . Our story has two threads: One starts with particular functions on called derivations. The other starts with the center ( ) of , that is, the subalgebra of all elements commuting with every 780
doi:10.1090/noti2099 fatcat:cxdm5jiy4zeyjerhp7kkyjbvz4