The Wedderburn principal theorem and Shukla cohomology

Abstract

The Wedderburn principal theorem states that a finite-dimensional algebra A over a perfect field F is a vector space direct sum of its radical ideal J and a subalgebra S: A = S ⊕ J. The proof of this fact was deep for its time. In a conceptual breakthrough, Hochschild found a cohomological proof of Wedderburn's theorem. This proof makes a reduction to the case where J 2 = 0. The quotient map A → A J has a linear right inverse s. The s( xy) − s( x) s( y) defines a J valued 2-cocycle in Hochschild cohomology theory. Now A J is a separable F algebra, so has vanishing positive-dimensional cohomology groups; whence there exists a map g: A J → J such that s( xy) − s( x) s( y) = s( x) g( y) − g( xy) + g( x) s( y). Hence ψ = s + g is a homomorphism of algebras that is a right inverse of A → A J . Taking S to be the subalgebra ψ( A J ) , A = S ⊕ J is satisfied. If A is instead an algebra over a general commutative ring, a linear right inverse s might not exist: e.g., the natural surjection of Z -algebras, Z p 2 → Z p , where p is prime. However, a set-theoretic right inverse t for A → A J exists by the axiom of choice. Forming both t( xy) − t( x) t( y) and t( x + y) − t( x) − t( y), we show that these give a J valued 2-cocycle in a more refined cohomology theory of algebras due to Shukla (1961). I give an updated account of the nuances of Shukla's cohomology theory, then obtain a fully generalized cohomological version of Wedderburn's theorem, and discuss its role in ring theory.