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 circled plus J. The proof of this fact was deep for its time.
In a conceptual breakthrough, Hochschild found a cohomological proof of Wedderburnʹs theorem. This proof makes a reduction to the case where J2 = 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 circled plus 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 image-algebras, image, 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 Shuklaʹs cohomology theory, then obtain a fully generalized cohomological version of Wedderburnʹs theorem, and discuss its role in ring theory.
