Algebraic Cohomology and Deformation Theory

Abstract

We should state at the outset that the present article, intended primarily as a survey, contains many results which are new and have not appeared elsewhere. Foremost among these is the reduction, in §28, of the formal deformation theory of a smooth compact complex algebraic variety χ to that of a single ring built from χ. Others include the relationship between the classical Hodge decomposition of the cohomology of an analytic manifold and the more recent Hodge decomposition of the cohomology of a commutative algebra, the invariance of the Euler characteristic of an algebra under deformation, the correspondence between the deformation theories for Morita equivalent algebras, much of the work on the deformation of presheaves (diagrams) of algebras, and the explicit description of the (algebraic) Hodge decomposition for regular affine algebras. However, in line with the goals of a survey article, we have tried to maximize the exposition, including details only in so far as they aid in this purpose. Many proofs are sketched; many others, including the most difficult, are omitted altogether.