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Andr'eQuillen cohomology of monoid algebras Klaus Altmann Arne B. Sletsje

Summary: Andr'e­Quillen cohomology of monoid algebras
Klaus Altmann Arne B. Sletsjøe
We compute the Andr'e­Quillen (or Harrison) cohomology of an affine toric variety. The
best results are obtained either in the general case for the first three cohomology groups, or
in the case of isolated singularities for all cohomology groups, respectively.
1 Introduction
(1.1) Let k be a field of characteristic 0. For any finitely generated k­algebra A the so­called
cotangent complex yielding the Andr'e­Quillen cohomology T n
A = T n (A; A; k) (n – 0) may be
defined. The first three of these A­modules are important for the deformation theory of A or its
geometric equivalent SpecA: T 1
A equals the set of infinitesimal deformations, T 0
A describes their
automorphisms, and T 2
A contains the obstructions for lifting infinitesimal deformations to larger
base spaces. Apart from occurring in long exact sequences no meaning of the higher cohomology
groups seems to be known when studying the deformation theory of closed subsets of SpecA. A very
readable reference for the definition of Andr'e­Quillen cohomology and its relations to Hochschild
and Harrison cohomology is Loday's book [Lo]. For applications in deformation theory see for


Source: Altmann, Klaus - Fachbereich Mathematik und Informatik & Institut für Mathematik, Freie Universität Berlin


Collections: Mathematics