Derived representation schemes with arbitrary coefficients and associative smoothness

Abstract

We study associative smoothness through representation homology with coefficients in arbitrary finite-dimensional algebras. We prove that the higher representation homology of a finitely generated formally smooth algebra with arbitrary finite-dimensional coefficients vanishes. We further show that allowing arbitrary coefficients yields a strictly stronger smoothness test: suitable non-matrix coefficients detect nonsmoothness in situations where matrix coefficients do not. These results raise the question of whether representation homology with arbitrary coefficients furnishes a homotopical characterization of associative smoothness in the spirit of the derived Kontsevich-Rosenberg principle. For finite-dimensional algebras, we prove that representation homology with arbitrary coefficients completely characterizes formal smoothness, providing supporting evidence for this philosophy.

14 pages