Moduli, Deformations and Algebraic K-Theory of Graded PI-Algebras

Abstract

Over a field of characteristic zero, for a finite grading group and a finitely generated graded $T$-ideal, we construct relatively free sheaves and finite-rank quotient stacks of algebra laws with prescribed identities. Their tangent complexes are PI-restricted Hochschild complexes, and the second variations of the identities give explicit lifting obstructions. The first-order deformation groupoid carries a natural relative algebraic K-theory functor; for its square-zero extensions, the relative Chern character identifies rational relative K-theory with negative cyclic homology. Additional identities determine closed PI-strata and localisation fibre sequences in K-theory with Azumaya coefficients. On the Azumaya substack, the degree annihilates the Brauer class, and Morita transport yields a base-change-compatible K-theory of PI-Azumaya families together with a K-theoretic monodromy equivalence.

31 pages . New title. New results have been included. Comments are welcome