Pseudotraces on Almost Unital and Finite-Dimensional Algebras

Abstract

We introduce the notion of almost unital and finite-dimensional (AUF) algebras, which are associative $\mathbb{C}$-algebras that may be non-unital or infinite-dimensional, but have sufficiently many idempotents. We show that the pseudotrace construction, originally introduced by Hattori and Stallings for unital finite-dimensional algebras, can be generalized to AUF algebras. Let $A$ be an AUF algebra. Suppose that $G$ is a projective generator in the category ${\mathrm{Coh}}_{\mathrm{L}}(A)$ of finitely generated left $A$-modules that are quotients of free left $A$-modules, and let $B={\mathrm{End}}_{A,-}(G{)}^{\mathrm{opp}}$. We prove that the pseudotrace construction yields an isomorphism between the spaces of symmetric linear functionals $\mathrm{SLF}(A)\stackrel{\simeq }{\to }\mathrm{SLF}(B)$, and that the non-degeneracies on the two sides are equivalent.

32 pages. Final version. To appear in J. Algebra