The Cartan determinant conjecture for representation-finite algebras

Abstract

Let $A$ be a finite-dimensional representation-finite algebra over an algebraically closed field $k$, and let $M$ be a finite-dimensional $A$-module such that ${\mathrm{End}}_A(M)$ has finite global dimension. We prove that ${\mathrm{End}}_A(M)$ has Cartan determinant one if $A$ has finite global dimension or $\mathrm{char}k\ne 2$; in particular, the Cartan determinant conjecture holds for representation-finite algebras over an algebraically closed field.

5 pages, comments welcome!