The Cartan determinant conjecture for representation-finite algebras

Comment

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 $\operatorname{End}_A(M)$ has finite global dimension. We prove that $\operatorname{End}_A(M)$ has Cartan determinant one if $A$ has finite global dimension or $\operatorname{char} k \neq 2$. In particular, the Cartan determinant conjecture holds for representation-finite algebras over an algebraically closed field.