# Testing the Cartan determinant conjecture via Gorenstein algebras

Let $A$ be a Gorenstein algebra (of infinite global dimension) with finitely many indecomposable Gorenstein projective modules and $X$ the basic direct sum of all indecomposable Gorenstein projective modules. Let $C_A$ be the Gorenstein-Cartan matrix of $A$, that is defined as the matrix with entries $Hom_A(G_i,G_j)$ for two indecomposable Gorenstein projective modules $G_i , G_J$. Let $B:=End_A(X)$, then it is known that $B$ has finite global dimension. There is an equivalence between $add(X)$ and $proj-B$ and thus the Gorenstein-Cartan matrix should be up to permutation the same as the cartan matrix of $B$. The Cartan determinant conjecture states that the determinant of the Cartan matrix of an algebra of finite global dimension is always 1. Is this conjecture verified for algebra of the form $B$ as above? If not, is there a proof at least when $A$ is a Nakayama algebra? Is there in general some survey article on the status of this conjecture (which seems to be independent of all the other homological conjectures)?