The Cartan matrix $C$ of a finite quiver algebra $A$ with points $e_i$ is defined as the matrix having entries $c_{i,j}=\dim(e_i A e_j)$. The Cartan determinant is defined as the determinant of the Cartan matrix.

Question. Who proved first that the Cartan determinant is an invariant of the derived category of the algebra? Is there a reference?


I suspect that this is one of those things that was essentially well-known to many people before anybody wrote it down, so it will be hard to pin down the first person to prove it.

But the main idea goes back to

Happel, Dieter, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62, 339-389 (1987). ZBL0626.16008.

There Happel considered the bilinear form on $K_0\left(D^b(\text{mod }A)\right)$, for an algebra $A$ of finite global dimension, given by $\left\langle[X],[Y]\right\rangle=\sum_i(-1)^i\dim\text{Hom}(X,Y[i])$. To get the statement about the Cartan determinant just requires the idea of doing the same for $K_0\left(K^b(P_A)\right)$ for an arbitrary finite dimensional algebra, where $K^b(P_A)$ is the triangulated subcategory of $D^b(\text{mod }A)$ consisting of bounded complexes of finitely generated projective modules.

The derived invariance of the Cartan determinant is explicitly stated as Proposition 1.5 in

Bocian, Rafał; Skowroński, Andrzej, Weakly symmetric algebras of Euclidean type., J. Reine Angew. Math. 580, 157-199 (2005). ZBL1099.16002,

where they credit me with suggesting the proof. But I don't remember doing so, and I suspect it may have been in a conversation at some conference many, many years earlier. I also suspect that I was not the only person at the conference who would have suggested exactly the same proof.


Check out these references:

$q$-Cartan matrices and combinatorial invariants of derived categories for skewed-gentle algebras, Christine Bessenrodt (Hannover), Thorsten Holm

Algebra invariants for finite directed graphs with relations, Christine Bessenrodt


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