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.



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> **Question.** Who proved first that the Cartan determinant is an invariant of the derived category of the algebra? Is there a reference?
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Derived invariance of the Cartan determinant
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**](http://dx.doi.org/10.1007/BF02564452), Comment. Math. Helv. 62, 339-389 (1987). [ZBL0626.16008](https://zbmath.org/?q=an:0626.16008).


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.**](http://dx.doi.org/10.1515/crll.2005.2005.580.157), J. Reine Angew. Math. 580, 157-199 (2005). [ZBL1099.16002](https://zbmath.org/?q=an:1099.16002),


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.


