Let $A$ be a finite dimensional algebra with simple modules $S_i$ and projective indecomposable modules $P_l$ and global dimension $g< \infty$ (and $n$ is the number of simple modules).

I noted that the number $\sum\limits_{l=1}^{n}{\sum\limits_{i=0}^{g}{(-1)^i Ext_A^i(S_l,P_l)}}$ is a derived invariant of the algebra.

In case I made no mistake, the proof goes by showing that this is the negative trace of the transpose of the inverse of the Coxeter transformation of the algbra, which is a derived invariant. One can also express the other coefficients of the Coxeter polynomial like this. Did this appear somewhere before?

Question: Is this still true when $A$ is just assumed to be Gorenstein? (that is the injective dimension of $A$ is finite on both sides).

Restricted to selfinjective algebras this would be implies by a positive answer to the question whether the Nakayama permutation is a derived invariant, which I forgot whether it is true.