Consider a finite-dimensional $k$-algebra $A$ of finite global dimension. Then it is known that the Serre functor on $D^b(mod-A)$ exists and is given by the Nakayama functor. The proof goes something like this:

The $k$-duality $(-)^*=R\underline{\text{Hom}}_k(-,k)$ gives an equivalence $D^b(mod-A)\to D^b(A-mod).$ The $A$-duality $(-)^\vee=R\underline{\text{Hom}}_A(-,A)$ maps the category of perfect complexes $Perf(A)$ to $Perf(A^{op})$, because $A^\vee$ is isomorphic to $A$ in $A-mod$. Then one proves that for $M\in Perf(A)$, $N\in D^b(mod-A)$ we have

$$Hom_{D(Mod-A)} (M,N)^*\cong Hom_{D(Mod-A)} (N,M^{\vee*}).$$

Finally, we notice that for finite-dimensional algebras of finite global dimension $Perf(A)=D^b(A-mod).$

But what if $\text{r.gl.dim}(A)=\infty$? The only reason I see why the above proof might not give us a Serre functor on $Perf(A)$ is that $(-)^*$ may not map $Perf(A)$ to $Perf(A^{op})$, since $A$ is not necessarily isomorphic to $A^*$ as a left $A$-module. In this case, is it enough to have $A\cong A^{op}$?


1 Answer 1


The assumption on the finite global dimension is just needed to have $Perf(A)=D^b(A-mod),$ which does not hold more generality.

It is indeed true that $Perf(A)$ has a Serre functor (given in the same way) when $A$ just is a Gorenstein algebra, that is the regular module $A$ has finite injective dimension as a left and right $A$-modules.

For example any selfinjective non-semisimple algebra (such as $K[x]/(x^n)$ for $n \geq 2$) is a Gorenstein algebra of infinite global dimension.

Good references are: "Representation Theory: A Homological Algebra Point of View" by Zimmermann and the book by Happel on Triangulated Categories.

  • $\begingroup$ Thanks! Do you have a reference for the statement about the case of Gorenstein algebras? $\endgroup$
    – IDC
    Dec 21, 2020 at 20:26

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