4
$\begingroup$

Let $R$ be a local Gorenstein ring of Krull dimension $d$ with an isolated singularity. Defined $D_{sing}(R)$ as the Verdier quotient $D^b(R)/Perf(R)$). Then, a famous result of Auslander says that the shift by $[d-1]$ is a Serre functor for $D_{sing}(R)$.

Let $A$ be a $\mathbb{N}$-graded Gorenstein algebra satisfying some "nice finiteness" conditions and let $$D_{sing}^{gr}(A) = D^b(gr-A)/Perf(gr-A),$$ the graded derived category of singularities of $A$. I was wondering if there is a general description of the Serre functor of $D_{sing}^{gr}(A)$? Or Perhaps under some extra-hypotheses on A?

I know that various results of Orlov and others show that $D_{sing}^{gr}(A)$ can be sometimes seen as a semi-orthogonal component of $D^b(Coh(X))$, where $X = Proj(A)$. But unless we are in the very special case where $D_{sing}^{gr}(A) = D^b(Coh(X))$, I haven't been able to find a description of the Serre functor of $D_{gr}^{sing}(A)$.

Thanks in advance!

Improve this question
$\endgroup$
5
  • $\begingroup$ It's almost the same (assuming said nice finiteness conditions) but you also need to shift the grading by the Gorenstein parameter (or its inverse, the sign isn't always the same in the literature and I find it hard to keep track in any case), which is, up to said sign, the grading shift that shows up on the copy of $E(k)$, the injective envelope of the residue field, in the last term of the minimal injective resolution of $A$. $\endgroup$
    – Greg Stevenson
    Commented Dec 3, 2017 at 9:35
  • $\begingroup$ @GregStevenson Thanks for the comment. Do you have any reference? $\endgroup$
    – Libli
    Commented Dec 3, 2017 at 10:44
  • $\begingroup$ Not off the top of my head. I don't recall having seen it written down explicitly for what that's worth (which is not much). The proof is more or less the same as in the ungraded case though and you can guess the answer from the graded analogue of local duality. $\endgroup$
    – Greg Stevenson
    Commented Dec 3, 2017 at 12:11
  • $\begingroup$ @GregStevenson : I don't believe that. I have the feeling it is infinitely more difficult to computer the Serre functor of the singularity category in the graded case than in the local case. $\endgroup$
    – Libli
    Commented Dec 3, 2017 at 15:17
  • $\begingroup$ @Libli On M.SE someone asked for a reference of the Auslander "famous" result; see here. Could you help him? $\endgroup$
    – user26857
    Commented Jun 8, 2023 at 4:59

0

Reset to default

You must log in to answer this question.