(Quoted from mathoverflow:) Let $R$ be a local Gorenstein ring of Krull dimension $d$ with an isolated singularity. Define the singularity category $D_{\rm sg}(R)$ as the Verdier quotient $D^{\rm b}(R)/{\rm Perf}(R)$. Then, a famous result of Auslander says that the shift by $[d-1]$ is a Serre functor for $D_{\rm sg}(R)$.
I've tried my best to find a reference of Auslander's original work, while I've only found a paper by Murfet which referred Auslander's result to [Aus78], which has hundreds of pages and I have no idea how it may relate to isolated singularities or Serre functors.
I think I should look for a formula like this: $${\rm Hom}_{{\rm stable} L(\Lambda)}(M, N)\cong {\rm Hom}_{{\rm stable}L(\Lambda)}(N, \Omega^{d-1}M)',$$ but I couldn't find it.