The following statement is a consequence of a theorem of Buchweitz and results on idempotent completions of triangulated categories: Let \(X\) be an algebraic variety with isolated Gorenstein singularities \(Z=\text{Sing}(X)=\{x_1,\dots,x_p\}\). Then there is an equivalence of triangulated categories
\[ \left(\frac{D^b(\text{Coh}(X))}{\text{Perf}(X)}\right)^\omega\overset\sim\rightarrow\bigvee_{i=1}^p \underline{\text{MCM}}\left(\hat{\mathcal O}_{X,x_i}\right). \] The left hand side stands for the idempotent completion of the Verdier quotient \(\frac{D^b(\text{Coh}(X))}{\text{Perf}(X)}\), on the right-hand side \(\underline{\text{MCM}}(\hat{\mathcal O}_{X,x_i})\) denotes the stable category of maximal Cohen-Macaulay modules over \(\hat{\mathcal O}_{X,x_i}=:\hat O_i\). The main goal of this article is to generalize this construction as follows. Let \(\mathcal F^\prime\in\text{Coh}(X),\) \(\mathcal F:=\mathcal O\oplus\mathcal F^\prime\) and \(\mathcal A:=\mathcal End_X(\mathcal F)\). Consider the ringed space \(\mathbb X:=(X,\mathcal A)\). It is well known that the functor \(\mathcal F\overset{\mathbb L}\otimes_X-:\text{Perf}(X)\longrightarrow D^b(\text{Coh}(\mathbb X))\) is fully faithful. If \(\text{gl.dim}(\text{Coh}(\mathbb X))<\infty\) then \(\mathbb X\) can be viewed as a non-commutative (or categorical) resolution of singularities of \(X\), and the authors suggest to study the triangulated category \(\Delta_X(\mathbb X):=(\frac{D^b(\text{Coh}(\mathbb X))}{\text{Perf}(X)})^\omega\) called the relative singularity category. Assuming \(\mathcal F\) to be locally free on \(U=X\backslash Z\), an analogue of the “localization equivalence” is proved for the category \(\Delta(\mathbb X)\). In addition, the Grothendieck group \(\Delta(\mathbb X)\) is described.
The main result of the article is a complete description of \(\Delta_U(\mathbb Y)\) in case \(Y\) is an arbitrary curve with nodal singularities and \(\mathcal F^\prime:=\mathcal I_Z\) is the ideal sheaf of the singular locus of \(Y\). It is proved that \(\Delta_Y\) splits into a union of \(p\) blocks: \(\Delta_Y(\mathbb Y)\overset\sim\rightarrow\bigvee_{i=1}^p\Delta_i\), where \(p\) is the number of singular points of \(Y\). Each block is equivalent to the category \(\Delta_{\text{nd}}=\frac{\text{Hot}^b(\text{pro}(A_{nd}))}{\text{Hot}^b(\text{add}(P_\ast))},\) where \(A_{nd}\) is the completed path algebra of the quiver with nodes \(-,\ast,+\) and arrows \(\alpha=(-\ast),\beta=(\ast-),\delta=(\ast+),\gamma=(+\ast)\) and relations \(\delta\alpha=0\), \(\beta\gamma=0\), and \(P_\ast\) is the indecomposable projective \(A_{nd}\)-module corresponding to the vertex \(\ast\). The authors prove that the category \(\Delta_{nd}\) is idempotent complete and \(\text{Hom}\)-finite, and moreover, they give a complete classification of indecomposable objects of \(\Delta_{nd}\). \(\Delta_{nd}\overset\sim\rightarrow(\frac{D^b(\Lambda-\text{mod})}{\text{Band}(\Lambda)})^\omega\) where \(\Lambda\) is the path algebra of the quiver with nodes \(1,2,3\), arrows \(a,c\) from \(1\) to \(2\) and \(b,d\) from \(2\) to \(3\) and relations \(ba=0\), \(dc=0\), and \(\text{Band}(\Lambda)\) is the category of the band objects in \(D^b(\Lambda-\text{mod})\), i.e. the objects which are invariant under the Auslander-Reiten translation in \(D^b(\Lambda-\text{mod})\). Thus the Auslander-Reiten quiver of \(\Delta_{\text{nd}}\) is described.
Let \(\text{P}(X)\) be the essential image of \(\text{Perf}(X)\) under the fully faithful functor \(\mathbb F:=\mathcal F\overset{\mathbb L}\otimes-:\text{Perf}(X)\rightarrow D^b(\text{Coh}(\mathbb X))\). This category is described in the following intrinsic way: \(\text{Ob}(\text{P}(X))=\{\mathcal H^\bullet\in \text{Ob}(D^b(\text{Coh}(\mathbb X)))|\mathcal H^\bullet\in\text{Im}(\text{Hot}^b(\text{add}(\mathcal F_x))\rightarrow D^b(\mathcal A_X-\text{mod}))\}\). In the above notations, the relative singularity category \(\Delta_X(\mathbb X)\) is the idempotent completion of the Verdier quotient \(\text{D}^b(\text{Coh}(\mathbb X))/\text{P}(X)\), and it has a natural structure of triangulated category.
Some of the main results are thus: Let \(D^b_Z(\text{Coh}(\mathbb X))\) be the full subcategory of \(D^b(\text{Coh}(\mathbb X))\) consisting of complexes whose cohomology is supported in \(Z\) and \(\text{P}_Z(X)\cap D^b_Z(\text{Coh}(\mathbb X))\). Then the canonical functor \(\mathbb H:\frac{D^b_Z(\text{Coh}(\mathbb X))}{\text{P}_Z(X)}\rightarrow\frac{D^b(\text{Coh}(\mathbb X))}{\text{P}(X)}\) is fully faithful. With the same notations, the authors prove that the induced functor \(\mathbb H^\omega:(\frac{D^b_Z(\text{Coh}(\mathbb X))}{\text{P}_Z(X)})^\omega\rightarrow(\frac{D^b(\text{Coh}(\mathbb X))}{\text{P}(X)})^\omega \) is an equivalence of triangulated categories.
In addition to much more, the authors answer the questions:
Is the category \(\Delta_Y(\mathbb Y)\) \(\text{Hom}\)-finite? What are the indecomposable objects?
What is the Grothendieck group of \(\Delta_Y(\mathbb Y)\)?
Assume that \(E\) is a plane nodal cubic curve. What is the relation of \(\Delta_E(\mathbb E)\) with the “quiver description” of \(D^b(\text{Coh}(\mathbb E))\)?
To conclude in line with the authors: Let \(Y\) be a nodal algebraic curve, \(Z\) its singular locus, \(\mathcal I=\mathcal I_Z\) and \(\mathbb Y=(Y,\mathcal A)\) for \(\mathcal A=\mathcal End_Y(\mathcal O\oplus I)\). Similarly, let \(O=k[[u,v]]/(uv)\), \(\mathfrak m=(u,v)\) and \(A=\text{End}_O(O\oplus\mathfrak m)\). Then the following results are true:
The category \(\Delta_Y(\mathbb Y)\) splits into a union of \(p\) blocks \(\Delta_{nd}\), where \(p\) is the number of singular points of \(Y\) and \(\Delta_{nd}=\Delta_O(A)\).
The category \(\Delta_{nd}\) is \(\text{Hom}\)-finite and representation discrete. In particular its indecomposable objects and the morphism spaces between them are explicitly known. Moreover, one can compute its Auslander-Reiten quiver.
It is proved that \(K_0(\Delta_{nd})\cong\mathbb Z^2\).
The category \(\Delta_{nd}\) admits an alternative ”quiver description” in terms of representations of a certain gentle algebra \(\Lambda\).
A nontrivial article, not very self contained, but the ideas are easy to follow and very interesting.