Related papers: Relative Singularity Categories

In this paper, we consider the singularity category $D_{sg}(\mod A)$ and the $\mathbb{Z}$-graded singularity category $D_{sg}(\mod^{\mathbb Z} A)$ for a Gorenstein monomial algebra $A$. Firstly, for a positively graded $1$-Gorenstein…

In this paper we generalize the definitions of singularities of pairs and multiplier ideal sheaves to pairs on arbitrary normal varieties, without any assumption on the variety being Q-Gorenstein or the pair being log Q-Gorenstein. The main…

Using the new approach to analytic geometry developed by Clausen and Scholze by means of condensed mathematics, we prove that for every affinoid analytic adic space $X$, pseudocoherent complexes, perfect complexes, and finite projective…

We study (not necessarily connected) Z-graded A-infinity-algebras and their A-infinity-modules. Using the cobar and the bar construction and Quillen's homotopical algebra, we describe the localisation of the category of A-infinity-algebras…

Let X be an algebraic variety with an action of an algebraic group G. Suppose X has a full exceptional collection of sheaves, and these sheaves are invariant under the action of the group. We construct a semiorthogonal decomposition of…

Let G be a reductive groups over an algebraically closed field k. Let P^{(i)} be associated parabolic subgroups, and X^{(i)}:=T^*G/P^i. The bounded derived categories of coherent sheaves on X^{(i)} are equivalent, but there is no canonical…

We consider the quotient of an exact or one-sided exact category $\mathcal{E}$ by a so-called percolating subcategory $\mathcal{A}$. For exact categories, such a quotient is constructed in two steps. Firstly, one localizes $\mathcal{E}$ at…

We give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga-Gorenstein ring. We then apply this result to the Frobenius category of special Cohen-Macaulay modules…

We introduce and investigate (dual) relative split objects with respect to a fully invariant short exact sequence in abelian categories. We compare them with (dual) relative Rickart objects, and we study their behaviour with respect to…

We apply the machinery of relative tensor triangular Chow groups to the action of the derived category of quasi-coherent sheaves on a noetherian scheme $X$ on the derived category of quasi-coherent $\mathcal{A}$-modules, where $\mathcal{A}$…

We compare closed and rigid monoidal categories. Closedness is defined by the tensor product having a right adjoint: the internal hom functor. Rigidity, on the other hand, generalises the duality of finite-dimensional vector spaces. In the…

We show the existence of a full exceptional collection in the graded stable derived category of a Gorenstein isolated quotient singularity using a result of Orlov (arXiv:math/0503632). We also show that the equivariant graded stable derived…

We complete the discrete cluster categories of type $\mathbb{A}$ as defined by Igusa and Todorov, by embedding such a discrete cluster category inside a larger one, and then taking a certain Verdier quotient. The resulting category is a…

Let $A$ be a Nakayama algebra. We give a description of the singularity category of $A$ inside its stable module category. We prove that there is a duality between the singularity category of $A$ and the singularity category of its opposite…

For a commutative Gorenstein Noetherian ring $R$, we construct an affine scheme $X$ solely from DG singularity category $S_{dg}(R)$ of $R$ such that there is a finite surjective morphism $X \rightarrow \mathrm{Spec}(R /I)$, where…

The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier's…

We introduce smooth L^\infty differential forms on a singular (semialgebraic) set X in R^n. Roughly speaking, a smooth L^\infty differential form is a certain class of equivalence of 'stratified forms', that is, a collection of smooth forms…

This paper studies several singularity categories of a locally bounded $k-$linear category $\mathscr{C}$ with radical square zero. Following the work of Bautista and Liu [6], we give a complete description of $D^{b}_{sg}(\mathscr{C})$,…

In this article we construct various models for singularity categories of modules over differential graded rings. The main technique is the connection between abelian model structures, cotorsion pairs and deconstructible classes, and our…

The singularity space consists of all germs $(X,x)$, with $X$ a Noetherian scheme and $x$ a point, where we identify two such germs if they become the same after an analytic extension of scalars. This is a Polish space for the metric given…