Related papers: Homotopy finiteness of some DG categories from alg…

Let $V$ be a finite-dimensional symplectic vector space over a field of characteristic 0, and let $G \subset Sp(V)$ be a finite subgroup. We prove that for any crepant resolution $X \to V/G$, the bounded derived category $D^b(Coh(X))$ of…

On one hand, together with Pelle Steffens, we recently characterized the infinity category of derived manifolds up to equivalence by a universal property. On the other hand, it is shown in recent work of Behrend-Liao-Xu that the category of…

We prove in this paper that for a quasi-compact and semi-separated (non necessarily noetherian) scheme X, the derived category of quasi-coherent sheaves over X, D(A_qc(X)), is a stable homotopy category in the sense of Hovey, Palmieri and…

In this paper we prove first a general theorem on semiorthogonal decompositions in derived categories of coherent sheaves for flat families over a smooth base. Based on the results of math.AG/0510670, we then show that the derived…

Let $X_N$ be the second infinitesimal neighborhood of a closed point in $N$-dimensional affine space. In this note we study $D^b(coh\, X_N)$, the bounded derived category of coherent sheaves on $X_N$. We show that for $N\geq 2$ the lattice…

We prove that the dg category of perfect complexes on a smooth, proper Deligne-Mumford stack over a field of characteristic zero is geometric in the sense of Orlov, and in particular smooth and proper. On the level of triangulated…

We give a generalization of Gabriel's Theorem on coherent sheaves to the case of coherent twisted sheaves on a smooth variety X over a field k. We show that the category Coh(X,\alpha) determines the scheme structure of X for \alpha in the…

This elementary survey article was prepared for a talk at the 2016 Superschool on Derived Categories and D-branes. The goal is to outline an identification of the bounded derived category of coherent sheaves on a Calabi-Yau threefold with…

Let X be a quasi-compact scheme, equipped with an open covering by affine schemes. A quasi-coherent sheaf on X gives rise, by taking sections over the covering sets, to a diagram of modules over the various coordinate rings. The resulting…

Building on the concept of a smooth DG algebra we define the notion of a smooth derived category. We the propose the definition of a categorical resolution of singularities. Our main example is the derived category $D(X)$ of quasi-coherent…

A famous theorem of D. Orlov describes the derived bounded category of coherent sheaves on projective hypersurfaces in terms of an algebraic construction called graded matrix factorizations. In this article, I implement a proposal of E.…

We introduce new enhancements for the bounded derived category $D^b(Coh(X))$ of coherent sheaves on a suitable scheme $X$ and for its subcategory $Perf(X)$ of perfect complexes. They are used for translating Fourier-Mukai functors to…

Given an algebraic stack $X$, one may compare the derived category of quasi-coherent sheaves on $X$ with the category of dg-modules over the dg-ring of functions on $X$. We study the analogous question in stable homotopy theory, for derived…

Let $\mathbf{X}$ be an Adams geometric stack. We show that $D(A_{qc}(\mathbf{X}))$, its derived category of quasi-coherent sheaves, satisfies the axioms of a stable homotopy category defined by Hovey, Palmieri and Strickland. Moreover we…

In this paper we study the homotopy limits of cosimplicial diagrams of dg-categories. We first give an explicit construction of the totalization of such a diagram and then show that the totalization agrees with the homotopy limit in the…

We prove that the bounded derived category of coherent sheaves with proper support is equivalent to the category of locally-finite, cohomological functors on the perfect derived category of a quasi-projective scheme over a field. We…

In this paper we study the derived categories of coherent sheaves on Grassmannians $\operatorname{Gr}(k,n),$ defined over the ring of integers. We prove that the category $D^b(\operatorname{Gr}(k,n))$ has a semi-orthogonal decomposition,…

The main purpose of this work is the study of the homotopy theory of dg-categories up to quasi-equivalences. Our main result provides a natural description of the mapping spaces between two dg-categories $C$ and $D$ in terms of the nerve of…

In this paper we introduce and study the formal punctured neighborhood of infinity, both in the algebro-geometric and in the DG categorical frameworks. For a smooth algebraic variety $X$ over a field of characteristic zero, one can take its…

We prove that under a certain mild hypothesis, the DG category of D-modules on a quasi-compact algebraic stack is compactly generated. We also show that under the same hypothesis, the functor of global sections on the DG category of…