Related papers: Cartesian modules in small categories

Let $(\mathcal C,\otimes,1)$ be an abelian symmetric monoidal category satisfying certain conditions and let $X$ be a scheme over $(\mathcal C,\otimes,1)$ in the sense of To\"en and Vaqui\'{e}. In this paper we show that when $X$ is…

Motivated by applications to the categorical and geometric local Langlands correspondences, we establish an equivalence between the category of filtered $\mathcal{D}$-modules on a smooth stack $X$ and the category of $S^1$-equivariant…

We study spaces of conformal blocks associated with line bundles over elliptic curves, with coefficients in a vertex algebra. For vertex algebras satisfying suitable finiteness and semisimplicity conditions, which are met by all admissible…

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…

We give an alternate formulation of pseudo-coherence over an arbitrary derived stack X. The full subcategory of pseudo-coherent objects forms a stable sub-infinity-category of the derived category associated to X. Using relative…

For a semi-separated noetherian scheme, we show that the category of cotorsion Gorenstein flat quasi-coherent sheaves is Frobenius and a natural non-affine analogue of the category of Gorenstein projective modules over a noetherian ring. We…

In arXiv:1807.09038 we formulated a conjecture describing the derived category D-mod(Gr$_{GL(n)}$) of (all) D-modules on the affine Grassmannian of the group $GL(n)$ as the category of quasi-coherent sheaves on a certain stack (it is…

Gerstenhaber and Schack ([GS]) developed a deformation theory of presheaves of algebras on small categories. We translate their cohomological description to sheaf cohomology. More precisely, we describe the deformation space of (admissible)…

The first goal of this paper is to provide an abstract framework in which to formulate and study local duality in various algebraic and topological contexts. For any stable $\infty$-category $\mathcal{C}$ together with a collection of…

We characterize all fields of definition for a given coherent sheaf over a projective scheme in terms of projective modules over a finite-dimensional endomorphism algebra. This yields general results on the essential dimension of such…

The main goal of this paper is to establish close relations among sheaves of modules on atomic sites, representations of categories, and discrete representations of topological groups. We characterize sheaves of modules on atomic sites as…

Let X be a quasi-compact and quasi-separated (not necessarily semiseparated) scheme. The category QcoX of all quasi-coherent sheaves of OX-modules has several diferent pure derived categories. Recently, categorical pure derived categories…

We construct a quasi-coherent sheaf of associative algebras which controls a category of $AV$-modules over a smooth quasi-projective variety. We establish a local structure theorem, proving that in \'etale charts these associative algebras…

We present a general construction of model category structures on the category $\mathbb{C}(\mathfrak{Qco}(X))$ of unbounded chain complexes of quasi-coherent sheaves on a semi-separated scheme $X$. The construction is based on making…

We leverage the results of the prequel in combination with a theorem of D. Orlov to yield some results in Hodge theory of derived categories of factorizations and derived categories of coherent sheaves on varieties. In particular, we…

We introduce graded $\mathbb{E}_{\infty}$-rings and graded modules over them, and study their properties. We construct projective schemes associated to connective $\mathbb{N}$-graded $\mathbb{E}_{\infty}$-rings in spectral algebraic…

We give a systematic construction of semiorthogonal decompositions of derived categories of coherent sheaves on quasi-smooth derived algebraic stacks over $\mathbb{C}$, where the summands are subcategories defined by weight conditions, and…

We introduce a notion of quasi-lisse vertex algebras, which generalizes admissible affine vertex algebras. We show that the normalized character of an ordinary module over a quasi-lisse vertex operator algebra has a modular invariance…

We define the class of rigid Frobenius algebras in a (non-semisimple) modular category and prove that their categories of local modules are, again, modular. This generalizes previous work of A. Kirillov, Jr. and V. Ostrik [Adv. Math. 171…

We show that every flat quasi-coherent sheaf on a quasi-compact quasi-separated scheme is a directed colimit of locally countably presentable flat quasi-coherent sheaves. More generally, the same assertion holds for any countably…