We prove that the tensor category of quasi-coherent modules $\mathsf{Qcoh}(X \times_S Y)$ on a fiber product of quasi-compact quasi-separated schemes is the bicategorical pushout of $\mathsf{Qcoh}(X)$ and $\mathsf{Qcoh}(Y)$ over $\mathsf{Qcoh}(S)$ in the $2$-category of cocomplete linear tensor categories. In particular, $\mathsf{Qcoh}(X \times Y)$ is the bicategorical coproduct of $\mathsf{Qcoh}(X)$ and $\mathsf{Qcoh}(Y)$. For this we introduce idals, which can be seen as non-embedded ideals, and use them to study localizations of cocomplete tensor categories in general.
The principal aim of this note is to give an elementary proof of the fact that any two fiber functors of a Tannakian category are locally isomorphic. This builds on an idea of Deligne concerning scalar extensions of Tannakian categories and implements a proof strategy which Deligne attributes to Grothendieck. Besides categorical generalities, the proof merely relies on basic properties of exterior powers and the classification of finitely generated modules over a principal ideal domain. Using related ideas (but less elementary means) we also present an alternative characterization of Tannakian…
For a large class of geometric objects, the passage to categories of quasi-coherent sheaves provides an embedding in the 2-category of abelian tensor categories. The notion of weakly Tannakian categories introduced by the author gives a characterization of tensor categories in the image of this embedding. However, this notion requires additional structure to be present, namely a fiber functor. For the case of classical Tannakian categories in characteristic zero, Deligne has found intrinsic properties - expressible entirely within the language of tensor categories - which are necessary and suf…
We study fiber functors on Tannakian categories which are equipped with a grading or a filtration. Our goal is to give a comprehensive set of foundational results about such functors. A main result is that each filtration on a fiber functor can be split by a grading fpqc-locally on the base scheme.
We prove that the localizations of the categories of dg categories, of cohomologically unital and strictly unital $A_\infty$ categories with respect to the corresponding classes of quasi-equivalences are all equivalent. Moreover we show that the last two localizations are equivalent to the corresponding quotients by the relation of being isomorphic in the cohomology of the $A_\infty$ category of $A_\infty$ functors. As an application we give a complete proof of a claim by Kontsevich stating that the category of internal Homs for two dg categories can be described as the category of strictly un…
We extend some classical results of Bousfield on homology localizations and nilpotent completions to a presentably symmetric monoidal stable $\infty$-category $\mathscr{M}$ admitting a multiplicative left-complete $t$-structure. If $E$ is a homotopy commutative algebra in $\mathscr{M}$ we show that $E$-nilpotent completion, $E$-localization, and a suitable formal completion agree on bounded below objects when $E$ satisfies some reasonable conditions.
The present paper is devoted to study the homotopy category associated with a simplicial descent category (D,s,E) (arXiv:0808.3684v2). We prove that the class E of equivalences has a calculus of left fractions over a quotient category of D modulo homotopy. We study the fiber/cofiber sequences induced by a (co)simplicial descent structure. Examples of such fiber/cofiber sequences are deduced for (commutative) differential graded algebras, simplicial sets or topological spaces. We prove that the homotopy category of a stable simplicial descent category is triangulated. In addition, these triangu…
Using the localization property, we construct a triangulated category of motives over quasi-projective T-schemes for any coefficient where T is a noetherian separated scheme, and we prove the Grothendieck six operations formalism. We also construct integral étale realization of motives.
This paper is a sequel to math.AG/0203287. A generalization of the Mukai flop has been studied by E. Markman. Here we call it a stratified Mukai flop. In this paper, we observe that, for a stratified Mukai flop: $X \to \bar{X} \leftarrow X^+$, the fiber product $X \times_{\bar{X}}X^+$ defines an isomorphism of Grothendieck groups $K(X) \to K(X^+)$, but, at the level of derived categories, the fiber product does not generally give an equivalence.
We introduce a new perspective on the $K$-theory of exact categories via the notion of a CGW-category. CGW-categories are a generalization of exact categories that admit a Qullen $Q$-construction, but which also include examples such as finite sets and varieties. By analyzing Quillen's proofs of dévissage and localization we define ACGW-categories, an analogous generalization of abelian categories for which we prove theorems analogous to dévissage and localization. In particular, although the category of varieties is not quite ACGW, the category of reduced schemes of finite type is; applying d…
In this paper we demonstrate that 'non-commutative localizations' of arbitrary additive categories (generalizing those defined by Cohn for rings) are closely (and naturally) related with weight structures. Localizing an arbitrary triangulated $C$ by a set $S$ of morphisms in the heart of a weight structure $w$ for it one obtains a triangulated category endowed with a weight structure $w'$. The heart of $w'$ is a certain idempotent completion of the non-commutative localization of the heart of $w$ by $S$. The latter is the natural categorical version of Cohn's localizations of rings i.e. the fu…
Augmentations, reduced ideal point gluings and compact type degenerations of curves
University of Georgia（佐治亚大学）；Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）；NRU Higher School of Economics（国立研究大学高等经济学院）
We provide an explicit formula for localizing $A^1$-homotopy invariants of topological Fukaya categories of marked surfaces. Following a proposal of Kontsevich, this differential $\mathbb Z$-graded category is defined as global sections of a constructible cosheaf of dg categories on any spine of the surface. Our theorem utilizes this sheaf-theoretic description to reduce the calculation of invariants to the local case when the surface is a boundary-marked disk. At the heart of the proof lies a theory of localization for topological Fukaya categories which is a combinatorial analog of Thomason-…
Let $R$ be a commutative Noetherian ring. We introduce the notion of localization functors $λ^W$ with cosupports in arbitrary subsets $W$ of $\text{Spec}\, R$; it is a common generalization of localizations with respect to multiplicatively closed subsets and left derived functors of ideal-adic completion functors. We prove several results about the localization functors $λ^W$, including an explicit way to calculate $λ^W$ by the notion of Cech complexes. As an application, we can give a simpler proof of a classical theorem by Gruson and Raynaud, which states that the projective dimension of a f…
Let X and Y be two smooth Deligne-Mumford stacks and consider a function f, resp. g, on X, resp. Y. Assume that there exists a complex F of sheaves on the fiber product of X and Y over A^1 (induced by f and g), such that the Fourier-Mukai transform with the kernel F gives an equivalence between the bounded derived categories of coherent sheaves on X and Y. If X_0 Y_0 are the fibers of f and g over zero, respectively, we show that the singular derived categories of X_0 and Y_0 are also equivalent. We apply this statement in the setting of McKay correspondence, and generalize a result of Orlov o…
As we proved earlier, for a triangulated category $\underline{C}$ endowed with a weight structure $w$ and a triangulated subcategory $\underline{D}$ of $\underline{C}$ (strongly) generated by cones of a set of morphisms $S$ in the heart $\underline{Hw}$ of $w$ there exists a weight structure $w'$ on the Verdier quotient $\underline{C}'=\underline{C}/\underline{D}$ such that the localization functor $\underline{C} \to \underline{C}'$ is weight-exact (i.e., "respects weights"). The goal of this paper is to find conditions ensuring that for any object of $\underline{C}'$ of non-negative (resp. no…
The aim of these notes is to collect and motivate the basic localization toolbox for the geometric study of ``spaces'', locally described by noncommutative rings and their categories of one-sided modules. We present the basics of Ore localization of rings and modules in much detail. Common practical techniques are studied as well. We also describe a counterexample for a folklore test principle. Localization in negatively filtered rings arising in deformation theory is presented. A new notion of the differential Ore condition is introduced in the study of localization of differential calculi. T…
It is now well known that the K-theory of a Waldhausen category depends on more than just its (triangulated) homotopy category (see [Schlichting]). The purpose of this note is to show that the K-theory spectrum of a (good) Waldhausen category is completely determined by its Dwyer-Kan simplicial localization, without any additional structure. As the simplicial localization is a refined version of the homotopy category which also determines the triangulated structure, our result is a possible answer to the general question: ``To which extent $K$-theory is not an invariant of triangulated derived…
This note extends some recent results on the derived category of a geometric invariant theory quotient to the setting of derived algebraic geometry. Our main result is a structure theorem for the derived category of a derived local quotient stack which admits a stratification of the kind arising in geometric invariant theory. The use of derived algebraic geometry leads to results with pleasingly few hypotheses, even when the stack is not smooth. Using the same methods, we establish a "virtual non-abelian localization theorem" which is a K-theoretic analog of the virtual localization theorem in…
We study the topological K-theory spectrum of the dg singularity category associated to a weighted projective complete intersection. We calculate the topological K-theory of the dg singularity category of a weighted projective hypersurface in terms of its Milnor fiber and monodromy operator, and, as an application, we obtain a lift of the Atiyah-Bott-Shapiro construction to the level of spectra.
We define notions of direct and inverse limits in an $n$-category. We prove that the $n+1$-category $nCAT'$ of fibrant $n$-categories admits direct and inverse limits. At the end we speculate (without proofs) on some applications of the notion of limit, including homotopy fiber product and homotopy coproduct for $n$-categories, the notion of $n$-stack, representable functors, and finally on a somewhat different note, a notion of relative Malcev completion of the higher homotopy at a representation of the fundamental group.
