Rendiconti del Seminario Matematico della Università di Padova, 1997
We study localization properties of some algebraic differential complexes associated to an arbitr... more We study localization properties of some algebraic differential complexes associated to an arbitrary commutative algebra which are higher order (in the sense of differential operators) analogues of the ordinary de Rham complex. These results should be considered, in the spirit of [ 11 ], as preliminaries to the study of the cohomological invariants provided by these higher de Rham complexes for singular varieties. Notations and Conventions. K: a commutative ring with unit; A : a commutative, associative K-algebra with unit; A-Mod : the category of A-modules; K Mod : the category of K-modules; DIFFA : the category whose objects are A-modules and whose morphisms are differential operators (Section 1), of any (finite) order, between them; Ens: the category of sets; [C, C] : the category of functors C ~ C, C being any category; Ob ( C) : the objects of C, C being any category; we will write C E e Ob(C) to mean that C is an object of C; (A, A )-B iMod : the category of (A, A)-bimodules, ...
These are expanded notes of some talks given during the fall 2002, about homotopical algebraic ge... more These are expanded notes of some talks given during the fall 2002, about homotopical algebraic geometry with special emphasis on its applications to derived algebraic geometry and derived deformation theory. We use the general framework developed in [HAG-I], and in particular the notions of model topology, model sites and stacks over them, in order to define various derived moduli functors and
In this paper we prove formal glueing, along an arbitrary closed subscheme $Z$ of a scheme $X$, f... more In this paper we prove formal glueing, along an arbitrary closed subscheme $Z$ of a scheme $X$, for the stack of pseudo-coherent, perfect complexes, and $G$-bundles on $X$ (for $G$ a smooth affine algebraic group). By iterating this result, we get a decomposition of these stacks along an arbitrary nonlinear flag of subschemes in $X$. By taking points over the base field, we deduce from this both a formal glueing, and a flag-related decomposition formula for the corresponding derived $\infty$-categories of pseudo-coherent and perfect complexes. We finish the paper by highlighting some expected progress in the subject matter of this paper, that might be related to a Geometric Langlands program for higher dimensional varieties. In the Appendix we also prove a localization theorem for the stack of pseudo-coherent complexes, which parallels Thomason's localization results for perfect complexes.
This is the first in a series of papers about foliations in derived geometry. After introducing d... more This is the first in a series of papers about foliations in derived geometry. After introducing derived foliations on arbitrary derived stacks, we concentrate on quasi-smooth and rigid derived foliations on smooth complex algebraic varieties and on their associated formal and analytic versions. Their truncations are classical singular foliations. We prove that a quasi-smooth rigid derived foliation on a smooth complex variety X is formally integrable at any point, and, if we suppose that its singular locus has codimension ≥ 2, then the truncation of its analytification is a locally integrable singular foliation on the associated complex manifold X h. We then introduce the derived category of perfect crystals on a quasi-smooth rigid derived foliation on X, and prove a Riemann-Hilbert correspondence for them when X is proper. We discuss several examples and applications.
We present an $\ell$-adic trace formula for saturated and admissible dg-categories over a base mo... more We present an $\ell$-adic trace formula for saturated and admissible dg-categories over a base monoidal dg-category. Moreover, we prove K\"unneth formulas for dg-category of singularities, and for inertia-invariant vanishing cycles. As an application, we prove a version of Bloch's Conductor Conjecture (stated by Spencer Bloch in 1985), under the additional hypothesis that the monodromy action of the inertia group is unipotent.
This is the second of series of papers on the study of foliations in the setting of derived algeb... more This is the second of series of papers on the study of foliations in the setting of derived algebraic geometry based on the central notion of derived foliation. We introduce sheaf-like coefficients for derived foliations, called quasi-coherent crystals, and construct a certain sheaf of dg-algebras of differential operators along a given derived foliation, with the property that quasi-coherent crystals can be interpreted as modules over this sheaf of differential operators. We use this interpretation in order to introduce the notion of good filtrations on quasi-coherent crystals, and define the notion of characteristic cycle. Finally, we prove a Grothendieck-Riemann-Roch (GRR) formula expressing that formation of characteristic cycles is compatible with push-forwards along proper and quasi-smooth morphisms. Several examples and applications are deduced from this, e.g. a GRR formula for D-modules on possibly singular schemes, and a foliated index formula for weakly Fredholm operators.
We present an $\ell$-adic trace formula for saturated and admissible dg-categories over a base mo... more We present an $\ell$-adic trace formula for saturated and admissible dg-categories over a base monoidal dg-category. As an application, we prove (a version of) Bloch's conductor conjecture, under the additional hypothesis that the inertia group acts with unipotent monodromy.
Rendiconti del Seminario Matematico della Università di Padova, 1997
We study localization properties of some algebraic differential complexes associated to an arbitr... more We study localization properties of some algebraic differential complexes associated to an arbitrary commutative algebra which are higher order (in the sense of differential operators) analogues of the ordinary de Rham complex. These results should be considered, in the spirit of [ 11 ], as preliminaries to the study of the cohomological invariants provided by these higher de Rham complexes for singular varieties. Notations and Conventions. K: a commutative ring with unit; A : a commutative, associative K-algebra with unit; A-Mod : the category of A-modules; K Mod : the category of K-modules; DIFFA : the category whose objects are A-modules and whose morphisms are differential operators (Section 1), of any (finite) order, between them; Ens: the category of sets; [C, C] : the category of functors C ~ C, C being any category; Ob ( C) : the objects of C, C being any category; we will write C E e Ob(C) to mean that C is an object of C; (A, A )-B iMod : the category of (A, A)-bimodules, ...
These are expanded notes of some talks given during the fall 2002, about homotopical algebraic ge... more These are expanded notes of some talks given during the fall 2002, about homotopical algebraic geometry with special emphasis on its applications to derived algebraic geometry and derived deformation theory. We use the general framework developed in [HAG-I], and in particular the notions of model topology, model sites and stacks over them, in order to define various derived moduli functors and
In this paper we prove formal glueing, along an arbitrary closed subscheme $Z$ of a scheme $X$, f... more In this paper we prove formal glueing, along an arbitrary closed subscheme $Z$ of a scheme $X$, for the stack of pseudo-coherent, perfect complexes, and $G$-bundles on $X$ (for $G$ a smooth affine algebraic group). By iterating this result, we get a decomposition of these stacks along an arbitrary nonlinear flag of subschemes in $X$. By taking points over the base field, we deduce from this both a formal glueing, and a flag-related decomposition formula for the corresponding derived $\infty$-categories of pseudo-coherent and perfect complexes. We finish the paper by highlighting some expected progress in the subject matter of this paper, that might be related to a Geometric Langlands program for higher dimensional varieties. In the Appendix we also prove a localization theorem for the stack of pseudo-coherent complexes, which parallels Thomason's localization results for perfect complexes.
This is the first in a series of papers about foliations in derived geometry. After introducing d... more This is the first in a series of papers about foliations in derived geometry. After introducing derived foliations on arbitrary derived stacks, we concentrate on quasi-smooth and rigid derived foliations on smooth complex algebraic varieties and on their associated formal and analytic versions. Their truncations are classical singular foliations. We prove that a quasi-smooth rigid derived foliation on a smooth complex variety X is formally integrable at any point, and, if we suppose that its singular locus has codimension ≥ 2, then the truncation of its analytification is a locally integrable singular foliation on the associated complex manifold X h. We then introduce the derived category of perfect crystals on a quasi-smooth rigid derived foliation on X, and prove a Riemann-Hilbert correspondence for them when X is proper. We discuss several examples and applications.
We present an $\ell$-adic trace formula for saturated and admissible dg-categories over a base mo... more We present an $\ell$-adic trace formula for saturated and admissible dg-categories over a base monoidal dg-category. Moreover, we prove K\"unneth formulas for dg-category of singularities, and for inertia-invariant vanishing cycles. As an application, we prove a version of Bloch's Conductor Conjecture (stated by Spencer Bloch in 1985), under the additional hypothesis that the monodromy action of the inertia group is unipotent.
This is the second of series of papers on the study of foliations in the setting of derived algeb... more This is the second of series of papers on the study of foliations in the setting of derived algebraic geometry based on the central notion of derived foliation. We introduce sheaf-like coefficients for derived foliations, called quasi-coherent crystals, and construct a certain sheaf of dg-algebras of differential operators along a given derived foliation, with the property that quasi-coherent crystals can be interpreted as modules over this sheaf of differential operators. We use this interpretation in order to introduce the notion of good filtrations on quasi-coherent crystals, and define the notion of characteristic cycle. Finally, we prove a Grothendieck-Riemann-Roch (GRR) formula expressing that formation of characteristic cycles is compatible with push-forwards along proper and quasi-smooth morphisms. Several examples and applications are deduced from this, e.g. a GRR formula for D-modules on possibly singular schemes, and a foliated index formula for weakly Fredholm operators.
We present an $\ell$-adic trace formula for saturated and admissible dg-categories over a base mo... more We present an $\ell$-adic trace formula for saturated and admissible dg-categories over a base monoidal dg-category. As an application, we prove (a version of) Bloch's conductor conjecture, under the additional hypothesis that the inertia group acts with unipotent monodromy.
Uploads
Papers by Gabriele Vezzosi