Grothendieck topologies and deformation Theory II

Starting from a sheaf of associative algebras over a scheme we show thatits deformation theory is described by cohomologies of a canonical object,called the cotangent complex, in the derived category of sheaves ofbi-modules over this sheaf of algebras. The passage from deformations tocohomology is based on considering a site which is naturally constructed outof our sheaf of algebras. It turns out that on the one hand, cohomology ofcertain sheaves on this site control deformations, and on the other hand,they can be rewritten in terms of the category of sheaves of bi-modules.     

A Quillen-type spectral sequence for the K-theory of varieties with isolated singularities

We construct a spectral sequence to compute the algebraic K-theory of any quasiprojective scheme X, when X has isolated singularities, using an explicit flasque resolution of the K-theory sheaves. This is a generalization of Quillen's construction for nonsingular varieties. The explicit resolution makes it possible to relate K-theory to intersection theory on singular schemes.Partially supported by NSF grants.Dedicated to A. Grothendieck on his sixtieth birthday     

Categorically Algebraic Foundations for Homotopical Algebra

We investigate a structure for an abstract cylinder endofunctor I whichproduces a good basis for homotopical algebra. It essentially consists ofthe usual operations (faces, degeneracies, connections, symmetries, verticalcomposition) together with a transformation w: I² I², whichwe call lens collapse after its realisation in the standard topologicalcase. This structure, if somewhat heavy, has the interest of beingcategorically algebraic, i.e., based on operations onfunctors. Consequently, it can be naturally lifted from a category A to itscategories of diagrams A^S and its slice categories A\X,A/X.Further, the dual structure, based on a cocylinder (or path) endofunctor Pcan be lifted to the category of A-valued sheaves on a site, wheneverthe path functor P preserves limits, and to the category Mon A of internalmonoids, with respect to any monoidal structure of A consistent with P.     

The Sebastiani–Thom Isomorphism in the Derived Category

We prove that the Sebastiani–Thom isomorphism for Milnor fibres and their monodromies exists as a natural isomorphism between vanishing cycles in the derived category.     

Parametrized spaces model locally constant homotopy sheaves

We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopy-theoretic version of the classical identification of covering spaces with locally constant sheaves. We also prove a new version of the classical result that spaces parametrized over X are equivalent to spaces with an action of ΩX. This gives a homotopy-theoretic version of the correspondence between covering spaces and π₁-sets. We then use these two equivalences to study base change functors for parametrized spaces.     

Torsionfree sheaves over a nodal curve of arithmetic genus one

We classify all isomorphism classes of stable torsionfree sheaves on an irreducible nodal curve of arithmetic genus one defined over ℂ. Let X be a nodal curve of arithmetic genus one defined over ℝ, with exactly one node, such that X does not have any real points apart from the node. We classify all isomorphism classes of stable real algebraic torsionfree sheaves over X of even rank. We also classify all isomorphism classes of real algebraic torsionfree sheaves over X of rank one.     

Faisceaux sans torsion et faisceaux quasi localement libres sur les courbes multiples primitives

This paper is devoted to the study of some coherent sheaves on non reduced curves that can be locally embedded in smooth surfaces. If Y is such a curve and n is its multiplicity, then there is a filtration C₁ = C ? C₂ ? … ? C_n = Y such that C is the reduced curve associated to Y, and for every P ∈ C, if z ∈ O_Y,P is an equation of C then (zⁱ) is the ideal of C_i in O_Y,P. A coherent sheaf on Y is called torsion free if it does not have any non zero subsheaf with finite support. We prove that torsion free sheaves are reflexive. We study then the quasi locally free sheaves, i.e., sheaves which are locally isomorphic to direct sums of the O_Ci.We define an invariant for these sheaves, the complete type, and prove the irreducibility of the set of sheaves of given complete type. We study the generic quasi locally free sheaves, with applications to the moduli spaces of stable sheaves on Y (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sheaves on quantaloids

We generalize the well-known notions of a singleton, completeQ-set, presheaf, and sheaf over a complete Heyting algebra or a right-sided idempotent quantale to arbitrary quantaloids. We show that every completeQ-set can be viewed as a sheaf and vice versa. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 133–171, April–June, 2000.     

Formal and Tempered Solutions of Regular D-Modules

A new family of sheaves has been recently studied by M. Kashiwara and P. Schapira generalizing to constructible sheaves the notion of moderate and formal cohomology. We prove comparison theorems when we regard these sheaves as solutions of a D-module. These results are natural generalizations of those of Y. Laurent and the author.     

On the reducibility of characteristic varieties

We show that some monodromies in the Morse local systems of a conically stratified perverse sheaf imply that other Morse local systems for smaller strata do not vanish. This result is then used to explain the examples of reducible characteristic varieties of Schubert varieties given by Kashiwara and Saito in type and by Boe and Fu for the Lagrangian Grassmannian.