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24
TRIPLES, ALGEBRAS AND COHOMOLOGY
 REPRINTS IN THEORY AND APPLICATIONS OF CATEGORIES
, 2003
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Topological and smooth stacks
"... Abstract. We review the basic definition of a stack and apply it to the topological and smooth settings. We then address two subtleties of the theory: the correct definition of a “stack over a stack ” and the distinction between small stacks (which are algebraic objects) and large stacks (which are ..."
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Abstract. We review the basic definition of a stack and apply it to the topological and smooth settings. We then address two subtleties of the theory: the correct definition of a “stack over a stack ” and the distinction between small stacks (which are algebraic objects) and large stacks (which are generalized spaces). 1.
Comparison of Abelian Categories Recollements
 DOCUMENTA MATH.
, 2004
"... We give a necessary and sufficient condition for a morphism between recollements of abelian categories to be an equivalence. ..."
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We give a necessary and sufficient condition for a morphism between recollements of abelian categories to be an equivalence.
Grothendieck topologies and ideal closure operations
 In preparation
, 2006
"... Abstract. We relate closure operations for ideals and for submodules to nonflat Grothendieck topologies. We show how a Grothendieck topology on an affine scheme induces a closure operation in a natural way, and how to construct for a given closure operation fulfilling certain properties a Grothendi ..."
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Abstract. We relate closure operations for ideals and for submodules to nonflat Grothendieck topologies. We show how a Grothendieck topology on an affine scheme induces a closure operation in a natural way, and how to construct for a given closure operation fulfilling certain properties a Grothendieck topology which induces this operation. In this way we relate the radical to the surjective topology and the constructible topology, the integral closure to the submersive topology, to the proper topology and to Voevodsky’s htopology, the Frobenius closure to the Frobenius topology and the plus closure to the finite topology. The topologies which are induced by a Zariski filter yield the closure operations which are studied under the name of hereditary torsion theories. The Grothendieck topologies enrich the corresponding closure operation by providing cohomology theories, rings of global sections, concepts of exactness and of stalks.
Stacks for Everybody
"... Abstract. Let S be a category with a Grothendieck topology. A stack over S is a category fibered in groupoids over S, such that isomorphisms form a sheaf and every descent datum is effective. If S is the category of schemes with the étale topology, a stack is algebraic in the sense of DeligneMumfor ..."
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Abstract. Let S be a category with a Grothendieck topology. A stack over S is a category fibered in groupoids over S, such that isomorphisms form a sheaf and every descent datum is effective. If S is the category of schemes with the étale topology, a stack is algebraic in the sense of DeligneMumford (respectively Artin) if it has an étale (resp. smooth) presentation. I will try to explain the previous definitions so as to make them accessible to the widest possible audience. In order to do this, we will keep in mind one fixed example, that of vector bundles; if you know what pullback of vector bundles is in some geometric context (schemes, complex analytic spaces, but also varieties or manifolds) you should be able to follow this exposition. 1.
De Rham and infinitesimal cohomology in Kapranov’s model for noncommutative algebraic geometry
, 2001
"... The title refers to the nilcommutative or NCschemes introduced by M. Kapranov in Noncommutative geometry based on commutator expansions, J. reine angew. Math 505 (1998) 73118. The latter are noncommutative nilpotent thickenings of commutative schemes. We consider also the parallel theory of nilP ..."
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The title refers to the nilcommutative or NCschemes introduced by M. Kapranov in Noncommutative geometry based on commutator expansions, J. reine angew. Math 505 (1998) 73118. The latter are noncommutative nilpotent thickenings of commutative schemes. We consider also the parallel theory of nilPoisson or NPschemes, which are nilpotent thickenings of commutative schemes in the category of Poisson schemes. We study several variants of de Rham cohomology for NC and NPschemes. The variants include nilcommutative and nilPoisson versions of the de Rham complex as well as of the cohomology of the infinitesimal site introduced by Grothendieck in Crystals and the de Rham cohomology of schemes, Dix exposés sur la cohomologie des schémas, Masson & Cie, NorthHolland (1968) 306358. It turns out that each of these noncommutative variants admits a kind of Hodge decomposition which allows one to express the cohomology groups of a noncommutative scheme Y as a sum of copies of the usual (de Rham, infinitesimal) cohomology groups of the underlying commutative scheme X (Theorems 6.2, 6.5, 6.8). As a byproduct we obtain new proofs for classical results of Grothendieck (Corollary 6.3) and of FeiginTsygan (Corollary 6.9) on the relation between de Rham and infinitesimal cohomology and between the latter and periodic cyclic homology.
SUBMERSIONS AND EFFECTIVE DESCENT OF ÉTALE MORPHISMS
, 2007
"... Using the flatification by blowup result of Raynaud and Gruson, we obtain new results for submersive and subtrusive morphisms. We show that universally subtrusive morphisms, and in particular universally open morphisms, are morphisms of effective descent for the fibered category of étale morphism ..."
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Using the flatification by blowup result of Raynaud and Gruson, we obtain new results for submersive and subtrusive morphisms. We show that universally subtrusive morphisms, and in particular universally open morphisms, are morphisms of effective descent for the fibered category of étale morphisms. Our results extend and supplement previous treatments on submersive morphisms by Grothendieck, Picavet and Voevodsky. Applications include the universality of geometric quotients and the elimination of noetherian hypotheses in many instances.
A SheafTheoretic View Of Loop Spaces
 Theory Appl. Categ
, 2001
"... The context of enriched sheaf theory introduced in the author's thesis provides a convenient viewpoint for models of the stable homotopy category as well as categories of finite loop spaces. Also, the languages of algebraic geometry and algebraic topology have been interacting quite heavily ..."
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The context of enriched sheaf theory introduced in the author's thesis provides a convenient viewpoint for models of the stable homotopy category as well as categories of finite loop spaces. Also, the languages of algebraic geometry and algebraic topology have been interacting quite heavily in recent years, primarily due to the work of Voevodsky and that of Hopkins. Thus, the language of Grothendieck topologies is becoming a necessary tool for the algebraic topologist. The current article is intended to give a somewhat relaxed introduction to this language of sheaves in a topological context, using familiar examples such as nfold loop spaces and pointed Gspaces. This language also includes the diagram categories of spectra from [19] as well as spectra in the sense of [17], which will be discussed in some detail. 1.