Chern Character, Loop Spaces and Derived Algebraic Geometry (original) (raw)
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A note on Chern character, loop spaces and derived algebraic geometry
Eprint Arxiv 0804 1274, 2008
In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of modules on schemes, as well as its quasi-coherent and perfect versions. We also explain how ideas from derived algebraic geometry and higher category theory can be used in order to construct a Chern character for these categorical sheaves, which is a categorified version of the Chern character for perfect complexes with values in cyclic homology. Our construction uses in an essential way the derived loop space of a scheme X, which is a derived scheme whose theory of functions is closely related to cyclic homology of X. This work can be seen as an attempt to define algebraic analogs of elliptic objects and characteristic classes for them. The present text is an overview of a work in progress and details will appear elsewhere.
2006
The contents of these notes originated from a talk delivered to a diversified audience at the Departamento de Matemática of my own University. The present notes expand and give full details of statements made in that occasion. Moreover, I have sought to expand as well on several algebraic notions in order to pave the way towards a more stable version even though self-sufficiency is out of question in a material of this order. If one is to single a module that stands alone in its totally device-independent nature, one'd better take the module of differentials Ω(A/k). In fact, most invariants of an algebra or scheme are attached primevally to this module or "découlent" from it by various procedures. Thus, Segre classes in algebraic geometry are ultimately defined in terms of Ω(A/k) and even the ubiquitous canonical module (dualizing sheaf) is "normally" related to Ω(A/k) by double-dualizing its top-wedge module. My objective in this Course is panorama-driven, due to the short span of lectures. However, details and examples will be worked out and, possibly, a few arguments too. 10. * Considerations about a formula of Kleiman-Plücker-Teissier for the degree (class) of the dual variety to a complete intersection.
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1998
We prove that the cyclic homology of a scheme with an ample line bundle coincides with the cyclic homology of its category of algebraic vector bundles. As a byproduct of the proof, we obtain a new construction of the Chern character of a perfect complex on a ringed space.
Algebraic-homological constructions attached to differentials
The contents of these notes originated from a talk delivered to a diversified audience at the Departamento de Matemática of my own University. The present notes expand and give full details of statements made in that occasion. Moreover, I have sought to expand as well on several algebraic notions in order to pave the way towards a more stable version even though self-sufficiency is out of question in a material of this order. If one is to single a module that stands alone in its totally device-independent nature, one'd better take the module of differentials Ω(A/k). In fact, most invariants of an algebra or scheme are attached primevally to this module or "découlent" from it by various procedures. Thus, Segre classes in algebraic geometry are ultimately defined in terms of Ω(A/k) and even the ubiquitous canonical module (dualizing sheaf) is "normally" related to Ω(A/k) by double-dualizing its top-wedge module. My objective in this Course is panorama-driven, d...
THE DERIVED CATEGORY OF QUASI-COHERENT SHEAVES AND AXIOMATIC STABLE HOMOTOPY
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We prove that the dg category of perfect complexes on a smooth, proper Deligne-Mumford stack over a field of characteristic zero is geometric in the sense of Orlov, and in particular smooth and proper. On the level of triangulated categories, this means that the derived category of perfect complexes embeds as an admissible subcategory into the bounded derived category of coherent sheaves on a smooth, projective variety. The same holds for a smooth, projective, tame Artin stack over an arbitrary field. Contents 1. Introduction 1 2. Preliminaries 4 3. Root constructions 8 4. Semiorthogonal decompositions for root stacks 12 5. Differential graded enhancements and geometricity 20 6. Geometricity for dg enhancements of algebraic stacks 25 Appendix A. Bounded derived category of coherent modules 28 References 29
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