Negative cyclic homology

General Introduction
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Search results
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Online References
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Paper References
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Definition
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Properties
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Standard theorems
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Open Problems
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
Toen: Note of Chern character, loop spaces and derived algebraic geometry. File Toen web publ Abel2007.pdf. Notion of derived cat sheaves, a categorification of the notion of complexes of sheaves of Omodules on schemes (also quasicohomology and perfect versions). Chern character for these categorical sheaves, a categorified version of the Chern char for perfect complexes with values in cyclic homology. Using the derived loop space. "This work can be seen as an attempt to define algebraic analogues of of elliptic objects and char classes for them". 1. Motivations: Elliptic cohomology, geometric interpretations, chromatic level and ncategorical level, 2VBs. Maybe the typical generalized CT of chromatic level n should be related to ncats, more precisely cohomology classes should be rep by maps from X to a certain nstack. Rognes redshift conjecture: Intuitively saying that the Kth spectrum of a commutative ring spectrum of chrom level n is of chrom level (n+1). More on ell cohomology and 2cats. Idea of categorical sheaves: For X a scheme, should have a symmetric monoidal 2cat Cat(X) which is a categorification of Mod(X), in the sense that Mod(X) should be the cat of endomorphisms of the unit objects in Cat(X). More details. Notions of secondary cohomology and secondary Ktheory. Notion of derived categorical sheaves, more reasonable than nonderived version. Relation between equivariant functions on LX and negative cyclic homology. 2. Categorification of homological algebra and dgcats. 3. Loop spaces in DAG. More, including relations with variations of Hodge structures. Final remark on algebraic elliptic cohomology. "Algebraic Ktheory determines complex topological Ktheory", ref to Walker 2002.
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