Trace theories and localization

  title={Trace theories and localization},
  author={Kaledin Dmitry Borisovich},
  journal={arXiv: K-Theory and Homology},
We show how one can twist the definition of Hochschild homology of an algebra or a DG algebra by inserting a possibly non-additive trace functor. We then prove that many of the usual properties of Hochschild homology survive such a generalization. In some cases this even includes Keller's Localization Theorem. 
14 Citations
What do Abelian categories form?
  • D. Kaledin
  • Mathematics
    Russian Mathematical Surveys
  • 2022
Given two finitely presentable Abelian categories and , we outline a construction of an Abelian category of functors from to , which has nice 2-categorical properties and provides an explicit model
Topological Hochschild homology and higher characteristics
We show that an important classical fixed point invariant, the Reidemeister trace, arises as a topological Hochschild homology transfer. This generalizes a corresponding classical result for the
THH and Traces of Enriched Categories
  • John D. Berman
  • Mathematics
    International Mathematics Research Notices
  • 2020
We prove that topological Hochschild homology (THH) arises from a presheaf of circles on a certain combinatorial category, which gives a universal construction of THH for any enriched $\infty
Deformation Theory and Partition Lie Algebras
A theorem of Lurie and Pridham establishes a correspondence between formal moduli problems and differential graded Lie algebras in characteristic zero, thereby formalising a well-known principle in
Trace theories, Bokstedt periodicity and Bott periodicity
We flesh out the theory of "trace theories" and "trace functors" sketched in arXiv:1308.3743, extend it to a homotopical setting, and prove a reconstruction theorem claiming that a trace theory is
$K$-theory of endomorphisms, the $\mathit{TR}$-trace, and zeta functions
We show that the characteristic polynomial and the Lefschetz zeta function are manifestations of the trace map from the $K$-theory of endomorphisms to topological restriction homology (TR). Along the
When Ext is a Batalin–Vilkovisky algebra
  • N. Kowalzig
  • Mathematics
    Journal of Noncommutative Geometry
  • 2018
We show under what conditions the complex computing general Ext-groups carries the structure of a cyclic operad such that Ext becomes a Batalin-Vilkovisky algebra. This is achieved by transferring
Adjunction in 2-categories
We give an overview of the parts of arXiv:2004.04279 that deal with 2-categories, up to and including adjunction, and explain how the Segal-type approach to 2-categories adopted there is related to
Witt vectors as a polynomial functor
For every commutative ring A, one has a functorial commutative ring W(A) of p-typical Witt vectors of A, an iterated extension of A by itself. If A is not commutative, it has been known since the


Cyclic homology with coefficients
We propose a category which can serve as the category of coefficients for the cyclic homology HC ∗(A) of an associative algebra A over a field k. The construction is categorical in nature, and
Simplicial Homotopy Theory
Simplicial sets, model categories, and cosimplicial spaces: applications for homotopy coherence, results and constructions, and more.
Fixed Point Theory and Trace for Bicategories
The Lefschetz fixed point theorem follows easily from the identification of the Lefschetz number with the fixed point index. This identification is a consequence of the functoriality of the trace in
Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie
We use a version of the method of Deligne-Illusie to prove that the Hodge-to-de Rham, a.k.a. Hochschild-to-cyclic spectral sequence degenerates for a large class of associative, not necessariyl
Additive K-theory
  • Complexe Cotangent et Déformations, I, Lecture Notes in Math. 239, Springer, Berlin-New York
  • 1971