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Trace theories and localization
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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.
Publication based on the results of:
In book
Vol. 643: Stacks and Categories in Geometry, Topology, and Algebra. , American Mathematical Society, 2015.
Papayanov G., Математические заметки 2024 Т. 115 № 5 С. 797–799
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Added: December 2, 2024
Efimov A., Inventiones Mathematicae 2020 Vol. 222 No. 2 P. 667–694
We disprove two (unpublished) conjectures of Kontsevich which
state generalized versions of categorical Hodge-to-de Rham degeneration for
smooth and for proper DG categories (but not smooth and proper, in which
case degeneration is proved by Kaledin (in: Algebra, geometry, and physics in
the 21st century. Birkhäuser/Springer, Cham, pp 99–129, 2017). In particular,
we show that there exists a minimal 10-dimensional ...
Added: September 24, 2020
Kaledin D., Advances in Mathematics 2019 Vol. 351 P. 33–95
In a previous paper, we have defined polynomial Witt vectors functor from vector spaces over a perfect field k of positive characteristic p to abelian groups. In this paper, we use polynomial Witt vectors to construct a functorial HochschildWitt complex WCH_∗(A) for any associative unital k-algebra A, with homology groups WHH∗(A). We prove that the group WHH_0(A) coincides with the ...
Added: June 8, 2019
Shoikhet B., / Series arXiv "math". 2018.
Given two small dg categories C,D, defined over a field, we introduce their (non-symmetric) twisted tensor product C⊗∼D. We show that −⊗∼D is left adjoint to the functor Coh(D,−), where Coh(D,E) is the dg category of dg functors D→E and their coherent natural transformations. This adjunction holds in the category of small dg categories (not in the homotopy category of dg categories Hot). We show ...
Added: December 7, 2018
Kaledin D., Успехи математических наук 2018 Т. 73 № 1 С. 3–34
A review of the classical construction of Witt vectors is presented, and some recent generalizations of it to the non-commutative case are described. ...
Added: September 13, 2018
Petrov A., Vaintrob D., Vologodsky V., Selecta Mathematica, New Series 2018 Vol. 24 No. 1 P. 531–561
It is expected that the periodic cyclic homology of a DG algebra over C (and, more
generally, the periodic cyclic homology of a DG category) carries a lot of additional
structure similar to the mixedHodge structure on the deRhamcohomology of algebraic
varieties. Whereas a construction of such a structure seems to be out of reach at the
moment its ...
Added: March 13, 2018
Vologodsky V., Petrov A., Vaintrob D., / Series arXiv "math". 2017.
It is expected that the periodic cyclic homology of a DG algebra over the field of complex numbers (and, more generally, the periodic cyclic homology of a DG category) carries a lot of additional structure similar to the mixed Hodge structure on the de Rham cohomology of algebraic varieties. Whereas a construction of such ...
Added: November 8, 2017
Kaledin D., Труды Математического института им. В.А. Стеклова РАН 2015 Т. 290 С. 43–60
Строится некоммутативное обобщение изоморфизма Картье для любой ассоциативной унитальной алгебры над совершенным полем k нечетной положительной характеристики. Роль дифференциальных форм играют классы гомологий Хохшильда, а дифференциал де Рама заменяется на дифференциал Конна–Цыгана. ...
Added: April 10, 2017
Kaledin D., / Series arXiv "math". 2016.
In arxiv:1602.04254, we have defined polynomial Witt vectors functor from vector spaces over a perfect field k of positive characteristic p to abelian groups. In this paper, we use polynomial Witt vectors to construct a functorial Hochschild-Witt complex WCH∗(A) for any associative unital k-algebra A, with homology groups WHH∗(A). We prove that the group WHH0(A) ...
Added: May 18, 2016
Galkin S., Shinder E., / Series math "arxiv.org". 2015. No. 1506.05831.
We define a zeta-function of a pre-triangulated dg-category and investigate its relationship with the motivic zeta-function in the geometric case. ...
Added: June 23, 2015
Alexey Elagin, / Series math "arxiv.org". 2014. No. 1403.7027.
Consider a finite group $G$ acting on a triangulated category $\TTT$. In this paper we try to understand when the category $\TTT^G$ of $G$-equivariant objects in $\TTT$ is triangulated. We prove that it is so in two cases: the action on the derived category $\D^b(\AA)$ induced by an action on an abelian category $\AA$ and ...
Added: September 15, 2014
Kuznetsov A., Journal fuer die reine und angewandte Mathematik 2015 Vol. 2015 No. 708 P. 213–243
We define the normal Hochschild cohomology of an admissible subcategory of the derived category of coherent sheaves on a smooth projective variety $X$ --- a graded vector space which controls the restriction morphism from the Hochschild cohomology of $X$ to the Hochschild cohomology of the orthogonal complement of this admissible subcategory. When the subcategory is ...
Added: December 22, 2013
Kuznetsov A., / Series math "arxiv.org". 2012. No. 1211.4693.
We define the normal Hochschild cohomology of an admissible subcategory of the derived category of coherent sheaves on a smooth projective variety $X$ --- a graded vector space which controls the restriction morphism from the Hochschild cohomology of $X$ to the Hochschild cohomology of the orthogonal complement of this admissible subcategory. When the subcategory is ...
Added: October 4, 2013
Galkin S., Shinder E., / Series math "arxiv.org". 2012. No. 1210.3339.
We construct quasi-phantom admissible subcategories in the derived category of coherent sheaves on the Beauville surface S. These quasi-phantoms subcategories appear as right orthogonals to subcategories generated by exceptional collections of maximal possible length 4 on S. We prove that there are exactly 6 exceptional collections consisting of line bundles (up to a twist) and these collections ...
Added: September 14, 2013
Pirkovskii A. Y., Известия РАН. Серия математическая 2012 Т. 76 № 4 С. 65–124
We prove the equation w.dg A = w.db A for every nuclear Fréchet–Arens–Michael algebra A of finite weak bidimension, where w.dg A is the weak global dimension and w.db A is the weak bidimension of A. Assuming that A has a projective bimodule resolution of finite type, we establish the estimate dg A ≤ db ...
Added: September 19, 2012
Polishchuk A., Positselski L., Transactions of the American Mathematical Society 2012 Vol. 364 No. 10 P. 5311–5368
We define and study the Hochschild (co)homology of the second kind (known also as the Borel-Moore Hochschild homology and the compactly supported Hochschild cohomology) for curved DG categories. An isomorphism between the Hochschild (co)homology of the second kind of a CDG-category B and the same of the DG category C of right CDG-modules over B, ...
Added: June 27, 2012