Holmstrom Cyclic homology

Cyclic homology

Nistor on the cyclic homology of Hecke algebras

---

Cyclic homology

Connes defined cyclic homology. I think amother definition was given independently by Loday-Quillen and Feigin-Tsygan, that agrees with Connes’ original definition rationally.

---

Cyclic homology

MathSciNet

Google Scholar

Google

arXiv: Experimental full text search

arXiv: Abstract search

---

Cyclic homology

NCG (Algebra and noncommutative geometry)

---

Cyclic homology

See also Negative cyclic homology

---

Cyclic homology

For cyclic homology pf schemes, see Weibel, and also http://www.math.uiuc.edu/K-theory/0046 and Keller

For cyclic homology of exact categories, see Keller

Cyclic homology of commutative algebras: Cortinas

Something on cyclic homology of A_infty-algebras, by Khalkhali.

Foliation groupoids and their cyclic homology, by Marius Crainic and Ieke Moerdijk: http://www.math.uiuc.edu/K-theory/0446

Page 33 of Weibel

Puschnigg on excision

cycle category in nlab

Weibel, C.: The Hodge filtration and cyclic homology. Preprint, uiuc K-theory archive (1994)

Various articles of Kaledin might be interesting, including something about coefficients of cyclic homology.

Not so relevant: http://www.math.uiuc.edu/K-theory/0024, http://www.math.uiuc.edu/K-theory/0275

---

Cyclic homology

Kazhdan et al: http://www.math.uiuc.edu/K-theory/0222

Cyclic homology: Various papers by Cortinas, see MR

The obstruction to excision in K-theory and in cyclic homology, by Guillermo Cortiñas: http://www.math.uiuc.edu/K-theory/0524

Cyclic homology, cdh-cohomology and negative K-theory , by Guillermo Cortinas , Christian Haesemeyer , Marco Schlichting , and Charles A. Weibel: http://www.math.uiuc.edu/K-theory/0722

Lodder on comparison with Leibniz homology

Jones: Cyclic homology and equivariant homology (1987)

Kassel: Cyclic homology, comodules, and mixed complexes (1987)

Toen: Algebres simplicicales etc, file Toen web prepr rhamloop.pdf. Comparison between functions on derived loop spaces and de Rham theory. Take a smooth k-algebra, k aof char zero. Then (roughly) the de Rham algebra of A and the simplical algebra $S^1 \otimes A$ determine each other (functorial equivalence). Consequence: For a smooth k-scheme $X$, the algebraic de Rham cohomology is identified with $S^1$-equivariant functions on the derived loop space of $X$. Conjecturally this should follow from a more general comparison between functions on the derived loop space and cyclic homology. Also functorial and multiplicative versions of HKR type thms on decompositions of Hochschild cohomology, for any separated k-scheme.

---

Cyclic homology

Dwyer, Hopkins, Kan: The homotopy theory of cyclic sets. Model structure on the cat of cyclic sets, with a homotopy cat equivalent to that of spaces with a circle action, or to spaces over K(Z, 2). “This places cyclic homology in algebraic topology” or something like that

http://ncatlab.org/nlab/show/cyclic+operad

---

Cyclic homology

Something about pro-complexes: http://www.math.uiuc.edu/K-theory/0515

---

Matthew Morrow email Sep 2012: These days I work in “large enough characteristic”, which usually eventually mean characteristic 0. I am interested mainly in “pro-excision”, which means the following: If A –> B is a morphism of rings carrying an ideal J of A isomorphically to an ideal of B (e.g. normalise A and let J be the conductor), then it is well known that in general there isn’t a Mayer-Vietoris sequence relating the K-theory (or HH, or HC) of the rings A, B, A/J, B/J. A trend in recent years has been to take the limit over all powers of J and see what happens. I can show that one often gets a Mayer-Vietoris sequence of formal pro groups. Weaker results have already had many interesting applications to algebraic cycles on singular varieties, so that’s what I am looking at now. It’s really not number theory…

---

Does there exist a general statement of the universal coefficient theorem in the context of cyclic homology or Hochschild? Loday states a result along the lines $HC_*(A | k) \otimes_k K \cong HC_*(A | K)$ in the case that $K$ is a localization of $k$ and $A$ is a $K$-module, flat over $k$. However, the result can be shown in greater generality:

If $A$ is flat over $k$, there is a spectral sequence with $E^2_{p,q} := Tor^k_p( HC_q(A | k), B )$ converging to $HC_*(A \otimes_k B | B)$. (The proof is essentially the same as the proof that this works in the case of symmetric homology, as we recently discovered in the course of my thesis research).

If anyone knows of the general result (UCT for HC or HH) somewhere in print, I’d be much obliged for the reference.

-Shaun Ault The Ohio State University

---

---

Cyclic homology

arXiv:1211.1813 Pro excision and h-descent for K-theory fra arXiv Front: math.AG av Matthew Morrow In this paper it is proved that K-theory (and Hochschild and cyclic homology) satisfies pro versions of both excision for ideals (of commutative Noetherian rings) and descent in the h-topology in characteristic zero; this is achieved by passing to the limit over all infinitesimal thickenings of the ideal or exceptional fibre in question.

---

Cyclic homology

Chapter 9 in Weibel: An introduction to homological algebra.

Loday: Cyclic homology. Springer, 1992.

Connes: Noncommutative differential geometry (1985) (online?)

Loday, Quillen: Cyclic homology and the Lie algebra homology of matrices (1984)

Tsygan: Homology of matrix algebras over rings and the Hochschild homology (1983)

Hesselholt in K-theory handbook.

See maybe things in the folder Noncomm geom and Cstar-alg.

Loday: Cyclic homology. In Noncomm geom folder. Covers many aspects of Cyclic and Hochschild homology. Among many other topics: Secondary char classes (section 11.5), Homology of small categories (App C), periodic and negative cyclic homology, Andre-Quillen homology, Deligne cohomology. For the latter, the main point is that there a cyclic homology complex of Connes which computes integral coeffs reduced Deligne cohomology but which has strictly commutative products!! This is stated for smooth algebras over C, not sure if it can be generalized to more general schemes. Also I am not sure if this has any relevance for non-reduced Deligne cohomology.

nLab page on Cyclic homology