nLab additive K-theory

Term additive K-theory is a synonym for cyclic homology, used in early articles mainly of Russian and a bit by French school. Sometimes, additive K-theory denotes more specifially a different packing of cyclic homology, namely with appropriately shifted degrees (see the reference by Loday-Quillen at Loday-Quillen-Tsygan theorem and the article by Kuribayashi below).

Additive K-theory is also a title of a historical article

• Boris Tsygan, Boris Feigin, Additive K-theory, in K-theory, arithmetic and geometry, LNM 1289 (1987), edited by Yu. I. Manin, pp. 67–209, seminar 1984-1986 in Moscow), MR89a:18017

The additive K-theory is here studied in relation to the algebraic K-theory and Hochschild homology. Like there is a K-theory spectrum, one also constructs an additive K-theory/cyclic homology spectrum.

Contents of Tsygan-Feigin

• Introduction.
• Ch. 1. Additive K-functors.
• Ch. 2. Derived functors and relative additive K-functors.
• Ch. 3. Generalized free products.
• Ch. A. Lie algebra homology.
• Ch. 5. Operations in additive K-theory.
• Ch. 6. Additive K-functors of the commutative noetherian algebras.
• Ch. 7. Characteristic classes.
• Appendix. Cyclic objects.

Related articles include

• Б. Л. Фейгин, Б. Л. Цыган, “Аддитивная K-теория и кристальные когомологии”, Функц. анализ и его прил., 19:2 (1985), 52–-62, pdf, MR88e:18008; Engl. transl. in B. L. Feĭgin, B. L. Tsygan, Additive $K$-theory and crystalline cohomology, Functional Analysis and Its Applications, 1985, 19:2, 124–132.

• Katsuhiko Kuribayashi, Toshihiro Yamaguchi, On additive K-theory with the Loday-Quillen $\ast$-product, Math. Scand. 87, No 1 (2000) article doi