Additive K-theory is a concept in mathematics that refers to a version of algebraic K-theory in which, according to Spencer Bloch, the general linear group GL has been replaced everywhere by its Lie algebra gl. It is not a single theory but rather a method for creating additive or infinitesimal analogues of multiplicative K-theories.
Formulation
Following Boris Feigin and Boris Tsygan, let $A$ be an algebra over a field $k$ of characteristic zero and let $\mathfrak{gl}(A)$ be the algebra of infinite matrices over $A$ with only finitely many nonzero entries. The Lie algebra homology
$$ H_{\cdot}(\mathfrak{gl}(A), k) $$
has a natural structure of a Hopf algebra. The space of its primitive elements of degree $i$ is denoted by $K_i^{+}(A)$ and called the $i$-th additive K-functor of $A$.
The additive K-functors are related to cyclic homology groups by the isomorphism
$$ HC_i(A) \cong K_{i+1}^{+}(A). $$
Relation to Cyclic Homology
According to the nLab, the term "additive K-theory" is also used as a synonym for cyclic homology, particularly in early articles from the Russian and French schools. More specifically, additive K-theory can denote a different packing of cyclic homology with appropriately shifted degrees (see the Loday–Quillen–Tsygan theorem).
History
The concept was developed in a foundational article by Boris Feigin and Boris Tsygan, Additive K-theory, published in K-Theory, Arithmetic and Geometry (Lecture Notes in Mathematics 1289, Springer, 1987, edited by Yu. I. Manin, pp. 67–209). The work studies additive K-theory in relation to algebraic K-theory and Hochschild homology, and constructs an additive K-theory (or cyclic homology) spectrum.
Related Concepts
- Additive Chow groups: Spencer Bloch has also explored connections between additive K-theory and additive Chow groups.
- Cathelineau's version and Bloch–Esnault's version: Two infinitesimal (additive) versions of K-theory of a field exist, each equipped with a regulator map when the field is the field of complex numbers.
References
- Bloch, Spencer (2006-07-23). "Algebraic Cycles and Additive Chow Groups" (PDF). Dept. of Mathematics, University of Chicago.
- Feigin, B.; Tsygan, B. Additive K-theory, LNM 1289, Springer.
- Feigin, B. L.; Tsygan, B. L. "Additive K-theory and crystalline cohomology", Functional Analysis and Its Applications, 1985, 19:2, 124–132.
