Infinitesimal K-theory

Abstract

In section 1 we develop the language of categories with deformations. In section 2 we construct the localization C —> C[Def-1] for a category with deformations. A general criterion for the existence of derived functors with respect to this localization is established in section 3. In section 4 we show that both non-commutative de Rham cohomology and the rational 4—construction of the elementary group meet this criterion, and compute their derived functors. The sheaf theoretic approach is developed in section 5 where the character cr mentioned above is constructed. The isomorphism (6) is proven in section 6. Also in this section we conjecture that Hn(A, K®) = 0 for positive n, and show that this conjecture is related to finding a non commutative analogue of Grothendieck’s isomorphism (4). The map (1) is constructed in section 7; a more concrete interpretation of Free'nf as a category of integrable connections with as maps the gauge transformations is discussed.