Path-Connected Components of Affine Schemes and Algebraic K-Theory

Abstract

We introduce a functor $\mathfrak{M}:\mathbf{Alg}\times {\mathbf{Alg}}^{\mathrm{op}}\to \mathrm{pro}\text{-}\mathbf{Alg}$ constructed from representations of ${\mathrm{Hom}}_{\mathbf{Alg}}(A,B\otimes ?)$. As applications, the following items are introduced and studied: (i) Analogue of the functor ${\pi }_0$ for algebras and affine schemes. (ii) Cotype of Weibel's concept of strict homotopization. (iii) A homotopy invariant intrinsic singular cohomology theory for affine schemes with cup product. (iv) Some extensions of $\mathbf{Alg}$ that are enriched over idempotent semigroups. (v) Classifying homotopy pro-algebras for Cortiñas-Thom's KK-groups and Weibel's homotopy K-groups.

37 pages. The author would like to express his sincere gratitude to Professor Cortinas for many valuable comments, hints, and remarks on the early version of this manuscript

References in corpus (1)

• On the structure of noncommutative mapping schemes