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

TL;DR

This paper develops a new functor for algebras and schemes, introducing homotopy invariants and cohomology theories, and explores their applications in algebraic K-theory and related areas.

Contribution

It introduces a novel functor from algebra pairs to pro-algebras, extending classical concepts like $oldsymbol{ ext{pi}_0}$ and developing homotopy invariants for affine schemes.

Findings

Defined a functor $rak{M}$ for algebra pairs and schemes.

Established a homotopy invariant cohomology theory with cup product.

Connected homotopy pro-algebras to KK and K-groups.

Abstract

We introduce a functor $\mathfrak{M}:\mathbf{Alg}\times\mathbf{Alg}^\mathrm{op}\rightarrow\mathrm{pro}\text{-}\mathbf{Alg}$ 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 Corti\~{n}as-Thom's KK-groups and Weibel's homotopy K-groups.