A1-homotopy invariance of algebraic K-theory with coefficients and Kleinian singularities

TL;DR

This paper extends the A1-homotopy invariance of algebraic K-theory with coefficients from schemes to dg categories, enabling computations for complex structures like Kleinian singularities and dg cluster categories.

Contribution

It generalizes A1-homotopy invariance to dg categories and applies this to compute K-theory for Kleinian singularities and dg cluster categories.

Findings

A1-homotopy invariance holds for dg categories.

Complete K-theory computation for Kleinian singularities.

Vanishing and divisibility properties of algebraic K-theory.

Abstract

C. Weibel and Thomason-Trobaugh proved (under some assumptions) that algebraic K-theory with coefficients is A1-homotopy invariant. In this article we generalize this result from schemes to the broad setting of dg categories. Along the way, we extend Bass-Quillen's fundamental theorem as well as Stienstra's foundational work on module structures over the big Witt ring to the setting of dg categories. Among other cases, the above A1-homotopy invariance result can now be applied to sheaves of (not necessarily commutative) dg algebras over stacks. As an application, we compute the algebraic K-theory with coefficients of dg cluster categories using solely the kernel and cokernel of the Coxeter matrix. This leads to a complete computation of the algebraic K-theory with coefficients of the Kleinian singularities parametrized by the simply laced Dynkin diagrams. As a byproduct, we obtain some vanishing and divisibility properties of algebraic K-theory (without coefficients).