## A¹-homotopy invariance of algebraic K-theory with coefficients and du Val singularities

##### Author(s)

Trigo Neri Tabuada, Goncalo Jorge
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C. Weibel, and Thomason and Trobaugh, proved (under some assumptions) that algebraic K-theory with coefficients is A1-homotopy invariant. We generalize this result from schemes to the broad setting of dg categories. Along the way, we extend the Bass–Quillen 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 du Val singularities parametrized by the simply laced Dynkin diagrams. As a byproduct, we obtain vanishing and divisibility properties of algebraic K-theory (without coefficients). Keywords: A¹-homotopy, algebraic K-theory, Witt vectors, sheaf of dg algebras, dg orbit category, cluster category, du Val singularities, noncommutative algebraic geometry

##### Date issued

2016-09##### Department

Massachusetts Institute of Technology. Department of Mathematics##### Journal

Annals of K-Theory

##### Publisher

Mathematical Sciences Publishers

##### Citation

Tabuada, Gonçalo. “A¹-homotopy invariance of algebraic K-theory with coefficients and du Val singularities.” Annals of K-Theory 2, 1 (January 2017): 1–25 © 2017 Mathematical Sciences Publishers

Version: Original manuscript

##### ISSN

2379-1691

2379-1683