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$\mathbb A^1$-homotopy invariance of algebraic $K$-theory with coefficients and du Val singularities

Gon├žalo Tabuada

Vol. 2 (2017), No. 1, 1ÔÇô25
DOI: 10.2140/akt.2017.2.1

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).

$\mathbb{A}^1$-homotopy, algebraic $K$-theory, Witt vectors, sheaf of dg algebras, dg orbit category, cluster category, du Val singularities, noncommutative algebraic geometry
Mathematical Subject Classification 2010
Primary: 14A22, 14H20, 19E08, 30F50, 13F35
Received: 10 March 2015
Revised: 20 October 2015
Accepted: 4 November 2015
Published: 3 September 2016
Gon├žalo Tabuada
Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139
United States
Centro de Matem├ítica e Aplica├ž├Áes
Departamento de Matemática
Faculdade de Ciências e Tecnologia
Universidade Nova de Lisboa