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Abstract
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C. Weibel, and Thomason and Trobaugh, proved (under some assumptions) that algebraic
-theory with coefficients
is
-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
-homotopy
invariance result can now be applied to sheaves of (not necessarily commutative)
dg algebras over stacks. As an application, we compute the algebraic
-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
-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
-theory
(without coefficients).
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Keywords
$\mathbb{A}^1$-homotopy, algebraic $K$-theory, Witt
vectors, sheaf of dg algebras, dg orbit category, cluster
category, du Val singularities, noncommutative algebraic
geometry
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Mathematical Subject Classification 2010
Primary: 14A22, 14H20, 19E08, 30F50, 13F35
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Milestones
Received: 10 March 2015
Revised: 20 October 2015
Accepted: 4 November 2015
Published: 3 September 2016
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