  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).
