Name: Joseph Gubeladze
Title: Higher K-theory of toric varieties
Abstract:
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Abstact: A natural higher $K$-theoretic analogue of the triviality of vector
bundles on affine toric varieties is the conjecture on nilpotence of the
multiplicative action of the natural numbers on the $K$-theory of these
varieties. This includes both Quillen's fundamental result on $K$-homotopy
invariance of regular rings and the stable version of the triviality of
vector bundles on affine toric varieties. Moreover, it yields a similar
behavior of not necessarily affine toric varieties and, further, of their
equivariant closed subsets.
The conjecture is equivalent to the claim that
the relevant admissible morphisms of the category of vector bundles on an
affine toric variety can be supported by monomials not in a non-degenerate
corner subcone of the underlying polyhedral cone. Using Suslin-Wodzicki's
solution to the excision problem and Thomason's localization technique for
singular varieties, in combination with our previous results in this direction,
we prove that one can in fact eliminate all lattice points in such a subcone,
except maybe one point. The elimination of the last point is also possible
in 0 characteristic if the action of the big Witt vectors satisfies a very
natural condition. A partial result on this question (based on Goodwillie's
theorem) in the arithmetic case provides first non-simplicial
examples -- actually, an explicit infinite series of combinatorially different
affine toric varieties, verifying the conjecture for all higher groups
simultaneously.
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