2001/04/17 by Joseph Gubeladze, Gubeladze, Joseph · 1 citation
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #14F05 #14M25 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Alkaloids: synthesis and pharmacology #Commutative Algebra (math.AC) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Primary 19E08 #Secondary 13D15 #math.AC #math.AG #math.KT #msc:13D15 #msc:14F05 #msc:14M25 #msc:19E08
paper · pdf · doi:10.48550/arxiv.math/0104166
Final version, to appear in K-Theory
openalex publication_date 2001/04/17 · arxiv created 2003/04/09 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. 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 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.