1,090 research outputs found
The K-theory of toric varieties in positive characteristic
We show that if X is a toric scheme over a regular ring containing a field
then the direct limit of the K-groups of X taken over any infinite sequence of
nontrivial dilations is homotopy invariant. This theorem was known in
characteristic 0. The affine case of our result was conjectured by Gubeladze.Comment: Companion paper to arXiv:1106.138
The basic geometry of Witt vectors, I: The affine case
We give a concrete description of the category of etale algebras over the
ring of Witt vectors of a given finite length with entries in an arbitrary
ring. We do this not only for the classical p-typical and big Witt vector
functors but also for variants of these functors which are in a certain sense
their analogues over arbitrary local and global fields. The basic theory of
these generalized Witt vectors is developed from the point of view of commuting
Frobenius lifts and their universal properties, which is a new approach even
for the classical Witt vectors. The larger purpose of this paper is to provide
the affine foundations for the algebraic geometry of generalized Witt schemes
and arithmetic jet spaces. So the basics here are developed somewhat fully,
with an eye toward future applications.Comment: Final versio
Quasi-coherent Hecke category and Demazure Descent
Let G be a reductive algebraic group with a Borel subgroup B. We define the
quasi-coherent Hecke category for the pair (G,B). For any regular Noetherian
G-scheme X we construct a monoidal action of the Hecke category on the derived
category of B-equivariant quasi-coherent sheaves on X. Using the action we
define the Demazure Descent Data on the latter category and prove that the
Descent category is equivalent to the derived category of G-equivariant sheaves
on X.Comment: 12 pages. Changes suggested by the refere
The Steinberg group of a monoid ring, nilpotence, and algorithms
For a regular ring R and an affine monoid M the homotheties of M act
nilpotently on the Milnor unstable groups of R[M]. This strengthens the K_2
part of the main result of [G5] in two ways: the coefficient field of
characteristic 0 is extended to any regular ring and the stable K_2-group is
substituted by the unstable ones. The proof is based on a
polyhedral/combinatorial techniques, computations in Steinberg groups, and a
substantially corrected version of an old result on elementary matrices by
Mushkudiani [Mu]. A similar stronger nilpotence result for K_1 and algorithmic
consequences for factorization of high Frobenius powers of invertible matrices
are also derived.Comment: final version, to appear in Journal of Algebr
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