63 research outputs found
On the K-theory of planar cuspical curves and a new family of polytopes
Let k be a regular F_p-algebra, let A = k[x,y]/(x^b - y^a) be the coordinate
ring of a planar cuspical curve, and let I = (x,y) be the ideal that defines
the cusp point. We give a formula for the relative K-groups K_q(A,I) in terms
of the groups of de Rham-Witt forms of the ring k. At present, the validity of
the formula depends on a conjecture that concerns the combinatorial structure
of a new family of polytopes that we call stunted regular cyclic polytopes. The
polytopes in question appear as the intersections of regular cyclic polytopes
with (certain) linear subspaces. We verify low-dimensional cases of the
conjecture. This leads to unconditional new results on K_2 and K_3 which extend
earlier results by Krusemeyer for K_0 and K_1.Comment: 38 page
Topological Hochschild homology and the Hasse-Weil zeta function
We consider the Tate cohomology of the circle group acting on the topological
Hochschild homology of schemes. We show that in the case of a scheme smooth and
proper over a finite field, this cohomology theory naturally gives rise to the
cohomological interpretation of the Hasse-Weil zeta function by regularized
determinants envisioned by Deninger. In this case, the periodicity of the zeta
function is reflected by the periodicity of said cohomology theory, whereas
neither is periodic in general
On the K-theory of truncated polynomial algebras over the integers
We show that the K_{2i}(Z[x]/(x^m),(x)) is finite of order (mi)!(i!)^{m-2}
and that K_{2i+1}(Z[x]/(x^m),(x)) is free abelian of rank m-1. This is
accomplished by showing that the equivariant homotopy groups of the topological
Hochschild spectrum THH(Z) are finite, in odd degrees, and free abelian, in
even degrees, and by evaluating their orders and ranks, respectively.Comment: Journal of Topology (to appear
On the K-theory of local fields
The authors establish a connection between the Quillen K-theory of certain
local fields and the de Rham-Witt complex of their rings of integers with
logarithmic poles at the maximal ideal. They consider fields K that are
complete discrete valuation fields of characteristic zero with perfect residue
fields k of characteristic p > 2. They evaluate the K-theory with
Z/p^v-coefficients of K, and verify the Lichtenbaum-Quillen conjecture for K.Comment: Abstract added in migration; 113 pages, published versio
Algebraic -theory of planar cuspidal curves
In this paper, we evaluate the algebraic -groups of a planar cuspidal
curve over a perfect -algebra relative to the cusp point. A
conditional calculation of these groups was given earlier by Hesselholt,
assuming a conjecture on the structure of certain polytopes. Our calculation
here, however, is unconditional and illustrates the advantage of the new setup
for topological cyclic homology by Nikolaus-Scholze, which is used throughout.
The only input necessary for our calculation is the evaluation by the Buenos
Aires Cyclic Homology group and by Larsen of the structure of Hochschild
complex of the coordinate ring as a mixed complex, that is, as an object of the
infinity category of chain complexes with circle action.Comment: 9 page
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