K-theory of Matroids and Monoid Schemes
This paper continues the study of the $K$-theory of monoid schemes, using it to give a useful definition of the higher $K$-theory of a matroid via its Bergman fan.
math.KT↗
arXiv subjects
Charles A. Weibel
Publications and source records attributed to Charles A. Weibel.
$K_2$-regularity and normality
We take a fresh look at the relationship between $K$-regularity and regularity of schemes, proving two results in this direction. First, we show that $K_2$-regular affine algebras over fields of characteristic zero are normal. Second, we improve on Vorst's $K$-regularity bound in the case of local complete intersections; this is related to recent work on higher du Bois singularities.
math.AG↗
The $K'$- theory of monoid sets
This paper studies the K-theory of categories of partially cancellative monoid sets, which is better behaved than that of all finitely generated monoid sets. A number of foundational results are proved, making use of the formalism of CGW-categories due to Campbell and Zakharevich, and numerous example computations are provided.
math.KT↗
K-theory of line bundles and smooth varieties
We give a $K$-theoretic criterion for a quasi-projective variety to be smooth. If $\mathbb{L}$ is a line bundle corresponding to an ample invertible sheaf on $X$, it suffices that $K_q(X) = K_q(\mathbb{L})$ for all $q\le\dim(X)+1$.
math.KT↗
Toric varieties, monoid schemes and $cdh$ descent
We give conditions for the Mayer-Vietoris property to hold for the algebraic K-theory of blow-up squares of toric varieties in any characteristic, using the theory of monoid schemes. These conditions are used to relate algebraic K-theory to topological cyclic homology in characteristic p. To achieve our goals, we develop for monoid schemes many notions from classical algebraic geometry, such as separated and proper maps.
math.KT↗
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.
math.KT↗
$K$-theory of cones of smooth varieties
Let $R$ be the homogeneous coordinate ring of a smooth projective variety $X$ over a field $\k$ of characteristic~0. We calculate the $K$-theory of $R$ in terms of the geometry of the projective embedding of $X$. In particular, if $X$ is a curve then we calculate $K_0(R)$ and $K_1(R)$, and prove that $K_{-1}(R)=\oplus H^1(C,\cO(n))$. The formula for $K_0(R)$ involves the Zariski cohomology of twisted Kähler differentials on the variety.
math.KT↗
Norm Varieties and the Chain Lemma (after Markus Rost)
The goal of this paper is to present proofs of two results of Markus Rost: the Chain Lemma and the Norm Principle. These are the final steps needed to complete the publishable verification of the Bloch-Kato conjecture, that the norm residue maps are isomorphisms between Milnor K-theory $K_n^M(k)/p$ and etale cohomology $H^n(k,μ_p^n)$ for every prime p, every n and every field k containing 1/p. Our proofs of these two results are based on Rost's 1998 preprints, his web site and Rost's lectures at the Institute for Advanced Study in 1999-2000 and 2005.
math.KT↗
The Artinian Berger Conjecture
We propose an Artinian version of Berger's Conjecture for curves, concerning the module of Kähler differentials of an algebra. Our version implies Berger's Conjecture in characteristic 0. We establish our Artinian Berger Conjecture in a number of cases, and prove that Berger's Conjecture holds for curve singularities whose conductor ideal contains the cube of a maximal ideal.
math.AC↗