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.
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arXiv subjects
Christian Haesemeyer
Publications and source records attributed to Christian Haesemeyer.
$K_2$-regularity is equivalent to smoothness for toric varieties
Recent work by Wanchun Shen shows that $K_2$-regular affine local complete intersections of any dimension in characteristic zero are smooth. In this note, we show that the same holds for affine toric varieties.
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$K_2$-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.
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The passage among the subcategories of weakly approximable triangulated categories
In this article we prove that all the inclusions between the 'classical' and naturally defined full triangulated subcategories of a weakly approximable triangulated category are intrinsic (in one case under a technical condition). This extends all the existing results about subcategories of weakly approximable triangulated categories. Together with a forthcoming paper about uniqueness of enhancements, our result allows us to generalize a celebrated theorem by Rickard which asserts that if $R$ and $S$ are left coherent rings, then a derived equivalence of $R$ and $S$ is "independent of the decorations". That is, if $D^?(R\text{-}\square)$ and $D^?(S\text{-}\square)$ are equivalent as triangulated categories for some choice of decorations $?$ and $\square$, then they are equivalent for every choice of decorations. But our theorem is much more general, and applies also to quasi-compact and quasi-separated schemes -- even to the relative version, in which the derived categories consist of complexes with cohomology supported on a given closed subscheme with quasi-compact complement.
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On The Topological Period-Index Problem over 8-manifolds
We establish upper bounds of the indices of topological Brauer classes over a closed orientable 8-manifolds. In particular, we verify the Topological Period-Index Conjecture (TPIC) for topological Brauer classes over closed orientable 8-manifolds of order not congruent to 2 mod 4. In addition, we provide a counter-example which shows that the TPIC fails in general for closed orientable 8-manifolds.
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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.
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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 $q\le\dim(X)+1$.
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The $K$-theory of toric schemes over regular rings of mixed characteristic
We show that if $X$ is a toric scheme over a regular commutative ring $k$ then the direct limit of the $K$-groups of $X$ taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was previously known for regular commutative rings containing a field. The affine case of our result was conjectured by Gubeladze. We prove analogous results when $k$ is replaced by an appropriate $K$-regular, not necessarily commutative $k$-algebra.
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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.
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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.
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The 0-th stable A^1-homotopy sheaf and quadratic zero cycles
We study the 0-th stable A^1-homotopy sheaf of a smooth proper variety over a field k assumed to be infinite, perfect and to have characteristic unequal to 2. We provide an explicit description of this sheaf in terms of the theory of (twisted) Chow-Witt groups as defined by Barge-Morel and developed by Fasel. We study the notion of rational point up to stable A^1-homotopy, defined in terms of the stable A^1-homotopy sheaf of groups mentioned above. We show that, for a smooth proper k-variety X, existence of a rational point up to stable A^1-homotopy is equivalent to existence of a 0-cycle of degree 1.
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Stable A^1-homotopy and R-equivalence
We prove that existence of a k-rational point can be detected by the stable A^1-homotopy category of S^1-spectra, or even a "rationalized" variant of this category.
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$K$-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.
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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.
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The $K$-theory of toric varieties
Recent advances in computational techniques for $K$-theory allow us to describe the $K$-theory of toric varieties in terms of the $K$-theory of fields and simple cohomological data.
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