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Register a new userThe $K$-theory of toric schemes over regular rings of mixed characteristic
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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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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.
Let $p$ be a prime. Let $V$ be a discrete valuation ring of mixed characteristic $(0,p)$ and index of ramification $e$. Let $f: G rightarrow H$ be a homomorphism of finite flat commutative group schemes of $p$ power order over $V$ whose generic fiber is an isomorphism. We provide a new proof of a result of Bondarko and Liu that bounds the kernel and the cokernel of the special fiber of $f$ in terms of $e$. For $e < p-1$ this reproves a result of Raynaud. Our bounds are sharper that the ones of Liu, are almost as sharp as the ones of Bondarko, and involve a very simple and short method. As an application we obtain a new proof of an extension theorem for homomorphisms of truncated Barsotti--Tate groups which strengthens Tates extension theorem for homomorphisms of $p$-divisible groups.
Let $p$ be a prime. Let $(R,ideal{m})$ be a regular local ring of mixed characteristic $(0,p)$ and absolute index of ramification $e$. We provide general criteria of when each abelian scheme over $Spec Rsetminus{ideal{m}}$ extends to an abelian scheme over $Spec R$. We show that such extensions always exist if $ele p-1$, exist in most cases if $ple ele 2p-3$, and do not exist in general if $ege 2p-2$. The case $ele p-1$ implies the uniqueness of integral canonical models of Shimura varieties over a discrete valuation ring $O$ of mixed characteristic $(0,p)$ and index of ramification at most $p-1$. This leads to large classes of examples of Neron models over $O$. If $p>2$ and index $p-1$, the examples are new.
We study the algebraic $K$-theory and Grothendieck-Witt theory of proto-exact categories of vector bundles over monoid schemes. Our main results are the complete description of the algebraic $K$-theory space of an integral monoid scheme $X$ in terms of its Picard group $operatorname{Pic}(X)$ and pointed monoid of regular functions $Gamma(X, mathcal{O}_X)$ and a description of the Grothendieck-Witt space of $X$ in terms of an additional involution on $operatorname{Pic}(X)$. We also prove space-level projective bundle formulae in both settings.
We introduce a functor $mathfrak{M}:mathbf{Alg}timesmathbf{Alg}^mathrm{op}rightarrowmathrm{pro}text{-}mathbf{Alg}$ constructed from representations of $mathrm{Hom}_mathbf{Alg}(A,Botimes ?)$. As applications, the following items are introduced and studied: (i) Analogue of the functor $pi_0$ for algebras and affine schemes. (ii) Cotype of Weibels concept of strict homotopization. (iii) A homotopy invariant intrinsic singular cohomology theory for affine schemes with cup product. (iv) Some extensions of $mathbf{Alg}$ that are enriched over idempotent semigroups. (v) Classifying homotopy pro-algebras for Corti~{n}as-Thoms KK-groups and Weibels homotopy K-groups.
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