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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.
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 affi
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
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 schem
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
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 stu