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Register a new userPurity results for $p$-divisible groups and abelian schemes over regular bases of mixed characteristic
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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.
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The isomorphism number (resp. isogeny cutoff) of a p-divisible group D over an algebraically closed field is the least positive integer m such that D[p^m] determines D up to isomorphism (resp. up to isogeny). We show that these invariants are lower semicontinuous in families of p-divisible groups of constant Newton polygon. Thus they allow refinements of Newton polygon strata. In each isogeny class of p-divisible groups, we determine the maximal value of isogeny cutoffs and give an upper bound for isomorphism numbers, which is shown to be optimal in the isoclinic case. In particular, the latter disproves a conjecture of Traverso. As an application, we answer a question of Zink on the liftability of an endomorphism of D[p^m] to D.
For a scheme $X$ defined over the length $2$ $p$-typical Witt vectors $W_2(k)$ of a characteristic $p$ field, we introduce total $p$-differentials which interpolate between Frobenius-twisted differentials and Buiums $p$-differentials. They form a sheaf over the reduction $X_0$, and behave as if they were the sheaf of differentials of $X$ over a deeper base below $W_2(k)$. This allows us to construct the analogues of Gauss-Manin connections and Kodaira-Spencer classes as in the Katz-Oda formalism. We make connections to Frobenius lifts, Borger-Weilands biring formalism, and Deligne--Illusie classes.
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.
Let $k$ be a perfect field of characteristic $p geq 3$. We classify $p$-divisible groups over regular local rings of the form $W(k)[[t_1,...,t_r,u]]/(u^e+pb_{e-1}u^{e-1}+...+pb_1u+pb_0)$, where $b_0,...,b_{e-1}in W(k)[[t_1,...,t_r]]$ and $b_0$ is an invertible element. This classification was in the case $r = 0$ conjectured by Breuil and proved by Kisin.
Let $p$ be a prime. Let $R$ be a regular local ring of dimension $dge 2$ whose completion is isomorphic to $C(k)[[x_1,ldots,x_d]]/(h)$, with $C(k)$ a Cohen ring with the same residue field $k$ as $R$ and with $hin C(k)[[x_1,ldots,x_d]]$ such that its reduction modulo $p$ does not belong to the ideal $(x_1^p,ldots,x_d^p)+(x_1,ldots,x_d)^{2p-2}$ of $k[[x_1,ldots,x_d]]$. We extend a result of Vasiu-Zink (for $d=2$) to show that each Barsotti-Tate group over $text{Frac}(R)$ which extends to every local ring of $text{Spec}(R)$ of dimension $1$, extends uniquely to a Barsotti-Tate group over $R$. This result corrects in many cases several errors in the literature. As an application, we get that if $Y$ is a regular integral scheme such that the completion of each local ring of $Y$ of residue characteristic $p$ is a formal power series ring over some complete discrete valuation ring of absolute ramification index $ele p-1$, then each Barsotti-Tate group over the generic point of $Y$ which extends to every local ring of $Y$ of dimension $1$, extends uniquely to a Barsotti-Tate group over $Y$.
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