Purity of level m stratifications

Marc-Hubert Nicole; Adrian Vasiu; Torsten Wedhorn

Annales scientifiques de l'École Normale Supérieure (2010)

  • Volume: 43, Issue: 6, page 925-955
  • ISSN: 0012-9593

Abstract

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Let k be a field of characteristic p > 0 . Let D m be a BT m over k (i.e., an m -truncated Barsotti–Tate group over k ). Let S be a k -scheme and let X be a BT m over S . Let S D m ( X ) be the subscheme of S which describes the locus where X is locally for the fppf topology isomorphic to D m . If p ≥ 5 , we show that S D m ( X ) is pure in S , i.e. the immersion S D m ( X ) ↪ S is affine. For p ∈ { 2 , 3 } , we prove purity if D m satisfies a certain technical property depending only on its p -torsion D m [ p ] . For p ≥ 5 , we apply the developed techniques to show that all level m stratifications associated to Shimura varieties of Hodge type are pure.

How to cite

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Nicole, Marc-Hubert, Vasiu, Adrian, and Wedhorn, Torsten. "Purity of level $m$ stratifications." Annales scientifiques de l'École Normale Supérieure 43.6 (2010): 925-955. <http://eudml.org/doc/272165>.

@article{Nicole2010,
abstract = {Let $k$ be a field of characteristic $p&gt;0$. Let $D_m$ be a $\operatorname\{BT\}_m$ over $k$ (i.e., an $m$-truncated Barsotti–Tate group over $k$). Let $S$ be a $k$-scheme and let $X$ be a $\operatorname\{BT\}_m$ over $S$. Let $S_\{D_m\}(X)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_m$. If $p\ge 5$, we show that $S_\{D_m\}(X)$ is pure in $S$, i.e. the immersion $S_\{D_m\}(X) \hookrightarrow S$ is affine. For $p\in \lbrace 2,3\rbrace $, we prove purity if $D_m$ satisfies a certain technical property depending only on its $p$-torsion $D_m[p]$. For $p\ge 5$, we apply the developed techniques to show that all level $m$ stratifications associated to Shimura varieties of Hodge type are pure.},
author = {Nicole, Marc-Hubert, Vasiu, Adrian, Wedhorn, Torsten},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {truncated Barsotti–Tate groups; affine schemes; group actions; $F$-crystals; stratifications; purity; and Shimura varieties},
language = {eng},
number = {6},
pages = {925-955},
publisher = {Société mathématique de France},
title = {Purity of level $m$ stratifications},
url = {http://eudml.org/doc/272165},
volume = {43},
year = {2010},
}

TY - JOUR
AU - Nicole, Marc-Hubert
AU - Vasiu, Adrian
AU - Wedhorn, Torsten
TI - Purity of level $m$ stratifications
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2010
PB - Société mathématique de France
VL - 43
IS - 6
SP - 925
EP - 955
AB - Let $k$ be a field of characteristic $p&gt;0$. Let $D_m$ be a $\operatorname{BT}_m$ over $k$ (i.e., an $m$-truncated Barsotti–Tate group over $k$). Let $S$ be a $k$-scheme and let $X$ be a $\operatorname{BT}_m$ over $S$. Let $S_{D_m}(X)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_m$. If $p\ge 5$, we show that $S_{D_m}(X)$ is pure in $S$, i.e. the immersion $S_{D_m}(X) \hookrightarrow S$ is affine. For $p\in \lbrace 2,3\rbrace $, we prove purity if $D_m$ satisfies a certain technical property depending only on its $p$-torsion $D_m[p]$. For $p\ge 5$, we apply the developed techniques to show that all level $m$ stratifications associated to Shimura varieties of Hodge type are pure.
LA - eng
KW - truncated Barsotti–Tate groups; affine schemes; group actions; $F$-crystals; stratifications; purity; and Shimura varieties
UR - http://eudml.org/doc/272165
ER -

References

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