Holmstrom Purity

Mentioned in Geisser in K-theory handbook. Also I think in Levine’s chapter, under Bloch-Ogus axioms.

Fujiwara: A proof of the absolute purity conjecture (following Gabber)

Mentioned by Scholl in some meeeting Oct/Nov 2008

Title: Notions of purity and the cohomology of quiver moduli Authors: Michel Brion, Roy Joshua http://front.math.ucdavis.edu/1205.0629

Scholl-Deninger: The Beilinson conjectures, p 5 in the double page version. Says that weak purity for Deligne cohomology holds, meaning that for a closed subscheme Y of X of pure codim q, we have $H^p_Y(X, R(q) ) = 0$ for $p 2q$.

arXiv:0909.0969 Purity results for $p$-divisible groups and abelian schemes over regular bases of mixed characteristic from arXiv Front: math.AG by Adrian Vasiu, Thomas Zink 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 $\Spec R\setminus{\ideal{m}}$ extends to an abelian scheme over $\Spec R$. We show that such extensions always exist if $e\le p-1$, exist in most cases if $p\le e\le 2p-3$, and do not exist in general if $e\ge 2p-2$. The case $e\le p-1$ 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 Néron models over $O$. If $p>2$ and index $p-1$, the examples are new.

arXiv:1108.6250 Proof of a conjecture of Colliot-Thélène from arXiv Front: math.AG by Jan Denef We prove a conjecture of Colliot-Thélène that implies the Ax-Kochen Theorem on p-adic forms. We obtain it as an easy consequence of a diophantine purity theorem whose proof forms the body of the present paper.

nLab page on Purity