2008
DOI: 10.1007/s00222-008-0133-y
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Sur la p-dimension des corps

Abstract: Let A be an excellent integral henselian local noetherian ring, k its residue field of characteristic p>0 and K its fraction field. Using an algebraization technique introduced by the first named author, and the one-dimension case already proved by Kazuya KATO, we prove the following formula: cd_p(K) = dim(A) + p-rank(k), if k is separably closed and K of characteristic zero. A similar statement is valid without those assumptions on k and K

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Cited by 15 publications
(8 citation statements)
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“…XVIII].) On the other hand, cd p (K) = 2 was shown by Kato, see [47], or [24] for a more general result. We now use results on Serre's conjecture II: By [28], if H is a semi-simple, simply connected quasi-split reductive group with no E 8 factors, then H 1 (K, H) = ( 1).…”
Section: 22mentioning
confidence: 94%
“…XVIII].) On the other hand, cd p (K) = 2 was shown by Kato, see [47], or [24] for a more general result. We now use results on Serre's conjecture II: By [28], if H is a semi-simple, simply connected quasi-split reductive group with no E 8 factors, then H 1 (K, H) = ( 1).…”
Section: 22mentioning
confidence: 94%
“…By the main result of [GO08] (which is due to [Kat82] in this case), it follows that the Galois group has -cohomological dimension at most . Moreover, by Theorem 4.8, it follows that if is any finite extension of and the ring of integers (which is excellent as a complete local ring), then is -truncated.…”
Section: Pro-galois Descentmentioning
confidence: 93%
“…Working stalkwise on and using the compatibility of étale cohomology with filtered colimits, we can now reduce to the case where is an excellent, strictly henselian normal local domain with residue field (and is contained in the maximal ideal), since excellence and normality are preserved by strict henselization (see [Gre76] for the former). The statement then becomes that for , which follows from the Gabber–Orgogozo bound [GO08, Theorem 6.1] (noting that the -dimension of the residue field is since is separably closed).…”
Section: Asymptotic -Localitymentioning
confidence: 99%
See 1 more Smart Citation
“…If is complete, contains a coefficient field [Mat86, Theorem 28.3], and a choice of a system of parameters yields a finite injective map . By Lemma 2.6 it now suffices to show that But in view of Lemma 2.5 this follows from the fact that if form a -basis of , then form a -basis of and of ; see [GO08, Lemma 2.1.5].…”
Section: Preliminaries On the P-dimensionmentioning
confidence: 99%