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In a paper I need to make reference to two conjectures by Gabber, from

  • Ofer Gabber, On purity for the Brauer group, in: Arithmetic Algebraic Geometry, MFO Report No. 37/2004, doi:10.14760/OWR-2004-37

(see Conjectures 2 and 3, page 1975)

  1. Let $R$ be a strictly henselian complete intersection noetherian local ring of dimension at least 4. Then $Br'(U_R) = 0$ (the cohomological Brauer group of the punctured spectrum is $0$).

  2. Let $R$ be a complete intersection noetherian local ring of dimension 3. Then $Pic(U_R)$ is torsion-free.

Does anyone know of any new developments on these conjectures beyond the Oberwolfach report above? I tried MathScinet but could not find anything. May be someone in the Arithmetic Geometry community happen to know some news on these? Thanks a bunch.

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    $\begingroup$ Have you tried e-mailing Gabber with your question? $\endgroup$
    – Ryan Budney
    Commented Jan 3, 2010 at 6:13
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    $\begingroup$ I am doing it right now! Stay tuned (: $\endgroup$
    – Hailong Dao
    Commented Jan 3, 2010 at 6:18
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    $\begingroup$ I'd also like to know more about this. Progress on the purity conjecture for the Brauer group would be really interesting! $\endgroup$
    – Clark Barwick
    Commented Jan 18, 2010 at 15:26

1 Answer 1

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Happy news: there have been some progress over the last 10 years. The hypersurface case of Conjecture 2 was proved in

  • H. Dao, Picard groups of punctured spectra of dimension three local hypersurfaces are torsion-free. Compositio Mathematica, 148(1) (2012) pp. 145-152. doi:10.1112/S0010437X11005513, arXiv:1004.0471,

Česnavičius and Scholze recently settled both Conjectures:

  • Kestutis Cesnavicius, Peter Scholze, Purity for flat cohomology, arXiv:1912.10932.
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