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](https://doi.org/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.



Two conjectures by Gabber on Brauer and Picard groups
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](https://doi.org/10.1112/S0010437X11005513), arXiv:[1004.0471](https://arxiv.org/abs/1004.0471),


Česnavičius and Scholze recently settled both Conjectures:


* Kestutis Cesnavicius, Peter Scholze, *Purity for flat cohomology*, arXiv:[1912.10932](https://arxiv.org/abs/1912.10932).


