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In a paper I need to make reference to 2 conjectures by Gabber

(see Conjectures 2 and 3, page 1975)

http://www.mfo.de/programme/schedule/2004/32/OWR_2004_37.pdf

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.

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.

In a paper I need to make reference to 2 conjectures by Gabber

(see Conjectures 2 and 3, page 1975)

http://www.mfo.de/programme/schedule/2004/32/OWR_2004_37.pdf

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. Of course, Gabber does not normally write up his results, so may be someone in the Arithmetic Geometry community happen to know some news on these? Thanks a bunch.

In a paper I need to make reference to 2 conjectures by Gabber

(see Conjectures 2 and 3, page 1975)

http://www.mfo.de/programme/schedule/2004/32/OWR_2004_37.pdf

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.