Questions tagged [brauer-groups]

Questions concerning Brauer groups of fields, rings, varieties, schemes or more general ringed spaces, invariants associated to Brauer classes such as index and period.

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38 votes
1 answer
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Why is there no Brauer scheme?

Let $X$ be a proper scheme over a base field $k$ (one could consider more general settings, but I am primarly interested in a "geometric" situation with $k$ being algebraically closed). Then the ...
user25309's user avatar
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6 votes
4 answers
3k views

Brauer group of projective space

I've read that $\text{Br} \mathbb{P}^n_k$ (here $\text{Br}$ is the cohomological Brauer group, i.e. $H^2_{ét}(-,\mathbb{G}_m)$) is just isomorphic to $\text{Br} k$. As proof of this fact seems to be ...
Wanderer's user avatar
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29 votes
2 answers
2k views

Making $\mathbb{Q}$-cohomology integral

Let $X$ be an algebraic variety (say, smooth and projective) over $\mathbb{C}$, and fix $$\alpha\in H^i(X^{\text{an}}, \mathbb{Q})$$ with $i>0$. Does there always exist a variety $Y$ and a ...
Daniel Litt's user avatar
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9 votes
1 answer
672 views

Software for detecting Brauer-Manin obstructions?

In the context of another MO question, the following question arose: Does there exist any software for detecting Brauer–Manin obstructions to the existence of integer solutions to a single polynomial ...
Timothy Chow's user avatar
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6 votes
0 answers
669 views

Brauer group of a rational variety

This is a follow-up question to this question. There and here $X$ is a normal projective rational surface over $\mathbb{C}$ with finitely generated divisor class group $\text{Cl}(X)$. My question is: ...
user269218's user avatar
5 votes
1 answer
229 views

Countably many isomorphism classes of reductive groups over a field with countable Brauer and Witt groups

Assume a field has a countable Brauer group and a countable Witt group. Are there countably many isomorphism classes of reductive groups over it?
guriz's user avatar
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5 votes
0 answers
484 views

Brauer groups of a local ring and of its residue field

This is a question of DeMeyer (see the last paragraph of [1]): What's an example of a local ring $A$ with residue field $k$ such that the restriction map on Brauer groups $\varphi : \operatorname{...
Minseon Shin's user avatar
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4 votes
2 answers
521 views

Brauer group of $\mathbb{Z}_{(p)}$

This may be a well known result but I could not find it in the standard references. What is the Brauer group of the local ring $\mathbb{Z}_{(p)}$ (the ring of integers localized at $p$)?
user123's user avatar
  • 81
2 votes
0 answers
158 views

Interpretation of some maps involving cohomology groups

I've asked some questions on Math Stackexchange regarding areas around my research but I received very little success with responses, so I thought I might try to share some of my other problems here ...
oleout's user avatar
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0 votes
1 answer
1k views

Explicit bijection between Azumaya algebras and Brauer-Severi schemes

This is kind of the relative version of this question. Even though I made extensive enquiries, I couldn't find good references for this and it seems to me that these questions are pretty well ...
Ben's user avatar
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