16
votes
Accepted
Smooth projective models of Severi-Brauer varieties over a DVR are also Severi-Brauer varieties
I wrote up some notes on this in 2004. There have been some developments since then that I will indicate below.
Denote the smooth, proper morphism as follows, $$\pi:\mathcal{X}\to \text{Spec}\ R.$$ ...
12
votes
Accepted
Brauer groups and field extensions
No: the conic $C:X^2+Y^2+1=0$ splits over the field $L=\mathbb{Q}(x)[y]/(x^2+y^2+1)$, since $(X,Y)=(x,y)$ is an $L$-point of $C$. However $L$ has no subfields algebraic over $\mathbb{Q}$ other than $\...
- 4,665
11
votes
Brauer group of a curve over non-algebraically closed field
Just for completeness: The "correct" way to understand the Brauer group of $X$ using its codimension $1$ points is via residue maps.
Specifically: Let $X$ be a regular integral noetherian scheme. ...
- 20.6k
11
votes
Accepted
Software for detecting Brauer-Manin obstructions?
I strongly disagree with the assertion that "the language of modern algebraic geometry [...] is unfamiliar to many people who might otherwise have the right skills to write the software". ...
- 35.8k
11
votes
Accepted
Brauer group of rational numbers
What reference are you reading?
For a field $K$, every finite-dimensional central simple $K$-algebra $A$ is isomorphic to ${\rm M}_n(D)$ where $n$ is a positive integer and $D$ is a division ring with ...
- 49.1k
10
votes
Accepted
Are local fields $C_{2}$?
No: see Guy Terjanian, "Un contre-example à une conjecture d'Artin", C. R. Acad. Sci. Paris Sér. A–B 262 (1966) A612 for an example of homogeneous form of degree $4$ in $18$ variables over the $2$-...
- 28.7k
10
votes
Some questions on division algebras
Questions 2 and 3 are addressed in section 11 of
Auel, Asher; Brussel, Eric; Garibaldi, Skip; Vishne, Uzi, Open problems on central simple algebras., Transform. Groups 16, No. 1, 219-264 (2011). ...
- 1,486
10
votes
Accepted
Do $PGL_n$-torsors induce elements of the Brauer group
Your first statement is not quite true. A $\mathrm{PGL}_2$-torsor does indeed give an element of order $2$ in the Brauer group, but there can be elements of order $2$ in the Brauer group of a general ...
- 4,155
10
votes
Brauer group of $\mathbb{Z}_{(p)}$
Lemma. Let $K$ be a number field with ring of integers $\mathcal O_K$, and let $S \subseteq \Omega_K^f$ be a set of finite places of $K$. Then there is a canonical short exact sequence
$$0 \to \...
- 26.5k
8
votes
Accepted
Making $\mathbb{Q}$-cohomology integral
As Jason Starr remarks in the comments, this answer to a question of his implies the answer to both of my questions is "no." For the latter, one may take:
$X=\mathbb{P}^1, \mathcal{L}=\mathcal{O}(1)$...
8
votes
Can base-change be non-surjective on Brauer groups?
For finite fields, the Brauer group is zero ( It comes from Wedderburn's theorem), so the answer is NO.
For number fields, the answer is YES. Following RP's question in the comments, I will prove the ...
- 1,629
8
votes
Accepted
Brauer group of a curve over non-algebraically closed field
I don't think your map is injective. Here is an attempt at a recipe for constructing a counterexample.
The ingredients are a $C_1$-field $F$ of characteristic zero and a smooth projective curve $X_0$...
- 4,155
8
votes
Category of modules over an Azumaya algebra and the Brauer group
$k$-linear cocomplete categories admit a "tensor product over $\text{Mod}(k)$" (thinking of them as module categories over $\text{Mod}(k)$) and the only thing you need to know about it to ...
- 114k
7
votes
Accepted
On a morphism from the Brauer group to the Picard group
It seems to me that the involution of $Q \otimes Q$ that exchanges $a \otimes b$ and $b\otimes a$ is inner, which means that the homomorphism you describe should always be trivial.
Here is the ...
- 26.7k
7
votes
Accepted
Calculating topological index
The index in this case is $8$. You can see that it divides $8$ as your space $X$ supports a tautological degree $8$ topological Azumaya algebra given by the map $B(SL_8/\mu_2) \rightarrow BPGL_8$. If ...
- 4,962
7
votes
Subspaces of $ A_{n}(\mathbb {Q})$ in which all nonzero matrices are invertible
For a field $\mathbb{F}$, let $\mu_\mathbb{F}(n)$ denote the maximal dimension of a subspace $N\subset A_n(\mathbb{F})$ such that all the nonzero elements of $N$ are invertible. For simplicity, I ...
- 106k
6
votes
Is any element in $H^2_{et}(X,\mathcal{O}_X^*)$ locally trivial in the Zariski topology?
An explicit simple counter-example is the following: just take the quaternion algebra $(x,y)$ over $k(x,y)$, where $k$ is a field with $\mathrm{char}(k) \neq 2$. This is non-zero on any open subset of ...
- 20.6k
6
votes
Explicit examples of Azumaya algebras
Another example I found: if $X=G$ is a linear algebraic group over an algebraically closed field $k$, then every Azumaya algebra is given by a projective representation
$$\pi_1G \ \longrightarrow \...
- 5,278
6
votes
Accepted
Purity of Brauer group for stacks
The answer seems to be positive and actually at least in the context of regular (locally) noetherian Deligne--Mumford stacks. (Actually, Artin stack should also be enough as we can compute the Brauer ...
- 11.9k
5
votes
Accepted
Are Azumaya algebras of trivial Brauer class isomorphic to $\mathcal{E}nd(\mathcal{H})$?
The answer is yes and one can take $$\mathcal{H}:=\mathcal{Hom}_{\mathcal{End}(\mathcal G)}(\mathcal F\otimes \mathcal A,
\mathcal G).$$ Here, $\mathcal F\otimes \mathcal A$ is viewed as a left $\...
- 2,836
5
votes
On a morphism from the Brauer group to the Picard group
Here is another way to see that the given map should be zero. It corresponds to a map of sheaves of grouplike $\mathbb{E}_\infty$-spaces $K(\mathbb{G}_m,2)\rightarrow K(\mathbb{G}_m,1)$. All such maps ...
- 4,962
4
votes
Brauer group of projective space
Here is a proof that works in all characteristics. The idea comes from Colliot-Thélène ["Formes quadratiques multiplicatives et variétés algébriques: deux compléments", Bull. Soc. Math. France, 108(2)...
- 4,155
4
votes
Accepted
What is known about lower etale cohomology of unirational varieties?
Concerning the $H^2$, for $X$ a smooth projective rationally chain connected variety over an algebraically closed field $k$ and $\ell \in k^\ast$, it follows from
Theorem 1.2 in https://arxiv.org/...
- 9,387
4
votes
Postnikov invariants of the Brauer 3-group
Let me see if I understand what Jacob says in the comments. I think his argument can be summarized as: the Brauer 3-group is étale-locally an Eilenberg-MacLane spectrum, hence étale-locally an $\...
- 114k
4
votes
Accepted
Cohomological Brauer group vs classical
Briefly, one uses the exact sequence
$$
H^{1}(X,GL_{n})\rightarrow H^{1}(X,PGL_{n})\rightarrow H^{2}(X,\mathbb{G}_{m})
$$
(etale cohomology). The set $H^{1}(X,PGL_{n})$ classifies the isomorphism ...
- 184
4
votes
Countably many isomorphism classes of reductive groups over a field with countable Brauer and Witt groups
$\newcommand\kalg{k\textrm-\mathbf{alg}}\newcommand\kalgf{\kalg_\text f}$Welcome new contributor. I am just writing my comment as an answer. Let $k$ be a field and denote by $\kalg$ the category ...
3
votes
Brauer group of a curve over non-algebraically closed field
Here's an explicit example (joint with A. Landesman). Let $k$ be an algebraically closed field of characteristic not $2$ or $3$, and let $X/k$ be a non-supersingular K3 with Neron-Severi rank $\geq 5$....
- 2,525
3
votes
Elementary proof that a central simple algebra over a field having a maximal subfield is a cyclic algebra
The proof uses as an essential ingredient Proposition 2.2.8, which itself relies on Lemma 2.2.9 telling you that the k-algebras in $M_n(k)$ isomorphic to $k^n$ are conjugate to the subalgebra of ...
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