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 $\...
- 4,746
12
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 ...
- 50.6k
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. ...
- 21.1k
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". ...
- 37k
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). ...
- 1,505
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 ...
- 4,185
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 $2$-...
- 32.4k
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 \...
- 28.7k
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$...
- 4,185
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 ...
- 1,766
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 ...
- 118k
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 ...
- 5,062
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 ...
- 27k
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 ...
- 108k
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 ...
- 21.1k
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 ...
- 12.1k
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 \...
- 5,701
6
votes
Accepted
Grothendieck purity for Brauer groups of stacks
The relevant result is now in the literature.
It first appeared as Proposition 8.1 in On Brauer groups of tame stacks by Anchenjang. Actually he proves the result for all algebraic stacks smooth over ...
- 388
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 ...
- 5,062
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 $\...
- 2,928
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)...
- 4,185
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 $\...
- 118k
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/...
- 9,497
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
Accepted
Involution action on Brauer group of an abelian variety
$\newcommand{\bG}{\mathbb{G}}$Let $X$ be any smooth scheme over an algebraically closed field $k$ of characteristic $p$. From the short exact sequence $0\to\mu_p\to \bG_m\to\bG_m\to 0$ of sheaves on ...
- 7,367
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 ...
- 1,017
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