Wiki. “Brauer 群” [Brauer群]

Azumaya 代数是平展景局部矩阵代数. “an algebra whose module category is invertible with respect to the Morita symmetric monoidal structure”.

In more geometric settings, the first example of an Azumaya algebra is the endomorphism algebra of a vector bundle, though these have trivial Brauer class. Locally, any Azumaya algebra is the endomorphism algebra of a vector bundle, but the vector bundles do not generally glue to a vector bundle on the total space. However, every Azumaya algebra is the endomorphism algebra of a twisted vector bundle, a perspective that has recently gained a great deal of importance. For instance, in the theory of moduli spaces of vector bundles, there is always a twisted universal vector bundle, and the class of its endomorphism algebra in the Brauer group is precisely the obstruction to the existence of a universal (non-twisted) vector bundle on the moduli space. Brauer groups and Azumaya algebras play an important role in many areas of mathematics, but especially in arithmetic geometry, algebraic geometry, and applications to mathematical physics. In arithmetic geometry, they are closely related to Tate’s conjecture on l-adic cohomology of schemes over finite fields, and they play a critical role in studying rational points of varieties through, for example, the Brauer Manin obstructions to the Hasse principle. In algebraic geometry, Azumaya algebras arise naturally when studying moduli spaces of vector bundles, and Brauer classes appear when considering certain constructions motivated from physics in homological mirror symmetry. The Brauer group was also used by Artin-Mumford (AM72) to construct one of the first examples of a non-rational unirational complex variety.

域

域 $K$ 的 Brauer 群 $\mathrm{Br}(\mathbb{R})$ 是其上的中心单代数的相似类关于张量积构成的群, 也即上同调 $$ H^2(\operatorname{Gal}(K_{\mathrm{sep}}/K),K^*_{\mathrm{sep}}) = H^2(\operatorname{Spec}K_{\mathrm{et}},\mathbb G_{\mathrm{m}}). $$ 两者都能推广到概形, 但不相同.

设 $A,B$ 为 $K$ 上的有限维中心单代数, 若存在 $n,m\in\mathbb{Z}_+$ 使得 $M_n(A)\simeq M_m(B)$ (作为 $K$-代数), 则称 $A$ 与 $B$ 相似.

$A$ 与 $B$ 相似当且仅当存在可除代数 $D$ 使得 $A,B$ 均为 $D$ 上的矩阵代数.

例. $\mathrm{Br}(\mathbb{R})\simeq \{[\mathbb{R}],[\mathbb H]\}$. $\mathbb H\otimes\mathbb H\simeq M_4(\mathbb{R})$.

定理. 有限可除代数是域.

推论. 有限域的 Brauer 群平凡.

环

与模范畴的关系

设 $R$ 为交换环, 考虑 $2$-范畴 $\mathsf{Alg}_R$, 其对象为 $R$-代数 (即 $R$-模范畴中的幺半群), 态射为双模, 态射的复合为张量积, $2$-态射为双模的同态.

考虑 $\mathsf{Alg}_R$ 的核 (极大子群胚) $\mathrm{Core}(\mathsf{Alg}_R)$, 其对象为 $R$-代数, 态射为 Morita 等价, $2$-态射为双模的同构.

This may be understood as the $2$-groupoid of (generalized) line $2$-bundles over $\operatorname{Spec}R$, inside that of all $2$-vector bundles.

• $\pi_0(\mathbf{Br}(R))$ 为 $R$ 的 Brauer 群;
• $\pi_1(\mathbf{Br}(R))$ 为 $R$ 的 Picard 群;
• $\pi_2(\mathbf{Br}(R))$ 为 $R$ 的乘法可逆元群.

平展上同调的定义

概形 $X$ 的 Brauer 群 $\mathrm{Br}(X)$ 基本上是平展上同调 $H^2_{\text{\'et}}(X,\mathbb G_m)$.

“范畴化” 的定义

Brauer 群某种意义上是 Picard 群的范畴化.

对于概形 $X$ 可以谈论 “$X$ 上的拟凝聚叠”, 又叫 “$X$-线性稳定 ∞-范畴”; 其构成的范畴记为 $\mathsf{Cat}_X$. 一个与 Brauer 群相关的群是所谓 “导出 Brauer 群”, 定义为 $\mathsf{Cat}_X$ 的 Picard 群 $$ \mathrm{Br}^{\mathrm{der}}(X) := \operatorname{Pic}(\mathsf{Cat}_X). $$