Brauer Groups: When Do They Differ?

Hey guys! Ever wondered when different kinds of Brauer groups actually disagree? It's a super interesting question in algebraic geometry, and today we're diving deep into it. We're going to explore the relationships between the Azumaya Brauer group, the torsion part of the Grothendieck Brauer group, and the full Grothendieck Brauer group. Buckle up; it's going to be a wild ride!

Understanding Brauer Groups

Let's start with the basics. Brauer groups, in general, are used to classify certain algebraic objects. In our case, we're talking about the Brauer group of a scheme, which is a powerful invariant in algebraic geometry. To understand when two Brauer groups disagree, it's crucial to know what each of them represents.

The Azumaya Brauer Group: $Br_{Az}(X)$

The Azumaya Brauer group, denoted as $Br_{Az}(X)$, classifies Azumaya algebras over a scheme $X$. Azumaya algebras are essentially generalizations of matrix algebras over a field, but defined over a scheme. Think of them as algebras that locally look like matrix algebras in the étale topology. The Azumaya Brauer group is relatively well-behaved and easier to compute compared to other Brauer groups. It represents the classes of Azumaya algebras modulo matrix algebras. Understanding the Azumaya Brauer group involves studying how Azumaya algebras behave under different conditions, such as changes in the base scheme or under various algebraic operations. The structure of $Br_{Az}(X)$ can reveal a lot about the underlying geometric properties of the scheme $X$. For instance, if $Br_{Az}(X)$ is trivial, it suggests that Azumaya algebras over $X$ are, in a sense, not too different from matrix algebras, indicating a certain simplicity in the algebraic structure of $X$. Conversely, a non-trivial $Br_{Az}(X)$ points to the existence of more complex Azumaya algebras that cannot be easily reduced to matrix algebras, which can be linked to more intricate geometric features of $X$.

The Grothendieck Brauer Group: $Br_{Gr}(X)$

The Grothendieck Brauer group, $Br_{Gr}(X)$, is a broader classification that includes more general objects than just Azumaya algebras. It is defined using étale cohomology, specifically $H^2(X_{et}, \mathbb{G}_m)$, where $\mathbb{G}_m$ is the multiplicative group. This group classifies classes of torsors under $\mathbb{G}_m$, which are related to line bundles and more general algebraic objects. The Grothendieck Brauer group is often more difficult to compute than the Azumaya Brauer group because it involves higher cohomology and more abstract constructions. The Grothendieck Brauer group provides a deeper insight into the cohomological properties of the scheme $X$. It captures information about the twisting of vector bundles and other geometric objects on $X$. The difference between the Azumaya and Grothendieck Brauer groups lies in the type of objects they classify. While the Azumaya Brauer group focuses on Azumaya algebras, the Grothendieck Brauer group encompasses a wider range of objects described by étale cohomology. The Grothendieck Brauer group can reveal subtle algebraic and geometric structures that are not apparent from the Azumaya Brauer group alone. It is a powerful tool for studying the arithmetic and geometric properties of schemes, especially in cases where the Azumaya Brauer group is insufficient to capture the full complexity of the situation.

The Torsion Part of the Grothendieck Brauer Group: $Br_{Gr}(X)_{tor}$

Now, let's talk about the torsion part of the Grothendieck Brauer group, $Br_{Gr}(X)_{tor}$. This is the subgroup of $Br_{Gr}(X)$ consisting of elements of finite order. In other words, an element $\alpha \in Br_{Gr}(X)$ belongs to $Br_{Gr}(X)_{tor}$ if there exists a positive integer $n$ such that $n\alpha = 0$. Understanding the torsion part of the Grothendieck Brauer group is crucial because it often behaves differently from the entire group. Torsion elements can reveal interesting arithmetic properties of the scheme $X$, such as the presence of certain finite group actions or the existence of specific types of coverings. The torsion part of the Grothendieck Brauer group is particularly important in the study of algebraic cycles and their relation to cohomology. It plays a key role in various conjectures and theorems in algebraic geometry, such as the Tate conjecture and the Bloch-Kato conjecture. Analyzing $Br_{Gr}(X)_{tor}$ can provide deep insights into the structure of the scheme $X$ and its arithmetic properties, often complementing the information obtained from the Azumaya Brauer group and the full Grothendieck Brauer group. The interplay between these different Brauer groups highlights the rich algebraic and geometric structure of schemes and their invariants.

The Inclusions: $Br_{Az}(X) \subseteq Br_{Gr}(X)_{tor} \subseteq Br_{Gr}(X)$

Okay, so we have these inclusions: $Br_{Az}(X) \subseteq Br_{Gr}(X)_{tor} \subseteq Br_{Gr}(X)$. Let's break down what they mean and when they might not be equalities.

• $Br_{Az}(X) \subseteq Br_{Gr}(X)_{tor}$: This inclusion states that every Azumaya algebra class is a torsion element in the Grothendieck Brauer group. This is because Azumaya algebras are, in a sense,