[Paper Review] Comparison of arithmetic Brauer groups with geometric Brauer groups

Xinyi Yuan|arXiv (Cornell University)|Nov 25, 2020

Algebraic Geometry and Number Theory20 references4 citations

TL;DR

This paper compares the arithmetic Brauer group and geometric Brauer group of smooth projective varieties over finitely generated fields, establishing that the cokernel of the natural map from the arithmetic Brauer group modulo base field Brauer group to the Galois-invariant part of the geometric Brauer group is finite or has finite exponent, depending on the characteristic. The key method is a pull-back trick using cohomological techniques and Picard schemes, extending earlier results to all characteristics simultaneously.

ABSTRACT

Let $X$ be a projective and smooth variety over a field $k$. The goal of this paper is to prove that the cokernel of the canonical map $Br(X) o Br(X_{k^s})^{G_k}$ has a finite exponent. Both groups are natural invariants arising from consideration of the Tate conjecture of divisors over $X$.

Motivation & Objective

• To compare the arithmetic Brauer group and geometric Brauer group of smooth projective varieties over finitely generated fields.
• To understand the kernel and cokernel of the natural map from Br(X)/Br(k) to Br(X^s)^{G_k}.
• To extend the finiteness results of Colliot-Thélène and Skorobogatov from characteristic zero to arbitrary characteristic.
• To establish precise bounds on the cokernel, including its structure as a finite group or a group of finite exponent.
• To provide a complete and self-contained proof of the comparison theorem, clarifying known results in the literature.

Proposed method

• Uses the pull-back trick via restriction to hyperplane sections Ci of X, reducing the problem to lower-dimensional cases.
• Applies the spectral sequence for étale cohomology to relate Brauer groups to Picard groups and their Galois cohomology.
• Constructs a G_k-invariant splitting of the étale cohomology exact sequence using the cup-product pairing with respect to an ample line bundle.
• Reduces the problem to studying the morphism between Picard schemes of X and its hyperplane sections, particularly focusing on the Pic^0 components.
• Employs Poincaré’s complete reducibility theorem to analyze the kernel and cokernel of morphisms between abelian varieties associated to Picard schemes.
• Uses the weak Lefschetz theorem to show that the restriction map on Tate modules is injective, implying finiteness of kernels of Pic^0 morphisms.

Experimental results

Research questions

• RQ1 How does the cokernel of the map Br(X)/Br(k) → Br(X^s)^{G_k} behave in positive characteristic?
• RQ2 What is the precise structure of the cokernel when the base field has positive characteristic?
• RQ3 To what extent does the pull-back trick via hyperplane sections allow reduction of the Brauer group comparison to lower-dimensional cases?
• RQ4 Can the finiteness and exponent bounds on the cokernel be established uniformly across all characteristics?
• RQ5 How does the geometric Brauer group relate to the arithmetic Brauer group in the context of the Tate conjecture for divisors?

Key findings

• The kernel of the map Br(X)/Br(k) → Br(X^s)^{G_k} injects canonically into H^1(k, Pic(X^s)), with the cokernel having finite exponent.
• When the base field k has characteristic zero, the cokernel of the map is finite.
• When k has positive characteristic p, the cokernel is isomorphic to the direct sum of a finite group of order prime to p and a p-group of finite exponent.
• For k a finite field, both the kernel and cokernel of the map are finite.
• The comparison theorem holds uniformly across all characteristics, with the same method as in Colliot-Thélène and Skorobogatov’s characteristic zero result.
• The proof relies on constructing a finite kernel and cokernel morphism between Picard schemes via abelian subvarieties, using the weak Lefschetz theorem and non-degeneracy of intersection pairings.