Tate conjecture and finiteness of Brauer group

Question

What is the exact relation between the Tate conjecture for divisors on $X$ and finiteness of the Brauer group of $X$? And what is the reference for these relations?

More precisely, let $X$ be a smooth proper variety over a finite field $k$, and $\ell\neq p=\mathrm{char}(k)$. Then one can formulate a number of conjectures about its cohomologies:

$(T_{\ell})$ The cycle class map $NS(X)\otimes \mathbf{Q}_\ell \to \mathrm{H}^2_{\mathrm{et}}(X_{\overline{k}}, \mathbf{Q}_\ell(1))^{G_{k}}$ is surjective;

$(T_{p})$ The cycle class map $NS(X)\otimes \mathbf{Q}_p \to \mathrm{H}^2_{\mathrm{crys}}(X/W(k))[1/p]^{\phi=p}$ is surjective;

$(B_\ell)$ the $\ell$-power torsion group $Br(X)[\ell^\infty]$ is finite;

$(B_p)$ the $p$-power torsion group $Br(X)[p^\infty]$ is finite;

$(B)$ the Brauer group $Br(X)$ is finite.

What is the precise relation between all these conjectures? Are they equivalent?

1. If $X$ is a surface, it was proven in Milne's paper that all these claims are indeed equivalent.

2. I believe that it is not hard to see that $(T_\ell)$ and $(B_\ell)$ are equivalent for a fixed $\ell$.

3. Morrow showed that $(T_\ell)$ for a fixed $\ell$ is equivalent to all other $(T_{\ell'})$ and to $(T_p)$.

So basically the question is whether $T_p$ is equivalent to $B_p$ (or if there is an implication in one direction?) and whether $B_\ell$ (or $B_p$) is equivalent to $B$?

UPD: Actually, the argument in Milne's paper does not seem to use that $X$ is a surface to show that $(T_p)$ is equivalent to $(B_p)$. So the only real question is to show that any of these conjectures imply $(B)$. For this, Milne seems to use that there is an intersection pairing on the Neron-Severi group of $X$.

UPD2: Actually, using the methods from the proof of Theorem 4.3 in the paper of Morrow, one can show that finiteness of $Br(X)[\ell^\infty]$ implies finiteness of $Br(X)$ by reducing to the case of surfaces. But I would be still curious to see direct proof (or a reference).

Answer 1

This could be seen as an extended comment. Here is a way that they are related outside of surface case:

• $T(X,n)$ is the conjunction of the Tate’s conjecture for X in degree n together with Beilinson’s conjecture that rational and numerical equivalence on X agree with rational coefficients in codimension n.
• $L(X,n)$ is the Lichtenbaum conjecture, states that $H^i_W(X,Z(n))$ (The Weil-etale motivic cohomology) is finitely generated. Then according to Thomas Geisser's work here $L(X,n)$ implies $T(X,n)$ and if $T(X,n)$ holds for all smooth projective varieties over finite fields and all $n$ then $L(X,n)$ is implied.

Thomas Geisser in this, Proposition 5.1 proves that $L(X,n)$ is equivalent to Conjecture 3.4 for a fixed $n$, which is finiteness of the groups $H^i_{et}(X,\mathbb{Z}(n))$ for all $i$ except $2n, 2n+2$ (there is a typo in the link $2n+1$ should be $2n+2$.). For $i=2n$ it is being finitely generated and for $i=2n+2$ the condition is being cofinite.

Since $Br(X)=H^3(X, \mathbb{Z}(1))$, then finiteness of $Br(X)$ implies $L(X,1)$ and hence $T(X,1)$. Conversely you need all of $T(X,n)$ for all $n$ and $X$ to get finiteness of $Br(X)$.