The Picard-Brauer short exact sequence

Question

It seems to be a rather well understood fact that, given commutative rings $R,S$, and a homomorphism $R \to S$ there is a short exact sequence $$\text{Pic}(R) \to \text{Pic}(S) \to F_0 \to \text{Br}(R) \to \text{Br}(S)$$

relating the Picard and Brauer groups. In some sense this is not surprising, as Picard groups are related to first etale cohomology groups, and Brauer groups to the (torsion in the) second. (For example, see here, although I'm unsure what $F_0$ is)

Yet I can't seem to find this result proven in the literature, and it doesn't appear obvious to me how to prove this (clearly first and last maps arise from the fact that the Picard and Brauer groups are both functors $\text{Ring} \to \text{Ab}$, so I am interested in the middle 3 terms).

Is there a 'text-book' level reference for this result? Note that I can find various generalizations of this result using symmetric monoidal categories and fancy category theory, but I'm purely interested in a 'algebraic' proof of the case of rings.

Answer 1

Here is one way to do it. Under appropriate conditions on the ring (at the very least Milne's Étale Cohomology has some conditions listed in it, but I won't look them up now) we get that the Azumaya algebra definition of the Brauer group is the same as the étale cohomological one $$Br(R)=H^2(R, \mathbb{G}_m):=H^2_{et}(\operatorname{Spec}(R), \mathbb{G}_m).$$ We always have that $$Pic(R)=H^1(R, \mathbb{G}_m):=H^1_{et}(\operatorname{Spec}(R), \mathbb{G}_m).$$

Now given a ring homomorphism $f:R\to S$ we get a map (we'll just call it the same thing) $f:\operatorname{Spec}(S)\to \operatorname{Spec}(R)$. Consider the Leray spectral sequence for this map and the sheaf $\mathbb{G}_m$, i.e. $$H^p(R, R^qf_*\mathbb{G}_m)\Rightarrow H^{p+q}(S, \mathbb{G}_m)$$ (unfortunately, confusing due to $R$ repetition).

The long exact sequence of low degree terms gives us (using the cohomological interpretation of everything) $$Pic(R)\to Pic(S)\to H^0(R, R^1f_*\mathbb{G}_m)\to Br(R)\to Br(S)\to \cdots$$

This even tells us what the $F_0$ is. It is just the global sections of the first higher direct image of $\mathbb{G}_m$.

I guess you need to know when the cohomological Brauer group is the same as the Brauer group for this to work, and what conditions you need for the Leray spectral sequence to apply.