# Regarding a 'global' version of Chase-Harrison-Rosenberg exact sequence for rings

If $$R$$ is a commutative ring with identity with a 'nice' action of a finite group $$G$$, the subring $$R^G\subset R$$ gives a Galois extension of rings. In this case, S.U. Chase, D.K. Harrison, A. Rosenberg have shown that there exsits the following exact sequence (also knonwn as Chase-Harrison-Rosenberg exact sequence):

$$0\rightarrow H^1 (G, R^*)\rightarrow Pic(R^G)\rightarrow Pic(R)^G \rightarrow H^2(G, R^*) \rightarrow Br(R/R^G)\rightarrow H^1(G,Pic(R))$$ $$(*)$$

where $$Pic(R)$$ denotes the Picard group of $$R$$, and $$Br(R/R^G) := ker(Br(R^G)\rightarrow Br(G))$$ comes from the map on Brower groups.

My question is the following: I have come across an exactly similar exact sequence in a paper, but it comes in the context of a finite Galois morphism $$\pi: X\rightarrow Y$$ of two smooth projective varieties ( see (3.5) of This paper). There isn't much information in the paper on how it is obtained. I know that a finite Galois morphism of two varieties can be defined affine locally, in which case it essentially becomes Galois extension of rings, but it is not clear to me how to 'patch' the different local sequences $$(*)$$ to get a similar exact sequence for schemes as well.

The Picard and Brauer groups are not sheaves in the etale topology, so it is not clear to me if the concept of patching even makes sense here.

Can anyone give me a reference, or let me know why a 'scheme'-version of $$(*)$$ should also be true? Thanks in advance.

• I'm leaving this as a comment, because I haven't checked any of this. If $X\to Y$ is etale and Galois group $G$, then there should be a spectral sequence $H^p(G, H^q(X_{et}, \mathbb{G}_m))\Rightarrow H^{p+q}(Y_{et}, \mathbb{G}_m))$. Look at the terms of low degree. Oct 8, 2021 at 12:26
• Indeed the spectral sequence is the Hochschild-Serre spectral sequence for a finite Galois covering. See Milne, Etale Cohomology, III, 2.20. Oct 8, 2021 at 13:39

I'm converting my comment to an answer. Let $$\pi:X\to Y$$ be a Galois étale cover, with Galois group $$G$$. One has a Hochschild-Serre spectral sequence $$E_2 = H^p(G, H^q(X_{et},\mathbb{G}_m))\Rightarrow H^{p+q}(Y_{et}, \mathbb{G}_m)$$ (The reference is given in "A User"'s comment.) The associated exact sequence of low degree terms reads $$0\to H^1(G, H^0(X, \mathbb{G}_m))\to Pic(Y)\to Pic(X)^G \to H^2(G,H^0(X, \mathbb{G}_m))\to Br(Y)$$ where I'm using $$Br$$, a bit inaccurately, for the 2nd étale cohomology with coefficients in $$\mathbb{G}_m$$. In fact, the image should lie in $$Gr^2Br(Y)$$ wrt the filtration on the abuttment. And this seen to be $$\ker[\ker Br(Y)\to Br(X)]\to H^1(G,Pic(X))$$.