# Known techniques to compute flat cohomology after base change

Let $$f$$ be some homogenous polynomial of degree $$d>2$$.

Let $$X = \operatorname{Proj} (k[x,y,z]/(f))$$ where $$k$$ is algebraically closed field of characteristic $$p >0$$.

Now let $$R$$ be a $$k$$-algebra. I'm interested in the computation of $$H^1(X \times_{spec(k)} spec(R),\mathbb{Z}/p\mathbb{Z})$$. The usual way is to start with the Artin-schreir sequence

$$0 \to \mathbb{Z}/p\mathbb{Z} \to \mathbb{G}_a \to \mathbb{G}_a \to 0$$

So, we get

$$0 \to H^0(X \times_{spec(k)} spec(R), \mathbb{Z}/p\mathbb{Z}) \to R \to R \to H^1(X \times_{spec(k)} spec(R),\mathbb{Z}/p\mathbb{Z}) \to H^1(X , \mathbb{G}_a) \otimes R \to H^1(X , \mathbb{G}_a) \otimes R \to$$

If the map from $$R \to R$$ is surjective ( as when $$R=k$$) then $$H^1(X \times_{spec(k)} spec(R),\mathbb{Z}/p\mathbb{Z}) = \operatorname{ker}( H^1(X , \mathbb{G}_a) \otimes R \to H^1(X , \mathbb{G}_a) \otimes R$$.

Is there any known way to handle cases when the map $$R \to R$$ is is not surjective or any other technique to compute $$H^1(X \times_{spec(k)} spec(R),\mathbb{Z}/p\mathbb{Z})$$?

First of all, there's no need to use flat cohomology here. By Theorem III.3.9 in Milne's Etale Cohomology, the canonical map $$H^i_{\mathrm{et}}(X, G) \rightarrow H^i_{\mathrm{fppf}}(X, G)$$ is an isomorphism when $$G$$ is a sheaf represented by a smooth group scheme over $$X$$. In particular, this works for the constant group $$\mathbf{Z}/p\mathbf{Z}$$ and the vector group $$\mathbf{G}_a$$ (which is just the quasi-coherent sheaf $$\mathscr{O}_X$$ and thus its flat cohomology is isomorphic to its Zariski cohomology). For $$H^1$$ at least, this boils down to the statement that torsors for smooth group schemes always have sections over an étale cover.

Furthermore, the Artin-Schreier sequence is an exact sequence of étale sheaves, so we have, for any $$R$$: $$0 \rightarrow R/\mathscr{F}(R) \rightarrow H^1_{\mathrm{et}}(X_R, \mathbf{Z}/p\mathbf{Z}) \rightarrow (\ker \mathscr{F} \colon H^1_{\mathrm{Zar}}(X, \mathscr{O}_X) \rightarrow H^1_{\mathrm{Zar}}(X, \mathscr{O}_X))\otimes_k R \rightarrow 0$$

On the other hand, $$X \rightarrow \mathrm{Spec}(k)$$ is proper, so we can use the proper base change theorem - note that unlike the smooth base change theorem, this works for any (locally constant) abelian torsion sheaf without a hypothesis that the torsion orders are units. Let $$\pi \colon X \rightarrow \mathrm{Spec}(k)$$ and $$f \colon \mathrm{Spec}(R) \rightarrow \mathrm{Spec}(k)$$ be the structure morphisms. Then the proper base change theorem tells us that the canonical map $$f^* R^1 \pi_* \mathbf{Z}/p\mathbf{Z} \rightarrow R^1 (\pi_R)_* \mathbf{Z}/p\mathbf{Z}$$ is an isomorphism. (Note that the pullback of the constant sheaf $$\mathbf{Z}/p\mathbf{Z}$$ from $$X$$ to $$X_R$$ is still this constant sheaf). Now the étale sheaf on the left is the constant sheaf on $$\mathrm{Spec}(R)_{\mathrm{et}}$$ associated to the cohomology group $$H^1_\mathrm{et}(X, \mathbf{Z}/p\mathbf{Z}) = \ker \mathscr{F} \colon H^1_{\mathrm{Zar}}(X, \mathscr{O}_X) \rightarrow H^1_{\mathrm{Zar}}(X, \mathscr{O}_X)$$

Finally, we have a Leray spectral sequence $$E_2^{p, q} = H^p_{\mathrm{et}}(\mathrm{Spec}(R), R^q (\pi_R)_* \mathbf{Z}/p\mathbf{Z}) \implies H^{p+q}_{\mathrm{et}}(X_R, \mathbf{Z}/p\mathbf{Z})$$

Now, we have an exact sequence of low-degree terms: $$0 \rightarrow H^1_{\mathrm{et}}(\mathrm{Spec}(R), \mathbf{Z}/p\mathbf{Z}) \rightarrow H^1_{\mathrm{et}}(X_R, \mathbf{Z}/p\mathbf{Z}) \rightarrow H^1_{\mathrm{et}}(X, \mathbf{Z}/p\mathbf{Z})^{\pi_0(\mathrm{Spec}(R))} \rightarrow H^2_{\mathrm{et}}(\mathrm{Spec}(R), \mathbf{Z}/p\mathbf{Z}) \rightarrow H^2_{\mathrm{et}}(X_R, \mathbf{Z}/p\mathbf{Z})$$

This should let you compute $$H^1(X_R, \mathbf{Z}/p\mathbf{Z})$$ as long as you know a bit about $$\mathrm{Spec}(R)$$ and understand how $$\mathscr{F}$$ acts on the $$k$$-vector space $$H^1_{\mathrm{Zar}}(X, \mathscr{O}_X)$$.

(note that the above arguments do not work for $$\mathbf{Z}/p\mathbf{Z}$$ replaced by $$\mu_p$$, since the étale and flat cohomologies of this non-smooth group scheme differ. However, the Kummer sequence is exact in the flat topology, so you get some information. I wouldn't expect flat cohomology of something like this to play nearly as nicely with base change, and suspect it's very hard to say anything in a reasonable level of generality).