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Known techniques to compute flat cohomology after base change
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 isomo …