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If $f : X\to V$ is a smooth proper map of smooth schemes, what are the global sections of

  • $R^if_{fppf, *}\mu_p$
  • $R^if_{fppf, *}\mathbb{G}_{\rm m}$

I was reading Milne's book "Arithmetic duality", but this is never clearly explained. I know both sheaves are the fppf sheaves associated to the presheaves

$$W\mapsto H^i_{fppf}(X_W, A)$$ $A = \mu_p,\mathbb{G}_{\rm m}$, but the global sections of a this presheaf cannot be the global sections of $R^if_*A$. Am I right? There can be more than one presheaf, with different global sections, with same associated sheaf.

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    $\begingroup$ Let $i = 1$ and $A = \mathbf{G}_{\rm m}$. Then $R^1f_{\rm fppf, *}\mathbf{G}_{\rm m}$ is representable in algebraic spaces, by work of Artin. The representing object is denoted $\text{Pic}_{X/V}$, and it is a separated algebraic space locally of finite type over $V$. The global sections of $R^1f_{\rm fppf, *}\mathbf{G}_{\rm m}$ are $\text{Pic}_{X/V}(V)$, and I'm not sure I can give a better description. Note that not all sections come from the class of a relative line bundle on $X$, unless an obstruction in the Brauer gp of $X$ vanishes. $\endgroup$
    – user113453
    Commented Jan 16, 2018 at 6:08
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    $\begingroup$ As you can see, an answer to this question is already quite nontrivial for $i = 1$ and $A = \mathbf{G}_{\rm m}$. It is still an open problem, posed by Artin (in general) whether $R^1f_{\rm fppf, *}\mu_p$ is representable in algebraic spaces, although there should be papers of Lieblich addressing this. For higher degrees, total mystery. $\endgroup$
    – user113453
    Commented Jan 16, 2018 at 6:10
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    $\begingroup$ @user113453. "It is still an open problem, posed by Artin (in general) whether $R^1f_{\text{fppf},*}\mu_p$ is representable in algebraic spaces." I am just clarifying: do you mean that this is open for morphisms more general than those specified by the OP? Since the OP specifies that $f$ is smooth and proper, it is cohomologically flat in degree $0$. Thus the Kummer sequence realizes $R^1f_{\text{fppf},*}\mu_p$ as the $p$-torsion in the relative Picard, i.e., the fiber product of the zero section and the $p$-power morphism. $\endgroup$
    – Jason Starr
    Commented Jan 16, 2018 at 9:22
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    $\begingroup$ By the way, there is a theory of Kato that interprets part of the groups $R^i f_{\text{fppf},*}\mu_p,$ $i>1,$ in terms of something like the de Rham complex. Part of this is discussed in Serre's Galois cohomology where he discusses the $p$-torsion in Brauer groups, as well as his article, "Cohomologie galoisienne: progr`es et probl`emes." I believe that Philippe Gille is writing a book on Galois cohomology and cohomological dimension that explains Kato's theory. $\endgroup$
    – Jason Starr
    Commented Jan 16, 2018 at 9:28

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