Let \(X\) be a scheme and \(G\) a sheaf of groups over \(X\). One can ask the following question: Given a class \(\alpha \in H^n(X,G)\) for \(n > 0\), can one find a cover \(\pi: T \to X\) such that \(\pi^* \alpha=0\)? The answer is trivial if one takes \(T\) to be a cover in a Čech cocycle associated to \(\alpha\). But the question becomes interesting if one adds the requirement that the map \(\pi\) have certain properties such as finiteness or properness. In this interesting and well-written paper, the author studies the answer to this question for fppf cohomology when \(G\) is either a finite flat commutative group scheme or an abelian scheme. To state the results, let us assume that \(X\) is a noetherian excellent scheme. Then the first main theorem of the paper states that if \(G\) is a finite flat commutative group scheme over \(X\), classes in \(H^n_{fppf}(X,G)\) can be annihilated by finite surjective maps to \(X\) for \(n > 0\). The second main theorem of the paper states that if \(A\) is an abelian scheme over \(X\), classes in \(H^n_{fppf}(X,A)\) can be annihilated by proper surjective maps to \(X\) for \(n > 0\).
We now outline the proof of the first theorem. Assume first that \(X\) is the spectrum of a strictly henselian local ring. The author uses a result of Raynaud (Théorème 3.1.1, [P. Berthelot, L. Breen and W. Messing, Théorie de Dieudonne cristalline. II. Lecture Notes in Mathematics. 930. Berlin-Heidelberg-New York: Springer-Verlag. (1982; Zbl 0516.14015)]) to embed \(G\) into an abelian group scheme \(A\) as a closed subscheme over \(X\). This produces a short exact sequence \(0 \to G \to A \to A/G \to 0\) of abelian sheaves on \(X_{fppf}\); and hence a long exact sequence \(\cdots \to H^{n-1}_{fppf}(X,A/G) \to H^n_{fppf}(X,G) \to H^n_{fppf}(X,A) \to \cdots\).
By a theorem of A. Grothendieck, (Théorème 11.7, [Dix Exposes Cohomologie Schemas, Advanced Studies Pure Math. 3, 88–188 (1968; Zbl 0198.25901)]), fppf and étale cohomology coincide when the coefficients are smooth group schemes. This then implies that \(H^i_{et}(X,A)=H^i_{et}(X,A/G)=0\) for \(i>0\); and by the long exact sequence above, one has \(H^i(X,G)=0\) for \(i>1\). Hence if \(f: (Sch/X)_{fppf} \to (Sch/X)_{et}\) denotes the canonical morphism, one has \(R^i f_* G=0\) for \(i>1\).
The theorem is proved by induction on \(n\). For \(n=1\), it is easy to show that any fppf \(G\)-torsor \(T\) over \(X\) is killed by pullback to \(T\). For \(n > 1\), one first shows that it is possible to pass first to an étale; then using a result due to Gabber (Lemma 5, R. T. Hoobler [Brauer groups in ring theory and algebraic geometry, Proc., Antwerp 1981, Lect. Notes Math. 917, 231–244 (1982; Zbl 0491.14013)]), a Zariski cover. The author finishes the proof using the Čech spectral sequence associated to this cover. The proof of the second theorem also goes along these lines.
The author finishes the paper by giving a proof of a theorem of M. Hochster and C. Huneke [Ann. Math. (2) 135, No. 1, 53–89 (1992; Zbl 0753.13003)] using the first main theorem; and an example showing that the word “proper” cannot be replaced by “finite” in the second main theorem.