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Homomorphisms of Barsotti-Tate groups and crystals in positive characteristic. – Erratum. (English) Zbl 0929.14029
Invent. Math. 134, No. 2, 301-333 (1998); erratum ibid. 138, No. 1, 225 (1999).
Let \(R\) be a discrete valuation ring, \(K\) its fraction field, and \(G\) and \(H\) \(p\)-divisible groups over \(R\). In the case of \(\text{char} K=0\), J. T. Tate [in: “\(p\)-divisible groups” Proc. Conf. local Fields, NUFFIC Summer School Driebergen 1966, 158-183 (1967; Zbl 0157.27601)] proved that the natural map \(\text{Hom}_R(G,H)\to\text{Hom}_K(G_K,H_K)\) is a bijection.
In the present paper the author obtains the analogous result in positive characteristic as a consequence of a general result on \(F\)-crystals.
More precisely, if one considers the natural inclusion \(j:\eta\to S\), where \(S=\text{Spec} R\) and \(\eta=\text{Spec} K\), then the author proves theorem 1.1:
If \(\text{char} K=p>0\) and \(R\) has a \(p\)-basis, then the natural functor \[ j^*:\{\text{non-degenerate} F\text{-crystals}/S\} \to \{\text{non-degenerate} F\text{-crystals}/\eta\} \] is fully faithful.
From general facts of Dieudonn√© crystalline theory [cf. P. Berthelot and W. Messing, in: “The Grothendieck Festschrift”, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. I, Prog. Math. 86, 173-247 (1990; Zbl 0753.14041)], the author deduces as a corollary that the map \(\text{Hom}_R(G,H)\to\text{Hom}_K(G_K,H_K)\) is bijective, without assumptions on \(R\). As applications of this corollary, in section 2, the author proves in this situation two results about the relations among abelian varieties and their \(p\)-divisible groups.
Precisely he proves a criterion for good reduction: Let \(X_\eta\) be an abelian variety over \(\eta\) with \(p\)-divisible group \(G_\eta\). Then \(X_\eta\) has good reduction if and only if \(G_\eta\) has good reduction. The same holds for semi-stable reduction.
The second result is the following theorem. Let \(F\) be a field finitely generated over \({\mathbb F}_p\). Let \(X\) and \(Y\) be abelian varieties over \(F\) and denote by \(X[p^\infty]\) and \(Y[p^\infty]\) their \(p\)-divisible groups. Then there is an isomorphism \(\text{Hom}(X,Y)\otimes{\mathbb Z}_p\cong\text{Hom}(X[p^\infty],Y[p^\infty])\).
The sections 3-9 of the paper are devoted to the proof of theorem 1.1.
In the erratum [Invent. Math. 138, No. 1, 225 (1999)], the author points out a minor mistake in lemma 2.1, part III.

14L05 Formal groups, \(p\)-divisible groups
14F30 \(p\)-adic cohomology, crystalline cohomology
14G20 Local ground fields in algebraic geometry
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