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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 Dieudonné 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.

##### MSC:
 14L05 Formal groups, $$p$$-divisible groups 14F30 $$p$$-adic cohomology, crystalline cohomology 14G20 Local ground fields in algebraic geometry
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