ERRATUM

Homomorphisms of Barsotti-Tate groups and crystals in positive characteristic by A.J. de Jong, Inventiones Mathematicae 134

The assertion (iii) of Lemma 2.1 is false. A counter example was found by T. Zink and W. Messing. We quickly indicate the construction. Let G_{0} be a p-divisible group over a perfect field k, together with a closed immersion i : α_{p} → G_{0} over k. Set G_{1} = G_{0}/i(α_{p}). Further, let j : α_{p,k[[t]]} → (G_{0} ×G_{0})_{k[[t]]} be the embedding equal to i in the first coordinate and equal to t·i in the second. Set G := Coker(j), and H = G_{1,k[[t]]}. The counter example is the homomorphism γ : G → H induced by the mapping pr_{2} : G_{0} × G_{0} → G_{0}. Indeed, Ker(γ_{k((t))}) ≅ (G_{0})_{k((t))} is a p-divisible group, but Ker(γ_{0}) ≅ G_{1} × α_{p} is not.

Lemma 2.1 part (iii) is used only in Definition 2.2. To correct this, one should replace the text ”Note....over S.” in the definition of semi-stable reduction by the condition that G^{μ} = Ker(G_{1} → G_{2}) and G^{et} = Coker(G_{1} → G_{2}) are p-divisible groups over S. No other changes are necessary.

The mistake in the proof is in the module theoretic assertion ”e generates L implies (∗) is exact” on page 332. The reader can easily find a counter example. Messing and Zink point out that the proof does produce a kernel and cokernel for α in the category of p-divisible groups over R.