Erratum to Crystalline Dieudonné module theory via formal and rigid geometry

Lemma 7.1.13.1 of [1] is wrong. This was pointed out to the author by Brian Conrad. The only place where Lemma 7.1.13.1 is used in the article is in the construction of the maps β_{n} (7.1.13.2). We give a correct construction of β_{n} (for A such that π is not a zero divisor).

Let c be an integer such that

πA ∩ I^{n} ⊂ πI^{n−c}

for all n ≥ c. The existence of c is the Artin-Rees lemma. Then for any integer t ∈ N we have

π^{t}A ∩ (π^{t−1}I^{n} + π^{t−2}I^{2n} + ⋯ + I^{tn}) ⊂ π^{t}I^{n−c}.

Next, we come to the definition of β_{n}. Let n ≥ c. Any a ∈ A[I^{n}/π] can be written in the form

a = a_{0} + a_{1}/π + ⋯ + a_{t}/π^{t}, a_{i} ∈ I^{in}

for some t ∈ N. We simply put β_{n}(a) = a_{0} mod I^{n−c}. To show that β_{n} is well defined, suppose that a_{0} + a_{1}/π + ⋯ + a_{t}/π^{t} (a_{i} ∈ I^{in}) represents zero in A[I^{n}/π]. This means that π^{t}a_{0} + π^{t−1}a_{1} + ⋯ + a_{t} = 0, as A has no π-torsion. Thus

π^{t}a_{0} ∈ π^{t−1}I^{n} + π^{t−2}I^{2n} + ⋯ + I^{tn}

and by the above it follows that a_{0} ∈ I^{n−c}. Hence β_{n}(a_{0} + a_{1}/π + ⋯ + a_{t}/π^{t}) = a_{0} mod I^{n−c} = 0 as desired.

[1] A.J. de Jong, Crystalline Dieudonné module theory via formal and rigid geometry, Publ. Math. I.H.E.S. 82, 5–96 (1995).