Erratum for “Slope Filtrations Revisited”

Kiran S. Kedlaya

Received: September 20, 2007

Communicated by Peter Schneider

Abstract. Some corrections are given for the manuscript “Slope Filtrations Revisited”, Documenta Math., Vol. 10 (2005), 447-525

2000 Mathematics Subject Classification: 14F30.

Laurent Berger has pointed out that the construction of Teichmüller presentations in [3, Definition 2.5.1] is not valid: it fails to properly account for the nonlinearity of the Teichmüller map. This would appear to invalidate those results of [3] depending on the use of Teichmüller presentations, or on plusminus-zero presentations. Fortunately, these can be corrected by adapting the technique of strong semiunit decompositions from [2], as follows. Retain notation as in [3, § 2.5]. A strong semiunit presentation of x ∈ Γ_{I} is a convergent sum x = ∑_{i∈ℤ} u_{i}π^{i} in which:

(a) each nonzero u_{i} belongs to Γ and satisfies v_{n}(u_{i}) = v_{0}(u_{i}) for all n ≥ 0;

(b) if i > j and u_{i}, u_{j} are both nonzero, then v_{0}(u_{i}) < v_{0}(u_{j}).

Such a presentation always exists by the same proof as in [2, Proposition 3.14], but there is no uniqueness property. Nonetheless, in each of [3, Proposition 3.3.7(c), Proposition 4.2.2, Lemma 4.3.2], one may safely replace all references to Teichmüller presentations (including implicit references via plusminus-zero presentations) with strong semiunit presentations. (One should also disregard the parenthetical remark about canonicality in the proof of [3, Proposition 4.2.2].) This substitution does not suffice for the proof of surjectivity in [3, Lemma 4.3.1], which uses the uniqueness property of Teichmüller presentations. This is harmless for the rest of the paper, because this lemma is used nowhere. For completeness, we point out that the lemma is an immediate consequence of a result of Fourquaux [1, Corollaire 3.9.19] (applied with a = 1).

Ruochuan Liu points out that the proof of [3, Lemma 2.9.1] is incomplete: it is only valid in case f has no slopes in [s′, s), as otherwise we cannot choose the unit u in the first sentence of the proof. To complete the proof in general, first note that the existence of g satisfying (a) and (b) follows from [3, Lemma 2.6.7]. To prove (c), choose s′′ with s′ < s′′ < s such that f has no slopes in [s′′, s). By the proof of [3, Lemma 2.9.1] as written, f is divisible by g in Γ_{[s′′,r]}. However, since g has no slopes less than s, g is a unit in Γ_{[s′,s′′]}, so f is also divisible by g in that ring. Since the intersection Γ_{[s′,s′′]} ∩ Γ_{[s′′,r]} inside Γ_{[s′′,s′′]} is equal to Γ_{[s′,r]} by [3, Corollary 2.5.7], f is divisible by g in Γ_{[s′,r]} as desired. Liu also notes a gap in the proof of [3, Lemma 2.9.3]: it is necessary to ensure that x_{i+1} ∈ Γ_{r}[π^{−1}]. To fix this, we must replace g_{i+1} − x_{i} wherever it appears by some y_{i} ∈ Γ_{r} such that g_{i+1} − x_{i} − y_{i} is divisible by h_{i+1} in Γ_{i+1}; this can be carried out by an argument similar to [3, Lemma 2.9.2]. We also take this opportunity to point out two errata to [2]. First (as noted by Kevin Buzzard), in the introduction (p. 95), it is incorrectly asserted that “Γ_{con} consists of series which take integral values on some open annulus with outer radius 1.” In fact, an element of Γ_{con} acquires this property only after multiplication by a large power of u (and conversely). Second, in [2, Lemma 2.3], R should be taken to be a Bézout domain, not merely a Bézout ring.

References

[1] L. Fourquaux, Logarithme de Perrin-Riou pour des extensions associées à un groupe de Lubin-Tate, PhD thesis, Université de Paris 6, 2005;

   http://tel.archives-ouvertes.fr/tel-00011919.

[2] K.S. Kedlaya, A p-adic local monodromy theorem, Annals of Math. 160 (2004), 93–184.

[3] K.S. Kedlaya, Slope filtrations revisited, Doc. Math. 10 (2005), 447–525.

Kiran S. Kedlaya Dept. of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139 USA kedlaya@mit.edu