As noted in [Reference Howe and Klevdal2, Remark 1.2.2] the statement of [Reference Anschütz1, Lemma 3.25] is false. A counterexample is presented in [Reference Howe and Klevdal2, Example 4.3.4]. In this erratum we present this counterexample, discuss the failure of [Reference Anschütz1, Lemma 3.25] and its effects on the results of [Reference Anschütz1]. We thank Sean Howe for informing us about the error in [Reference Anschütz1, Lemma 3.25].
We use the notation from [Reference Anschütz1, Section 3], that is,
$C/\mathbb {Q}_p$
is a non-Archimedean, algebraically closed field,
$A_{\mathrm {inf}}$
Fontaine’s period ring for
$\mathcal {O}_C$
and
$\epsilon =(1,\underset {\neq 1}{\zeta _p},\ldots )\in C^\flat $
,
$\mu =[\epsilon ]-1$
,
$\tilde {\xi }:=\frac {\varphi (\mu )}{\mu }$
,
$t=\mathrm {log}([\epsilon ])$
.
Example 0.1 [Reference Anschütz1, Example 3.3].
For
$d\in \mathbb {Z}$
, the pair
$A_{\mathrm {inf}}\{d\}:=\mu ^{-d}A_{\mathrm {inf}}\otimes _{\mathbb {Z}_p}\mathbb {Z}_p(d)$
with Frobenius
$\varphi _{A_{\mathrm {inf}}\{d\}}=\tilde {\xi }^d\varphi _{A_{\mathrm {inf}}}$
is a Breuil–Kisin–Fargues module, and in fact each Breuil–Kisin–Fargues module of rank
$1$
is isomorphic to some
$A_{\mathrm {inf}}\{d\}$
([Reference Anschütz1, Lemma 3.12]). The corresponding
$B_{\mathrm {dR}}^+$
-latticed
$\mathbb {Q}_p$
-vector space (in the terminology of [Reference Howe and Klevdal2, Definition 4.2.1]) is
$(\mathbb {Q}_p,t^{-d}B^+_{\mathrm {dR}})$
. Each
$A_{\mathrm {inf}}\{d\}$
admits a canonical rigidification because
$\tilde {x}=u\cdot p$
in
$A_{\mathrm {crys}}$
for some unit (alternatively one can use [Reference Anschütz1, Lemma 4.3]).
According to [Reference Anschütz1, Lemma 3.28]
$$\begin{align*}\mathrm{Ext}^1_{\mathrm{BKF}^\circ_{\mathrm{rig}}}(A_{\mathrm{inf}},A_{\mathrm{inf}}\{d\})\cong B_{\mathrm{dR}}/t^dB^+_{\mathrm{dR}}. \end{align*}$$
Now, a counterexample to [Reference Anschütz1, Lemma 3.25] will be provided by the case
$d=0$
with extension corresponding to
$1/t$
. Explicitly the corresponding extension of
$B^+_{\mathrm {dR}}$
-latticed
$\mathbb {Q}_p$
-vector spaces is given by
$$\begin{align*}0\to (\mathbb{Q}_p\cdot e_1, B^+_{\mathrm{dR}}\cdot e_1)\to (\mathbb{Q}_p\cdot e_1\oplus \mathbb{Q}_p \cdot e_2, B^+_{\mathrm{dR}}\cdot e_1 \oplus B^+_{\mathrm{dR}}(\frac{1}{t}\cdot e_1+ e_2))\to (\mathbb{Q}_p\cdot e_2,B^+_{\mathrm{dR}}\cdot e_2)\to 0 \end{align*}$$
as presented in [Reference Howe and Klevdal2, Example 3.1.4]. Now, the fiber functor
$\omega _{\acute {e}t}\otimes C$
in [Reference Anschütz1, Lemma 3.25] from rigidifed Breuil–Kisin–Fargues modules to C-vector spaces factors over the functor to
$B^+_{\mathrm {dR}}$
-latticed
$\mathbb {Q}_p$
-vector spaces, and this functor is not exact as a filtered functor as noted in [Reference Howe and Klevdal2, Example 3.1.4]: The above exact sequence maps in
$\mathrm {gr}^0$
to
$$\begin{align*}0\to C\to 0\to C\to 0. \end{align*}$$
Indeed, the lattice
$B^+_{\mathrm {dR}} e_1\oplus B^+_{\mathrm {dR}}(\frac {1}{t}\cdot e_1+e_2)$
induces on
$V_C:=C\cdot e_1\oplus C\cdot e_2$
the filtration
$$\begin{align*}0\subseteq \mathrm{Fil}^1=C\cdot e_1\subseteq \mathrm{Fil}^0=V_C. \end{align*}$$
This example shows that the mistake in the ‘proof’ of [Reference Anschütz1, 3.25] lies in the last five lines: Even though the element
$v\otimes 1$
is part of some basis (e.g.,
$v\otimes 1=e_1$
in the above example), it need not be part of an adapted basis. As far as I can tell, this is the only mistake made.
We now discuss the effect of this mistake to the rest of the paper.
-
(1) In [Reference Anschütz1, Section 2], we fix a filtered fiber functor
$\omega _0\otimes C\colon \mathcal {T}\to \mathrm {Vec}_C$
stating that later we can apply the discussion to rigidified Breuil–Kisin–Fargues modules. This is not true, however, restricting to CM rigidified Breuil–Kisin–Fargues modules the fiber functor
$\omega _{\acute {e}t}$
with its functorial filtration over C is a filtered fiber functor. Indeed, any fiber functor on a semisimple Tannakian category, which is equipped with a functorial filtration compatible with tensor products is necessary a filtered fiber functor as each exact sequence splits. Hence, the general theory of this section can be applied on the full Tannakian subcategory of CM-objects. We note that the type of a CM-object ([Reference Anschütz1, Definition 2.9]) only requires a functorial filtration on a fiber functor compatible with tensor products (and in characteristic
$0$
these data will automatically yield a filtered fiber functor on the CM-objects as explained above). -
(2) The proof of [Reference Anschütz1, Lemma 3.27] cites [Reference Anschütz1, Lemma 3.25]; however, the claimed exactness is not used in the argument. Indeed, the claimed triviality of the filtration follows by the correct compatibility of the filtration with tensor products. A similar argument occurs in [Reference Howe and Klevdal2, Theorem 4.3.5].
-
(3) With the above adjustments, the results in [Reference Anschütz1, Section 4, Section 5] are not affected.
