ERRATUM TO THE PAPER ‘BREUIL–KISIN–FARGUES MODULES WITH COMPLEX MULTIPLICATION’

As noted in [2, Remark 1.2.2] the statement of [1, Lemma 3.25] is false. A counterexample is presented in [2, Example 4.3.4]. In this erratum we present this counterexample, discuss the failure of [1, Lemma 3.25] and its effects on the results of [1]. We thank Sean Howe for informing us about the error in [1, Lemma 3.25].

We use the notation from [1, 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 [1, 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 $A_{\mathrm {inf}}\{d\}$ ([1, Lemma 3.12]). The corresponding $B_{\mathrm {dR}}^+$ -latticed $\mathbb {Q}_p$ -vector space (in the terminology of [2, 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 [1, Lemma 4.3]).

According to [1, 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 [1, 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 [2, Example 3.1.4]. Now, the fiber functor $\omega _{\acute {e}t}\otimes C$ in [1, 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 [2, 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 [1, 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. (1) In [1, 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 ([1, 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. (2) The proof of [1, Lemma 3.27] cites [1, 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 [2, Theorem 4.3.5].

3. (3) With the above adjustments, the results in [1, Section 4, Section 5] are not affected.