Torsion graded pieces of Nyggard filtration for crystalline representation

The exactness of the sequence in (4) is equivalent to that $E\mathfrak{M}^{*}\cap\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}=E\mathop{\rm Fil}\nolimits^{i-1}\mathfrak{M}^{*}$, and this easily follows that construction of $\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}$ in (1). (3) easily follows from the construction of $\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}$ and $\mathop{\rm Fil}\nolimits^{i}\mathcal{D}$. [GLS14, Lem. 4.3] proves that $\mathrm{gr}^{i}\mathfrak{M}^{*}$ is finite $\mathcal{O}_{K}$-free and $\mathop{\rm Fil}\nolimits^{i}\mathcal{D}\cap\mathfrak{M}^{*}=\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}$. To see that $\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}$ is finite free, consider the exact sequence

Since $\mathop{\rm Fil}\nolimits^{i+1}\mathfrak{M}/E\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}=\mathrm{ev}_{p}(\mathop{\rm Fil}\nolimits^{i+1}\mathfrak{M}^{*})\subset D$ and $\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}/\mathop{\rm Fil}\nolimits^{i+1}\mathfrak{M}^{*}=\mathrm{gr}^{i}\mathfrak{M}^{*}$ are torsion free and hence finite free $\mathcal{O}_{K}$-modules, we conclude that $\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}/E\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}$ is finite free $\mathcal{O}_{K}$-module. Since $E^{i}\mathfrak{M}^{*}\subset\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}\subset\mathfrak{M}^{*}$ and $\mathfrak{M}^{*}$ is finite $\mathfrak{S}$-free, $\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}$ is finite $\mathfrak{S}$-free by NAK.

To prove (1), first note that $E^{i}\mathcal{D}\subset\mathop{\rm Fil}\nolimits^{i}\mathcal{D}$ and $E^{i}\mathfrak{M}^{*}\subset\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}$. Now given a $x=\sum_{j}f_{j}\otimes a_{j}\in\mathop{\rm Fil}\nolimits^{i}(S[\frac{1}{p}]\otimes_{\varphi,\mathfrak{S}}\mathfrak{M})$, with $f_{j}\in S[\frac{1}{p}]$, $a_{j}\in\mathfrak{M}^{*}$, by removing $E^{l}$-term with $l\geq i$, we may assume that $f_{j}\in W(\kappa)[u][\frac{1}{p}]$. Therefore $p^{n}x\in\mathfrak{M}^{*}$ for sufficient large $n$. Note that $p^{n}x\in\mathop{\rm Fil}\nolimits^{i}(S[\frac{1}{p}]\otimes_{\varphi,\mathfrak{S}}\mathfrak{M}^{*})$, by construction, we have $p^{n}x\in\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}$. Therefore, $x$ is $S[\frac{1}{p}]\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}\subset\mathop{\rm Fil}\nolimits^{i}\mathcal{D}$ as required. ∎

In this Lemma, we allow the base field $K$ to be finite ramified over $W(\kappa)[\frac{1}{p}]$. In such generality, $S_{p}$ is replaced by $S_{\pi}:=\{f\in S[\frac{1}{p}]|f(\varpi)\in\mathcal{O}_{K}\}$ for a fixed uniformizer $\varpi\in\mathcal{O}_{K}$. We use the same idea in [GLS14, Prop. 4.6] and [Liu12, Prop. 2.13]. As the above, we fix a uniformizer $\varpi\in\mathcal{O}_{K}$ with Eisenstein polynomial $E\in W(\kappa)[u]$, a compatible system $\{\varpi_{n}\}_{n\geq 0}$ of $p^{n}$-th roots with $\varpi_{0}=\varpi$ as well as a compatible system $\{\zeta_{n}\}_{n\geq 0}$ of $p^{n}$-th roots of $1$. Write $\underline{\varpi}^{\flat}\coloneqq(\varpi_{n})_{n\geq 0}$ and $\underline{\zeta}^{\flat}\coloneqq(\zeta_{n})_{n\geq 0}$ as elements in $\mathcal{O}_{\mathbb C_{p}}^{\flat}$ where $\mathbb C_{p}$ is the $p$-adic completion of $\overline{K}$. Embed $\mathfrak{S}\to{A_{\mathrm{inf}}}$ and $S\to A_{\mathrm{cris}}$ given by $u\to[\underline{\varpi}^{\flat}]$. Let $K_{\infty}=\bigcup_{n=1}^{\infty}K(\varpi_{n})$, $L=\bigcup_{n=1}^{\infty}K_{\infty}(\zeta_{p^{n}})$ and $K_{1^{\infty}}=\bigcup_{n=1}^{\infty}K(\zeta_{p^{n}})$. We may always assume that $K_{\infty}\cap K_{1^{\infty}}=K$ by selecting a suitable uniformizer $\varpi$ (this is only needed for $p=2$ by [Wan22, Lem.2.1]). Let $\tau$ be a topological generator of $\mathop{\rm Gal}\nolimits(L/K_{1^{\infty}})$. We will also use $\tau$ to denote a lifting of $\tau$ in $G_{K}$ when it is acting on an element fixed by $\mathop{\rm Gal}\nolimits(\overline{K}/L)$.

Write $T^{\vee}:=\mathop{\rm Hom}\nolimits_{\mathbb Z_{p}}(T,\mathbb Z_{p})$ the dual of $T$. Then exists an ${A_{\mathrm{inf}}}$-linear injection

so that $\iota_{\mathfrak{S}}$ is compatible with Frobenius on both sides and $g({A_{\mathrm{inf}}}\otimes_{\mathfrak{S}}\mathfrak{M})\subset{A_{\mathrm{inf}}}\otimes_{\mathfrak{S}}\mathfrak{M},\forall g\in G_{K}$ by using $G_{K}$-action from that on $T^{\vee}\otimes_{\mathbb Z_{p}}{A_{\mathrm{inf}}}$ via $\iota_{\mathfrak{S}}$. Also $\iota_{\mathfrak{S}}$ induces the following commutative diagram

Here $\iota_{\mathfrak{S}}^{*}={A_{\mathrm{inf}}}\otimes_{\varphi,{A_{\mathrm{inf}}}}\iota_{\mathfrak{S}}$, $\iota_{S}=A_{\mathrm{cris}}[\frac{1}{p}]\otimes_{{A_{\mathrm{inf}}}}\iota^{*}_{\mathfrak{S}}$, and the left vertical arrows are induced by $\mathfrak{M}^{*}\subset\mathcal{D}$, and all arrows here are injective. In particular, we may regard ${A_{\mathrm{inf}}}\otimes_{\mathfrak{S}}\mathfrak{M}^{*}$ (resp. $A_{\mathrm{cris}}\otimes_{S}\mathcal{D}$) as submodule of $T^{\vee}\otimes_{\mathbb Z_{p}}{A_{\mathrm{inf}}}$ (resp. $T^{\vee}\otimes A_{\mathrm{cris}}[\frac{1}{p}]$) via $\iota_{\mathfrak{S}}$ (resp. $\iota_{S}$). Note that the modules at the right columns have natural $G_{K}$-actions. As submodules, the left columns are stable under these $G_{K}$-actions. The $G_{K}$-action on $\mathcal{D}$ is explicitly given by the following formula (see formula (6.19) in [LL21]):

where $\gamma_{i}=\frac{(\bullet)^{i}}{i!}$ is $i$-th divided power. In particular, $G_{\infty}$-acts on $\mathcal{D}$-trivially. Pick a $\tau$ so that $\frac{g([\underline{\varpi}^{\flat}])}{[\underline{\varpi}^{\flat}]}=[\underline{\zeta}^{\flat}]$. The above formula is simplified to $\tau(x)=\sum\limits_{i=0}^{\infty}\nabla^{i}(x)\gamma_{i}(u([\underline{\zeta}^{\flat}]-1))$. Write $w:=\frac{[\underline{\zeta}^{\flat}]-1}{E}\in{A_{\mathrm{inf}}}$. We see that $\tau(x)\subset\mathcal{R}[\frac{1}{p}]\otimes_{S}\mathcal{D}$ where $\mathcal{R}\subset A_{\mathrm{cris}}$ is the $p$-adic completion of $\mathfrak{S}[w,\{\gamma_{i}(E)\}_{i\geq 1}]$.

Write $\mathcal{D}_{A_{\mathrm{cris}}}:=A_{\mathrm{cris}}\otimes_{S}\mathcal{D}$ and $\mathfrak{M}^{*}_{{A_{\mathrm{inf}}}}:={A_{\mathrm{inf}}}\otimes_{\mathfrak{S}}\mathfrak{M}^{*}$. Set $\mathop{\rm Fil}\nolimits^{i}\mathcal{D}_{A_{\mathrm{cris}}}:=A_{\mathrm{cris}}[\frac{1}{p}]\otimes_{\mathfrak{S}}\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}$ and $\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}_{{A_{\mathrm{inf}}}}:={A_{\mathrm{inf}}}\otimes_{\mathfrak{S}}\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}$. By a similar arguments as in Remark 2.2, we see that $\mathop{\rm Fil}\nolimits^{i}\mathcal{D}_{A_{\mathrm{cris}}}$, $\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}_{{A_{\mathrm{inf}}}}$ injects to $\mathcal{D}_{A_{\mathrm{cris}}}$, $\mathfrak{M}^{*}_{{A_{\mathrm{inf}}}}$ respectively. We claim that $g(\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}_{{A_{\mathrm{inf}}}})\subset\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}_{{A_{\mathrm{inf}}}},\forall g\in G_{K}$. To prove the claim, it suffices to show that

Consider $\mathfrak{M}^{*}$ as an $\mathfrak{S}$-submodule of $\mathfrak{M}$ via $1\otimes\varphi_{\mathfrak{M}}$. Then $\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}=\mathfrak{M}^{*}\cap E^{i}\mathfrak{M}$ and $F^{i}=({A_{\mathrm{inf}}}\otimes_{\mathfrak{S}}\mathfrak{M}^{*})\cap E^{i}({A_{\mathrm{inf}}}\otimes_{\mathfrak{S}}\mathfrak{M})$. Since ${A_{\mathrm{inf}}}$ is flat over $\mathfrak{S}$, we have $\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}_{{A_{\mathrm{inf}}}}=F^{i}$ as required.

As in the proof of [GLS14, Prop. 4.6] and [Liu12, Prop. 2.13], for any subring $A\subset A_{\mathrm{cris}}[\frac{1}{p}]$ such that $\varphi(A)\subset A$, let $I^{[i]}A=\{x\in A|\varphi^{n}(x)\in\mathop{\rm Fil}\nolimits^{i}B_{\mathrm{dR}},\forall n\in\mathbb N\},$ and $I^{[n]}:=I^{[n]}{A_{\mathrm{inf}}}=([\underline{\zeta}^{\flat}]-1)^{n}{A_{\mathrm{inf}}}$. Now we claim that $(\tau-1)^{n}(\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*})\subset\mathop{\rm Fil}\nolimits^{i-n}\mathfrak{M}^{*}\otimes_{\mathfrak{S}}I^{[n]}$. To prove this claim, we first prove that

By a similar argument in [Liu08, (5.1.2)], $(\tau-1)^{n}(x)=\sum\limits_{m\geq n}a_{mn}\nabla^{m}(x)\gamma_{m}(u([\underline{\zeta}^{\flat}]-1))$ with $a_{mn}\in\mathbb Z$. For $n\leq m\leq i$, we have $\nabla^{m}(x)\in\mathop{\rm Fil}\nolimits^{i-m}\mathcal{D}$ by Griffith transversality $\nabla(\mathop{\rm Fil}\nolimits^{i}\mathcal{D})\subset\mathop{\rm Fil}\nolimits^{i-1}\mathcal{D}$ and $[\underline{\zeta}^{\flat}]-1=Ew$. Since $E^{m-n}\mathop{\rm Fil}\nolimits^{i-m}\mathcal{D}\subset\mathop{\rm Fil}\nolimits^{i-n}\mathcal{D}$, $(\tau-1)^{n}(x)\in\mathop{\rm Fil}\nolimits^{i-n}\mathcal{D}\otimes_{S[\frac{1}{p}]}I^{[n]}{\mathcal{R}}[\frac{1}{p}]$.

Note that $\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}_{{A_{\mathrm{inf}}}}=\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}\otimes_{\mathfrak{S}}{{A_{\mathrm{inf}}}}$ and $\forall g\in G_{K},\ g(\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*})\subset\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}_{{A_{\mathrm{inf}}}}={A_{\mathrm{inf}}}\otimes_{\mathfrak{S}}\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}$. Then using that $\mathop{\rm Fil}\nolimits^{i-n}\mathfrak{M}^{*}$ is finite $\mathfrak{S}$-free, we have

This prove that claim.

Now for any $x\in\mathop{\rm Fil}\nolimits^{i}\mathfrak{M}^{*}$, we have

By the claim, we have

Note that $\mathrm{ev}_{\pi}:S\to\mathcal{O}_{K}$ is the same as the canonical projection $\theta:{A_{\mathrm{inf}}}\to\mathcal{O}_{\mathbb C_{p}}$. Now it suffices to check that $\theta(\frac{w^{n-1}}{n})\in\mathcal{O}_{\mathbb C_{p}}$. It turns out that $v_{p}(w)=\frac{1}{p-1}$ and this follows that $\frac{n-1}{p-1}\geq v_{p}(n)$. ∎