# A short note on$(\varphi,\Gamma)$-modules

Abstract . This is an introductory note concerning $(\varphi,\Gamma)$-modules.

1. Motivation: Katz’s Theorem

Theorem 1 (Katz). Let $k$ be a perfect field of characteristic $p$. If $X$ is a smooth scheme over $W_{n}(k)$, the ring of Witt vectors of length $n$, and if $F_{X}$ is a lift to $X$ of the Frobenius on $W_{n}(k)$, then there is an equivalence of categories between the category \’etale sheaves of locally free $\mathbb{Z}/p^{n}$-modules $E$ of finite rank, and the the category of locally free $\mathcal{O}_{X}$-module $\mathcal{E}$ of finite rank, equipped with an $\mathcal{O}_{X}$-linear isomorphism $F_{X}^{\ast}\mathcal{E}\simeq \mathcal{E}$.
Corollary 2. Let $k$ be a perfect field of characteristic $p$ and let $\bar{k}$ be an algebraic closure of $k$. Then the construction $V\mapsto (V\otimes_{\mathbb{F}_{p}}\bar{k})^{\text{Gal}(\bar{k}/k)}$ induces an equivalence from the category of finite-dimensional $\mathbb{F}_{p}$-vector spaces $V$ with a continuous action of $\text{Gal}(\bar{k}/k)$ to the category of finite-dimensional $k$-vector spaces $M$ equipped with a Frobenius-semilinear automorphism $\varphi_{M}$.

2. Galois Representations of fields of characteristic $p>0$ and $\varphi$-modules In this section, we assume that $E$ is a field of characteristic $p>0$. Denote the absolute Galois group of $E$ by $G_{E}:=\text{Gal}(E^{\text{sep}}/E)$. Set $\sigma=(a\mapsto a^{p})$ to be the $p$-power Frobenius of $E$. We are mainly concerned about the situation where $E=\mathbb{F}_{q}((t))$ is local field of characteristic $p$ in this note.

Definition 3. A $\varphi$-module over $E$ is a finite-dimensional $E$-vector space $M$ together with a $\sigma$-semilinear map $\varphi:M\to M$. We say that $M$ is \’etale if the linearization of $\varphi$ is an isomorphism, i.e. the $E$-linear map $\varphi^{\ast}M=M\otimes_{E,\sigma}E\to M,~m\otimes a\mapsto \varphi(m)a$ is an isomorphism.

Remark 1.
1. If $M$ is any $E$-vector space, then giving an $\sigma$-semilinear map $\varphi:M\to M$ is equivalent to giving the linearization of $\varphi$.
2. For a $\varphi$-module $M$ over $E$, if we fix a basis with $A=\text{Mat}(\varphi)$, then $M$ is \’etale $\iff \text{Mat}(\varphi)$ is invertible in $E$.

2.1. $\mathbb{F}_{p}$-representaions

Theorem 4 ($\mathbb{F}_{p}$-representations). Let $\text{Rep}_{\mathbb{F}_{p}}(G_{E})$ denote the category of continuous representations of $G_{E}$ on finite-dimensional $\mathbb{F}_{p}$-vector spaces and let $\mathcal{M}_{\varphi}^{\text{\’et}}(E)$ denote the category of \’etale $\varphi$-modules over $E$. Then the functor
$\mathbf{D}:\text{Rep}_{\mathbb{F}_{p}}(G_{E})\to \mathcal{M}_{\varphi}^{\text{\’et}}(E),~V\mapsto \mathbf{D}(V):=(V\otimes_{\mathbb{F}_{p}}E^{\text{sep}})^{G_{E}}$
defines an equivalence of categoires, with quasi-inverse functor defined by
$M\mapsto \mathbf{V}(M):=(M\otimes_{E}E^{\text{sep}})^{\varphi=1}$
where $\varphi:M\otimes_{E}E^{\text{sep}}\to M\otimes_{E}E^{\text{sep}}$ is $\varphi$ diagonally.

2.2. $\mathbb{Z}_{p}$-representaions and $\mathbb{Q}_{p}$-representaions Now we want to “lift” the situation to $\mathbb{Z}_{p}$-coefficients.

Definition 5. A Cohen ring for $E$ is a pair $(\mathcal{O}_{\mathcal{E}},\phi)$ where $\mathcal{O}_{\mathcal{E}}$ is a complete DVR with uniformizer $p$ with residue field $E$, and $\phi$ is a lift of $p$-power Frobenius on $E$.

Example 1.
1. If $E$ is perfect, then Witt vector $(W(E),\phi)$ is the unique possible Cohen ring, where $\phi$ is the unique lift of Frobenius.
2. In our case of $E=\mathbb{F}_{q}((t))$, we can take $\mathcal{O}_{\mathcal{E}}=W(\mathbb{F}_{q})((t))^{\wedge}$, where $\wedge$ means we take $p$-adic completion. More precisely,
\begin{align*}
\mathcal{O}_{\mathcal{E}}:=\varprojlim_{m}W(\mathbb{F}_{q})((X))/p^{m}= \left\lbrace \sum_{i\in \mathbb{Z}}a_{i}X^{i}:a_{i}\in W(\mathbb{F}_{q}), \lim_{i\to -\infty}a_{i}=0\right\rbrace
\end{align*}
If $v$ denotes the normalized discrete valuation of $W(\mathbb{F}_{q})$ then the discrete valuation $v_{ \mathcal{O}_{\mathcal{E}}}$ of $\mathcal{O}_{\mathcal{E}}$ is given by
$v_{ \mathcal{O}_{\mathcal{E}}}\left( \sum_{i\in \mathbb{Z}}a_{i}X^{i} \right):=\min_{i} v(a_{i}) .$
One can take $\phi$ to be the lift of Frobenius on $W(\mathbb{F}_{q})$ (which we don’t have any choice) and $t\mapsto t^{p}$ (which we have freedom to choose; we might as well take $t\mapsto (1+t)^{p}-1$, which turns out to to more useful).

Definition 6. Fix a Cohen ring $(\mathcal{O}_{\mathcal{E}},\phi)$ for $E$. We define as follows.
1. $\mathcal{E}:=\mathcal{O}_{\mathcal{E}}[1/p]$.
2. $\mathcal{E}^{\text{ur}}$ is the maximal unramified extension of $\mathcal{E}$.
3. $\widehat{\mathcal{E}^{\text{ur}}}$ is the completion of $\mathcal{E}^{ur}$.
4. $\mathcal{O}_{\widehat{\mathcal{E}^{\text{ur}}}}$ is the ring of integers of $\widehat{\mathcal{E}^{\text{ur}}}$.
The extension $\mathcal{E}^{\text{un}}/\mathcal{E}$ is Galois with Galois group $G_{F}$, and we have $\phi$ and $G_{F}$-actions on all the above rings.
Lemma 7.
1. $(\mathcal{O}_{\widehat{\mathcal{E}^{\text{ur}}}})^{G_{E}}\simeq \mathcal{O}_{\mathcal{E}}$.
2. $(\mathcal{O}_{\widehat{\mathcal{E}^{\text{ur}}}})^{\varphi=1}=\mathbb{Z}_{p}$.
Definition 8.
1. A $\varphi$-module over $\mathcal{O}_{\mathcal{E}}$ is an finitedly generated $\mathcal{O}_{\mathcal{E}}$-module $M$ equipped with a $\phi$-semilinear map $\varphi:M\to M$. It is called \’etale if the linearization of $\varphi$ is an isomorphism.
2. A $\varphi$-module over $\mathcal{E}$ is a finite-dimensional $\mathcal{E}$-vector space $D$ equipped with a $\phi$-semilinear map $\varphi:D\to D$. It is called \’etale if there exists an $\mathcal{O}_{\mathcal{E}}$-lattice $M$ of $D$ which is stable under $\varphi$ such that $M$ is an \’etale $\varphi$-module.
Theorem 9 ($\mathbb{Z}_{p}$-representations). Let $\text{Rep}_{\mathbb{Z}_{p}}(G_{E})$ denote the category of continuous $G_{E}$-representations on finitely generated $\mathbb{Z}_{p}$-modules and let $\mathcal{M}_{\varphi}^{\text{\’et}}(\mathcal{O}_{\mathcal{E}})$ denote the category of \’etale $\varphi$-modules over $\mathcal{O}_{\mathcal{E}}$. Then the functor
$\mathbf{D}:\text{Rep}_{\mathbb{Z}_{p}}(G_{E})\to \mathcal{M}_{\varphi}^{\text{\’et}}(\mathcal{O}_{\mathcal{E}}) ,~\Lambda \mapsto \mathbf{D}(\Lambda):=(\Lambda\otimes_{\mathbb{Z}_{p}}\mathcal{O}_{\widehat{\mathcal{E}^{\text{ur}}}} )^{G_{E}}$
defines an equivalence of categories, with quasi-inverse
$M\mapsto \mathbf{V}(M):=(M\otimes_{\mathcal{O}_{\mathcal{E}}}\mathcal{O}_{\widehat{\mathcal{E}^{\text{ur}}}})^{\varphi=1} .$
Theorem 10 ($\mathbb{Q}_{p}$-representations). Let $\text{Rep}_{\mathbb{Q}_{p}}(G_{E})$ denote the category of continuous $G_{E}$-representations on finite dimensonal $\mathbb{Q}_{p}$-vector spaces and let $\mathcal{M}_{\varphi}^{\text{\’et}}(\mathcal{E})$ denote the category of \’etale $\varphi$-modules over $\mathcal{E}$. Then the functor
$\mathbf{D}:\text{Rep}_{\mathbb{Q}_{p}}(G_{E})\to \mathcal{M}_{\varphi}^{\text{\’et}}(\mathcal{E}),~V\mapsto \mathbf{D}(V):=(V\otimes_{\mathbb{Q}_{p}}\widehat{\mathcal{E}^{\text{ur}}})^{G_{F}}$
defines an equivalence of categories, with quasi-inverse
$M\mapsto \mathbf{V}(M):=(M\otimes_{\mathcal{E}}\widehat{\mathcal{E}^{\text{ur}}})^{\varphi=1} .$

3. Galois Representations of local fields of characteristic zero and $(\varphi,\Gamma)$-modules Let $K$ be a $p$-adic field, i.e. $K$ is a finite extension of $\mathbb{Q}_{p}$. Denote $G_{K}$ to be the absolute Galois group of $K$.

Problem 1. How to describe continuous $G_{K}$-representations on finitely generated $\mathbb{F}_{p}$-vector spaces/ $\mathbb{Z}_{p}$-modules/ $\mathbb{Q}_{p}$-representations?

The key idea is to find a (deeply ramified, infinite) Galois extension $K_{\infty}/K$ such that
• One can transfer the situation to equicharacteristic local fields, i.e. there is an equicharacteristic local field $F$ such that $H_{K}:=G_{K_{\infty}}\simeq G_{F}$.
• The transferred continuous action of $G_{K_{\infty}}$ on $\mathcal{O}_{\widehat{\mathcal{E}^{\text{ur}}}}$ can extend to a continuous $G_{K}$-action on $\mathcal{O}_{\widehat{\mathcal{E}^{\text{ur}}}}$ commuting with $\phi$.
• $\Gamma:=\text{Gal}(K_{\infty}/K)$ is as simple possible.
Now \textbf{we assume that the above is ture}, then we get a continuous $\Gamma=G_{K}/H_{K}$ on $(\mathcal{O}_{\widehat{\mathcal{E}^{\text{ur}}}})^{H_{K}}=\mathcal{O}_{\mathcal{E}}$ commuting with $\phi$.

Definition 11.
1. A $(\varphi,\Gamma)$-module over $\mathcal{O}_{\mathcal{E}}$ ($\mathcal{E}$, resp.) is $\varphi$-module over $\mathcal{O}_{\mathcal{E}}$ ($\mathcal{E}$, resp.) with a semilinear $\Gamma$-action commuting with $\varphi$.
2. A $(\varphi,\Gamma)$-module over $\mathcal{O}_{\mathcal{E}}$ ($\mathcal{E}$, resp.) is \’etale if the underlying $\varphi$-modules is \’etale.

Theorem 12 ($\mathbb{Z}_{p}$-representations). Let $\text{Rep}_{\mathbb{Z}_{p}}(G_{K})$ denote the category of continuous $G_{K}$-representations on finitely generated $\mathbb{Z}_{p}$-modules and let $\mathcal{M}^{\text{\’et}}_{\varphi,\Gamma}(\mathcal{O}_{\mathcal{E}})$ denote the category of $(\varphi,\Gamma)$-modules over $\mathcal{O}_{\mathcal{E}}$. Then there is an equivalence of categories
$\text{Rep}_{\mathbb{Z}_{p}}(G_{K})\to \mathcal{M}^{\text{\’et}}_{\varphi,\Gamma}(\mathcal{O}_{\mathcal{E}}),~\Lambda\mapsto(\Lambda\otimes_{\mathbb{Z}_{p}}\mathcal{O}_{\widehat{\mathcal{E}^{\text{ur}}}})^{H_{K}}$
with quasi-inverse
$M\mapsto (M\otimes_{\mathcal{O}_{\mathcal{E}}}\mathcal{O}_{\widehat{\mathcal{E}^{\text{ur}}}})^{\varphi=1},$
where $G_{K}$ acts on the target diagonally. ($G_{K}$ acts via its quotient $\Gamma$ on $M$).

Proof . Everything thing is a formal consequence from Theorem ??.

Corollary 13 ($\mathbb{F}_{p}$-representations). There is an equivalence of categories
$\text{Rep}_{\mathbb{F}_{p}}(G_{K})\simeq \mathcal{M}_{\varphi,\Gamma}^{\text{\’et}}(k((t)))$
where $k$ is the residue class field of $K_{\infty}$.

Theorem 14 ($\mathbb{Q}_{p}$-representations). Let $\text{Rep}_{\mathbb{Q}_{p}}(G_{K})$ denote the category of continuous $G_{K}$-representations on finite-dimensional $\mathbb{Q}_{p}$-vector spaces and let $\mathcal{M}^{\text{\’et}}_{\varphi,\Gamma}(\mathcal{E})$ denote the category of $(\varphi,\Gamma)$-modules over $\mathcal{E}$. Then there is an equivalence of categories
$\text{Rep}_{\mathbb{Z}_{p}}(G_{K})\to \mathcal{M}^{\text{\’et}}_{\varphi,\Gamma}(\mathcal{E}),~\Lambda\mapsto(\Lambda\otimes_{\mathbb{Q}_{p}}\widehat{\mathcal{E}^{\text{ur}}})^{H_{K}}$
with quasi-inverse
$M\mapsto (M\otimes_{\mathcal{E}}\widehat{\mathcal{E}^{\text{ur}}})^{\varphi=1}.$

Proof . Everything thing is a formal consequence from Theorem ??.

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