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

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

Contents
Contents
 1.  Motivation: Katz’s Theorem
 2.  Galois Representations of fields of characteristic $ p>0 $ and $ \varphi $-modules
   2.1.  $ \mathbb{F}_{p} $-representaions
   2.2.  $ \mathbb{Z}_{p} $-representaions and $ \mathbb{Q}_{p} $-representaions
 3.  Galois Representations of local fields of characteristic zero and $ (\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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