Gal(-/F) as a Group valued Sheaf over a Site

What is Site? Example of a SiteUsing covering in this site to define algebraic extension.The Site we will useSheaf over Site Main Result: $\mathrm{Gal}(-/F):(\mathcal C,\mathrm{Cov}(\mathcal C))^{\mathrm{op}}\to\mathsf{Grp}$ is a Sheaf$\mathrm{Gal}(-/F)$ as presheafThe Sheaf Condition

What is Site?

Definition of a Site

A site is a pair $(\mathcal{C},\mathrm{Cov}(\mathcal{C}))$$(\mathcal{C},\,\mathrm{Cov}(\mathcal C))$ where:

• $\mathcal{C}$$\mathcal{C}$ is a category;

• $\mathrm{Cov}(\mathcal{C})$$\mathrm{Cov}(\mathcal C)$ assigns to each object $U\in \mathcal{C}$$U\in\mathcal{C}$ a collection $\mathrm{Cov}(U)$$\mathrm{Cov}(U)$ of families of morphisms

  $$\boxed{{\displaystyle \{U_i\to U{\}}_{i\in I}}}$$

  called coverings, such that:

Identity. For every $U$$U$, the singleton family

$$\boxed{{\displaystyle \{{\mathrm{id}}_U:U\to U\}}}$$

lies in $\mathrm{Cov}(U)$$\mathrm{Cov}(U)$.

Base‐change stability. If

$$\boxed{{\displaystyle \{U_i\to U\}\in \mathrm{Cov}(U)\text{and}V\to U}}$$

is any arrow, then the pullback family

$$\boxed{{\displaystyle \{V{\times }_U{U}_i\to V\}}}$$

lies in $\mathrm{Cov}(V)$$\mathrm{Cov}(V)$.

Transitivity. If

$$\boxed{{\displaystyle \{U_i\to U\}\in \mathrm{Cov}(U)\text{and for each }i,\{V_{ij}\to U_i\}\in \mathrm{Cov}(U_i)}}$$

then the composed family

$$\boxed{{\displaystyle \{V_{ij}\to U{\}}_{i,j}}}$$

lies in $\mathrm{Cov}(U)$$\mathrm{Cov}(U)$.

Equivalently, $\mathrm{Cov}(\mathcal{C})$$\mathrm{Cov}(\mathcal C)$ is a Grothendieck pretopology on $\mathcal{C}$$\mathcal{C}$.

Example of a Site

Consider the coslice category $\mathcal{C}:=F/\mathsf{Field}$$\mathcal{C}:=F/\mathsf{Field}$.

Define

$$\mathrm{Cov}(L)=\{\{K_i\hookrightarrow L{\}}_{i\in I}|\bigcup _{i\in I}{K}_i=L\}.$$

Here $\hookrightarrow$$\hookrightarrow$ means $\subseteq$$\subseteq$ in set theory.

One checks easily that this defines a Grothendieck topology.

Using covering in this site to define algebraic extension.

Definition and Proposition. An equivalent definition of algebraic extension via covering.

Let $L/F$$L/F$ be a field extension, then $L/F$$L/F$ is an algebraic extension iff $L$$L$ could be covered by a family of finite extension. i.e.

We have:

$$\boxed{{\displaystyle \{K_i\hookrightarrow L{\}}_{i\in I}\in \mathrm{Cov}(L)\text{ such that }[K_i:F]<\mathrm{\infty }}}$$

Proof.

If $L/F$$L/F$ is an algebraic extension, then $\{F(x)\hookrightarrow L{\}}_{x\in L}$$\{F(x)\hookrightarrow L\}_{x\in L}$ is such a covering in $\mathrm{Cov}(L)$$\mathrm{Cov}(L)$.

Conversely, assume $\boxed{{\displaystyle \{K_i\hookrightarrow L{\}}_{i\in I}\in \mathrm{Cov}(L),[K_i:F]<\mathrm{\infty }}},$$\boxed{\{K_i\hookrightarrow L\}_{i\in I}\in\mathrm{Cov}(L),[K_i:F]<\infty},$ for each $x\in L$$x\in L$, we could find a $K_i$$K_i$ such that $x\in K_i$$x\in K_i$. Then $x$$x$ is algebraic since it lies in an algebraic extension. $◻$$\square$

The Site we will use

Consider the coslice category $\mathcal{C}:=F/\mathsf{Field}$$\mathcal C:=F/\mathsf{Field}$.

$$\boxed{{\displaystyle \text{We would like to associate a Grothendieck topology to make }(\mathcal{C},\mathrm{Cov}(\mathcal{C}))\text{ become a site and }\mathrm{Gal}(-/F)\text{ become a sheaf}}}$$

Let us define $\mathrm{Cov}(\mathcal{C})$$\mathrm{Cov}(\mathcal C)$ now.

Definition and Proposition.

Let $L/F$$L/F$ be a field extension

$$\mathrm{Cov}(L)=\{\{K_i\hookrightarrow L{\}}_{i\in I}|\bigcup _{i\in I}{K}_i=L\text{and}\mathrm{\forall }\text{ two elemnts subset }X\subseteq L,\mathrm{\exists }i\in I:X\subseteq K_i\}.$$

Here $\hookrightarrow$$\hookrightarrow$ means $\subseteq$$\subseteq$ in set theory.

Proof.

Let us check the axiom of covering.

Identity.

If $L\hookrightarrow L$$L\hookrightarrow L$ then clearly we have $\boxed{{\displaystyle L\hookrightarrow L\in \mathrm{Cov}(L)}}$$\boxed{L\hookrightarrow L\in\mathrm{Cov}(L)}$.

Base Change.

If $\boxed{{\displaystyle \{K_i\hookrightarrow L{\}}_{i\in I}\in \mathrm{Cov}(L)\text{ and assume }E\hookrightarrow L}}$$\boxed{\{K_i\hookrightarrow L\}_{i\in I}\in\mathrm{Cov}(L)\text{ and assume } E\hookrightarrow L}$. Then we have $\boxed{{\displaystyle K_i{\times }_{L}E\cong K_i\cap E}}$$\boxed{K_i\times_{L}E\cong K_i\cap E}$.

Hence we have $\boxed{{\displaystyle \{K_i{\times }_{L}E\hookrightarrow E{\}}_{i\in I}\in \mathrm{Cov}(E)}}$$\boxed{\{K_i\times_L E\hookrightarrow{ E}\}_{i\in I}\in\mathrm{Cov}(E)}$ since we have $(\bigcup _{i\in I}{K}_i)\cap E=L\cap E=E$$\Bigl(\bigcup_{i\in I} K_i\Bigr)\cap E=L\cap E=E$ and for any finite $Y$$Y$ such that $Y\hookrightarrow E\hookrightarrow L$$Y\hookrightarrow E\hookrightarrow L$, there exits $i\in I:Y\hookrightarrow K_i$$i\in I:Y\hookrightarrow K_i$, hence $Y\hookrightarrow K_i{\times }_{L}E$$Y\hookrightarrow K_i\times_L E$.

Transitivity.

If $\boxed{{\displaystyle \{K_i\hookrightarrow L{\}}_{i\in I}\in \mathrm{Cov}(L)\text{ and }\{E_{i,j}\hookrightarrow K_i{\}}_{j\in J_i}\in \mathrm{Cov}(K_i)\text{ then }\{E_{i,j}\hookrightarrow L{\}}_{i,j}\in \mathrm{Cov}(L)}}$$\boxed{\{K_i\hookrightarrow L\}_{i\in I}\in\mathrm{Cov}(L) \text{ and } \{E_{i,j}\hookrightarrow K_i\}_{j\in J_i}\in\mathrm{Cov}(K_i)\text{ then } \{E_{i,j}\hookrightarrow L\}_{i,j}\in\mathrm{Cov}(L)}$

Easy to see that we have

$$\boxed{{\displaystyle L=\bigcup _{i\in I}{K}_i=\bigcup _{i\in I}\bigcup _{j\in J_i}{E}_{i,j}}}$$

For any two elements set $X\hookrightarrow L$$X\hookrightarrow L$, $\mathrm{\exists }i\in I,X\hookrightarrow K_i$$\exists i\in I,X\hookrightarrow K_i$. Now let $\{E_{i,j}\hookrightarrow K_i{\}}_{j\in J_i}\in \mathrm{Cov}(K_i)$$\{E_{i,j}\hookrightarrow K_i\}_{j\in J_i}\in\mathrm{Cov}(K_i)$, then $\mathrm{\exists }j\in J_i,X\hookrightarrow E_{i,j}$$\exists j\in J_i,X\hookrightarrow E_{i,j}$.

Hence $\mathrm{Cov}(\mathcal{C})$$\mathrm{Cov}(\mathcal C)$ is a covering, $\boxed{{\displaystyle (\mathcal{C},\mathrm{Cov}(\mathcal{C}))}}$$\boxed{(\mathcal C,\mathrm{Cov}(\mathcal C))}$ form a site. $◻$$\square$

Sheaf over Site

Definition.

Let $(\mathcal{C},\mathrm{Cov}(\mathcal{C}))$$(\mathcal{C},\mathrm{Cov}(\mathcal C))$ be any site. A $\mathsf{Set}\mathsf{,}\mathsf{Group}\mathsf{,}\mathsf{Ab}\mathsf{,}\mathsf{Ring}\mathsf{,}\mathsf{R}\text{-}\mathsf{Mod}...$$\mathsf{Set,Group,Ab,Ring,R\text{-}Mod}...$ valued presheaf $\mathcal{F}$$\mathcal F$ is called a sheaf if for every covering family

$$\{U_i\to U{\}}_{i\in I}\in \mathrm{Cov}(U)$$

the diagram

$$\mathcal{F}(U)\stackrel{p}{\to }\prod _i\mathcal{F}(U_i){\rightrightarrows }_{p_j}^{p_i}\prod _{i,j}\mathcal{F}(U_i\cap U_j)$$

is an equalizer. Here $U_i\cap U_j=U_i{\times }_U{U}_j$$U_i\cap U_j=U_i\times_U U_j$.

Remark. I would like to add more explanation for this definition.

Firstly,

$$p=\prod _i\mathcal{F}(U_i\to U),p_i=\prod _i\mathcal{F}(U_i\cap U_j\to U_i),p_j=\prod _j\mathcal{F}(U_i\cap U_j\to U_j)$$

The locally property of sheaf comes from the equalizer arrow $p$$p$ is injective. Hence we have:

$$\boxed{{\displaystyle p(s)=(s{|}_{U_i}{)}_i=(t{|}_{U_i}{)}_i=p(t)⟺s=t}}$$

The gluing property comes from, let $(s_i{|}_{U_i}{)}_i\in \prod _i\mathcal{F}(U_i)$$(s_i|_{U_i})_i\in\prod_{i}\mathcal F(U_i)$, if

$$\boxed{{\displaystyle p_i((s_i{)}_i)=(s_i{|}_{U_i\cap U_j}{)}_{i,j}=(s_j{|}_{U_i\cap U_J}{)}_{i,j}=p_j((s_j{)}_j)}}$$

then by the universal property of equalizer, there exists unique $s\in \mathcal{F}(U)$$s\in \mathcal{F}(U)$ such that $p(s)=(s_i{|}_{U_i}{)}_i$$p(s)=(s_i|_{U_i})_{i}$.

Main Result: $\mathrm{Gal}(-/F):(\mathcal{C},\mathrm{Cov}(\mathcal{C}){)}^{\mathrm{op}}\to \mathsf{Grp}$$\mathrm{Gal}(-/F):(\mathcal C,\mathrm{Cov}(\mathcal C))^{\mathrm{op}}\to\mathsf{Grp}$ is a Sheaf

$\mathrm{Gal}(-/F)$$\mathrm{Gal}(-/F)$ as presheaf

Before we check the sheaf condition, we need to explain why $\mathrm{Gal}(-/F)$$\mathrm{Gal}(-/F)$ is a group valued presheaf.

Firstly, we know that for each object $K$$K$ in $\mathcal{C}$$\mathcal C$, $\mathrm{Gal}(K/F)={\mathrm{Aut}}_F(K)$$\mathrm{Gal}(K/F)=\mathrm{Aut}_F(K)$ is a group. For a field extension $K\hookrightarrow L$$K\hookrightarrow L$, we have the natural restriction map

$$\boxed{{\displaystyle {\mathrm{res}}_K^L:\mathrm{Gal}(L/F)\to \mathrm{Gal}(K/F),\sigma ⟼\sigma {|}_K}}$$

Hence it is a presheaf.

The Sheaf Condition

Now let us check the sheaf condition. Let $\boxed{{\displaystyle \{K_i\hookrightarrow L{\}}_{i\in I}\in \mathrm{Cov}(L)}}$$\boxed{\{K_i\hookrightarrow L\}_{i\in I}\in\mathrm{Cov}(L)}$, then we need to check the equalizer diagram as follows:

$$\mathrm{Gal}(L/F)\stackrel{p}{\to }\prod _i\mathrm{Gal}(K_i/F){\rightrightarrows }_{p_j}^{p_i}\prod _{i,j}\mathrm{Gal}(K_i\cap K_j/F)$$

Firstly, $K_i\cap K_j\hookrightarrow K_i\hookrightarrow L=K_i\cap K_j\hookrightarrow K_j\hookrightarrow L$$K_{i}\cap K_j\hookrightarrow K_i\hookrightarrow L=K_{i}\cap K_j\hookrightarrow K_j\hookrightarrow L$, hence we have $p_i\circ p=p_j\circ p$$p_i\circ p=p_j\circ p$.

We check locally and gluing condition.

Locally.

Let $\tau ,\sigma$$\tau,\sigma$ be two elements in $\mathrm{Gal}(L/F)$$\mathrm{Gal}(L/F)$, then if $\tau {|}_{K_i}=\sigma {|}_{K_i}$$\tau|_{K_i}=\sigma|_{K_i}$ for all $K_i\hookrightarrow L\in \{K_i\hookrightarrow L{\}}_{i\in I}$$K_i\hookrightarrow L\in\{K_i\hookrightarrow L\}_{i\in I}$, we have $\mathrm{\forall }x\in L,\tau (x)=\sigma (x)$$\forall x\in L,\tau(x)=\sigma(x)$, since $\bigcup _{i\in I}{K}_i=L$$\bigcup_{i\in I} K_i=L$. Hence $\tau =\sigma$$\tau=\sigma$.

Gluing.

Assume that we have a family $({\sigma }_i{)}_i\in \prod _i\mathrm{Gal}(K_i/F)$$(\sigma_i)_{i}\in\prod_{i}\mathrm{Gal}(K_i/F)$ such that ${\sigma }_i{|}_{K_i\cap K_j}={\sigma }_j{|}_{K_i\cap K_j}$$\sigma_i|_{K_i\cap K_j}=\sigma_j|_{K_i\cap K_j}$ for all $i,j$$i,j$. Then we define $\sigma (x)=\sigma {|}_i(x)$$\sigma(x)=\sigma|_{i}(x)$ whenever $x\in K_i$$x\in K_i$. Compatibility on overlaps ensures this is uniquely well–defined. Also, $\sigma$$\sigma$ is bijective on $L$$L$ since $\sigma {|}_i:K_i\to K_i$$\sigma|_i:K_i\to K_i$ is bijective and $\bigcup _{i\in I}{K}_i=L$$\bigcup_{i\in I} K_i=L$.

Now we need to check that $\sigma \in \mathrm{Gal}(L/F)$$\sigma\in\mathrm{Gal}(L/F)$. Firstly, $\mathrm{\forall }t\in F,\sigma (t)=\sigma {|}_i(t)=t$$\forall t\in F,\sigma(t)=\sigma|_i(t)=t$.

For $\{x,y\}\subset L$$\{x,y\}\subset L$. Hence $\mathrm{\exists }i\in I,\{x,y\}\hookrightarrow K_i$$\exists i\in I,\{x,y\}\hookrightarrow K_i$, we have $\sigma (x+y)=\sigma {|}_i(x+y)=\sigma {|}_i(x)+\sigma {|}_i(y)=\sigma (x)+\sigma (y)$$\sigma(x+y)=\sigma|_i(x+y)=\sigma|_i(x)+\sigma|_i(y)=\sigma(x)+\sigma(y)$. Similarly we have $\sigma (xy)=\sigma (x)\sigma (y)$$\sigma(xy)=\sigma(x)\sigma(y)$. Therefore, we have $\sigma \in \mathrm{Gal}(L/F)$$\sigma\in\mathrm{Gal}(L/F)$ $◻$$\square$