4.2. G-丛

In this section, we are going to start with $G$-bundles/torsors.

Thus, let $X$ be a scheme and $\mathcal{C}(\mathsf{Sch}/X{)}_{fppf}$. We are going to start with a group scheme over $X$, say $G$, which is flat and locally of finite presentation over $X$.

定义 4.2.0.1. A (principal) $\mathbf{G}$-bundle is a scheme $\pi :P\to X$ over $X$, where $\pi$ is fppf, with an $G$-action $\rho :G\times P\to P$, such that the map $G{\times }_{X}P\stackrel{\sim }{\to }P{\times }_{X}P$ given by $(g,p)\mapsto (gp,p)$, is isomorphism, where $gp=\rho (g,p)$.

This $G{\times }_{X}P\stackrel{\sim }{\to }P{\times }_{X}P$ condition is equivalent to saying if $Y\to X$ and $P(Y)\ne \varnothing$, then the group action of $G(Y)$ acts on $P(Y)$ is simple and transitive, i.e. $G(Y)$ acts on $P(Y)$ has no stabilizers and for all $p,p^{\prime }\in P(Y)$ there exists $g\in G(Y)$ so $gp=p^{\prime }$, i.e. for all $p,p^{\prime }\in P(Y)$ there exists unique $g\in G(Y)$ so $p^{\prime }=gp$.

So, before we give examples, we talk about the idea of $G$-bundle. In particular, we can think of $P$ as a group without a choice of identity. Here, if we give two elements, we only care about the difference, not which particular $g$ we are working with(so it is similar to the idea of potential functions in physics, i.e. we only care about diff of potentials, not the initial value).

定义 4.2.0.2. A map of $\mathbf{G}$-bundles $P\to P^{\prime }$ is a map of $S$-schemes $f:P\to P^{\prime }$ such that

定义 4.2.0.3. Let $\mathcal{C}$ be a site and $\mu$ a sheaf of groups. A $\mathbf{\mu }$-torsor is a sheaf $\mathcal{P}$ with $\mu \times \mathcal{P}\to \mathcal{P}$ such that:

1.	for all $X\in \mathcal{C}$, there exists $\{X_i\to X\}\in \mathrm{Cov}(X)$ such that $\mathcal{P}(X_i)\ne \varnothing$ for all $i$.
2.	we have $\mu \times \mathcal{P}\stackrel{\sim }{\to }\mathcal{P}\times \mathcal{P}$ where the map is given by $(g,p)\mapsto (gp,p)$.

定义 4.2.0.4. A $\mu$-torsor is trivial if $\mathcal{P}\cong \mu$ as $\mu$-torsor.

命题 4.2.0.5. If $\mu$ is representable on $\mathcal{C}=(\mathsf{Sch}/X{)}_{fppf}$ by a group scheme $G$, then$$\begin{array}{c}\{\text{Principle }G\text{-bundle}\}\stackrel{ϵ}{\to }\{\mu \text{-torsor}\}\\ P\mapsto h_P\end{array}$$is fully faithful. And if $G\to X$ is affine then $ϵ$ is equivalence.

证明. Yoneda says $P\mapsto h_P$ is fully faithful. Why $h_P$ is $\mu$-torsor? In other word, in def of torsor, $(2)$ holds by definition, but why we have $(1)$?

ConsiderWe want fppf cover $Y^{\prime }\twoheadrightarrow Y$ such that $P(Y^{\prime })\ne \varnothing$. However, note we haveHence we are done.

Now assume $G\to X$ is affine and $\mathcal{P}/X$ is $\mu$-torsor, we want $\mathcal{P}=h_P$ where $P$ is a $G$-bundle.

Note by assumption, we can find fppf cover $\{X_i\twoheadrightarrow X\}$ such that $\mathcal{P}(X_i)\ne \varnothing$. But then $\mathcal{P}{|}_{X_i}\cong {\mu }_{X_i}$ since $\backslash mu(X_i)$ acts on $\mathcal{P}(X_i)$ simple and transitive where ${\mu }_{X_i}=h_{G{\times }_X{X}_i}$, i.e. $\mathcal{P}$ is locally representable.

Since $\mathcal{P}$ is a sheaf over $X$, $\mathcal{P}{|}_{X_i}$ have canonical descent data. On the other hand, $\mathcal{P}{|}_{X_i}\cong G{\times }_X{X}_i$ is scheme affine over $X_i$. So, this yields descent data for the $G{\times }_X{X}_i$. But last result we showed last class is descent data is effective for affine morphisms. Therefore, we get a scheme $P\to X$ that also defines a sheaf $h_P$ which must agree with $\mathcal{P}$ because they are sheaves with same descent data.

Last thing we need to check is that why $P\to X$ is a $G$-bundle.

We need to check there exists action $G\times P\to P$ such that the graph of the action $G{\times }_{X}P\stackrel{\sim }{\to }P{\times }_{X}P$ is isomorphism.

Why $G{\times }_{X}P\to P$ exists? Well, we have $G\times P\to P$ by Yoneda because we have $\mu \times \mathcal{P}\to \mathcal{P}$ action such that $\mu \times \mathcal{P}\stackrel{\sim }{\to }\mathcal{P}\times \mathcal{P}$ so by Yoneda we get $G{\times }_{X}P\to P$ with $G\times P\stackrel{P}{\to }\times P$.

$\square$

注 4.2.0.6. If $G\to X$ is smooth, then $P\to X$ is also smooth. This is by fppf descent, i.e. we getand hence the arrow $P\to X$ is smooth as well. Note here smooth does nothing and it can be changed to basically any property.

命题 4.2.0.7. Let $X$ be a scheme, $\mathcal{F}$ be a sheaf on $X$, and $\mu =\mathrm{Aut}(\mathcal{F})$. Then there is an equivalence of categories$$\begin{array}{c}\mu -\text{torsors on }X\end{array}\stackrel{\sim }{\to }\begin{array}{c}\text{sheaves locally}\\ \text{isomorphic to }\mathcal{F}\end{array}$$given by map$$\mathcal{P}\mapsto {\mathcal{H}}_{\mathcal{P}}:={\mathrm{Hom}}_{\mu }(\mathcal{P},\mathcal{F})$$$${\mathcal{P}}_{\mathcal{H}}:=\mathrm{Isom}(\mathcal{F},\mathcal{H})\text{leftmapsto}\mathcal{H}$$where ${\mathrm{Hom}}_{\mu }$ means $\mu$-equivariant maps.

证明. We first show ${\mathcal{P}}_{\mathcal{H}}$ is $\mu$-torsor. We deinfe a map $\mu \times {\mathcal{P}}_{\mathcal{H}}\to {\mathcal{P}}_{\mathcal{H}}$ by $\mu (Y)\times {\mathcal{P}}_{\mathcal{H}}(Y)\to {\mathcal{P}}_{\mathcal{H}}(Y)$ by $(f,\lambda )\mapsto f\circ \lambda$, i.e. we get a diagram

Why is this simple and transitive? If ${\mathcal{P}}_{\mathcal{H}}(Y)\ne \varnothing$, then for $\lambda ,{\lambda }^{\prime }\in {\mathcal{P}}_{\mathcal{H}}(Y)$ we geti.e. ${\lambda }^{\prime }=p\circ \lambda$ for a unique $p$.

Why ${\mathcal{H}}_{\mathcal{P}}$ is a sheaf? GivenThen for ${\varphi }^{\prime }\in {\mathcal{P}}_{\mathcal{H}}(Y^{\prime })$ with ${\varphi }^{\prime }:{\mathcal{F}}_{Y^{\prime }}\to {\mathcal{H}}_{Y^{\prime }}$ $\mu$-equivariant map such that $p_1^{\ast }{\varphi }^{\prime }=p_2^{\ast }{\varphi }^{\prime }$, since sheaves satisfy descent so we get a unique $\varphi :{\mathcal{F}}_Y\to {\mathcal{H}}_Y$ such that $f^{\ast }\varphi ={\varphi }^{\prime }$.

Why $\varphi$ is $\mu$-equivariant? Viz, we want commutative diagramHowever, diagram commutes fppf locally over $Y^{\prime }$, hence it commutes over $Y$.

Why ${\mathcal{H}}_{\mathcal{P}}\cong \mathcal{F}$ locally?

Locally we get $\mu$-equivariant isomorphism $\mathcal{P}\cong \mu$. So we claim ${\mathcal{H}}_{\mathcal{P}}$ is locally isomorphic to ${\mathrm{Hom}}_{\mu }(\mu ,\mathcal{F})$. Well, suppose we have $\mu \stackrel{\alpha }{\to }\mathcal{F}$. Then we see $\alpha (\zeta )=\alpha (\zeta \cdot 1)=\zeta \cdot \alpha (1)$ and hence ${\mathrm{Hom}}_{\mu }(\mu ,\mathcal{F})\cong \mathcal{F}$ given by $\alpha \mapsto \alpha (1)$. This concludes the proof.

$\square$

注 4.2.0.8. If $\mathcal{F}$ is an ${\mathcal{O}}_X$-module, and $\mu ={\mathrm{Aut}}_{{\mathcal{O}}_X}(\mathcal{F})$, then $\mu$-torsor are equivalent to ${\mathcal{O}}_X$-modules $\mathcal{H}$ which are locally isomorphic to $\mathcal{F}$ as ${\mathcal{O}}_X$-modules.

例 4.2.0.9. Take $\mathcal{F}={\mathcal{O}}_X$ as ${\mathcal{O}}_X$-modules. Then we are shown that line bundles are isomorphic to $\mu$-torsors, where $\mu ={\mathrm{Aut}}_{{\mathcal{O}}_X}({\mathcal{O}}_X)={\mathcal{O}}_X^{\ast }$, where ${\mathcal{O}}_X^{\ast }$ is representable by ${\mathbb{G}}_m=\mathrm{Spec}\mathbb{Z}[x,x^{-1}]$. Hence, we see the line bundles are the same as ${\mathbb{G}}_m$-bundles.

Clearly we don’t have to restrict to ${\mathcal{O}}_X$. In other word, we can take $\mathcal{F}={\mathcal{O}}_X^n$, then $\mu ={\mathrm{GL}}_n({\mathcal{O}}_X)$, which is representable by scheme ${\mathrm{GL}}_n=\mathrm{Spec}\mathbb{Z}[x_{ij},x_{ij}^{-1}:1\le i,j\le n]$. We can also define this as $M_n:=\mathrm{Spec}\mathbb{Z}[x_{ij},1\le i,j\le n]$ and then $\mathrm{deg}(x_{ij})$ is a polynomial and we look at ${\mathrm{GL}}_n:=M_n\backslash V(\det )$, which is affine. In this case, we just get rank $n$ vector bundles are equivalent to ${\mathrm{GL}}_n$-bundles.

例 4.2.0.10. We can also talk about Brauer-Sever varieties, which are locally isomorphic to ${\mathbb{P}}^n$. Those are ${\mathrm{PGL}}_n$-torsors, and it is related to Azumaya algebras and the Brauer groups, where the Brauer groups are deeply related to class field theory in number theory.