# 4.2. G-丛

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

Thus, let $X$ be a scheme and $C((Sch)/X)_{fppf}$. 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 (principle) $G$-bundle is a scheme $π:P→X$ over $X$, where $π$ is fppf, with an $G$-action $ρ:G×P→P$, such that the map $G×_{X}P∼ P×_{X}P$ given by $(g,p)↦(gp,p)$, is isomorphism, where $gp=ρ(g,p)$.

This $G×_{X}P∼ P×_{X}P$ condition is equivalent to saying if $Y→X$ and $P(Y)=∅$, 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 $p,p_{′}∈P(Y)$ there exists $g∈G(Y)$ so $gp=p_{′}$, i.e. for all $p,p_{′}∈P(Y)$ there exists unique $g∈G(Y)$ so $p_{′}=gp$.

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 $G$-bundles $P→P_{′}$ is a map of $S$-schemes $f:P→P_{′}$ such that

定义 4.2.0.3. Let $C$ be a site and $μ$ a sheaf of groups. A $μ$-torsor is a sheaf $P$ with $μ×P→P$ such that:

1. | for all $X∈C$, there exists ${X_{i}→X}∈Cov(X)$ such that $P(X_{i})=∅$ for all $i$. |

2. | we have $μ×P∼ P×P$ where the map is given by $(g,p)↦(gp,p)$. |

定义 4.2.0.4. A $μ$-torsor is trivial if $P≅μ$ as $μ$-torsor.

命题 4.2.0.5. If $μ$ is representable on $C=((Sch)/X)_{fppf}$ by a group scheme $G$, then${PrincipleG-bundle}ϵ {μ-torsor}P↦h_{P} $is fully faithful. And if $G→X$ is affine then $ϵ$ is equivalence.

**证明.** Yoneda says $P↦h_{P}$ is fully faithful. Why $h_{P}$ is $μ$-torsor? In other word, in def of torsor, $(2)$ holds by definition, but why we have $(1)$?

ConsiderWe want fppf cover $Y_{′}↠Y$ such that $P(Y_{′})=∅$. However, note we haveHence we are done.

Now assume $G→X$ is affine and $P/X$ is $μ$-torsor, we want $P=h_{P}$ where $P$ is a $G$-bundle.

Note by assumption, we can find fppf cover ${X_{i}↠X}$ such that $P(X_{i})=∅$. But then $P∣_{X_{i}}≅μ_{X_{i}}$ since $\mu(X_{i})$ acts on $P(X_{i})$ simple and transitive where $μ_{X_{i}}=h_{G×_{X}X_{i}}$, i.e. $P$ is locally representable.

Since $P$ is a sheaf over $X$, $P∣_{X_{i}}$ have canonical descent data. On the other hand, $P∣_{X_{i}}≅G×_{X}X_{i}$ is scheme affine over $X_{i}$. So, this yields descent data for the $G×_{X}X_{i}$. But last result we showed last class is descent data is effective for affine morphisms. Therefore, we get a scheme $P→X$ that also defines a sheaf $h_{P}$ which must agree with $P$ because they are sheaves with same descent data.

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

We need to check there exists action $G×P→P$ such that the graph of the action $G×_{X}P∼ P×_{X}P$ is isomorphism.

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注 4.2.0.6. If $G→X$ is smooth, then $P→X$ is also smooth. This is by fppf descent, i.e. we getand hence the arrow $P→X$ 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, $F$ be a sheaf on $X$, and $μ=Aut(F)$. Then there is an equivalence of categories$μ−torsors onX ∼ sheaves locallyisomorphic toF $given by map$P↦H_{P}:=Hom_{μ}(P,F)$$P_{H}:=Isom(F,H)\leftmapstoH$where $Hom_{μ}$ means $μ$-equivariant maps.

**证明.** We first show $P_{H}$ is $μ$-torsor. We deinfe a map $μ×P_{H}→P_{H}$ by $μ(Y)×P_{H}(Y)→P_{H}(Y)$ by $(f,λ)↦f∘λ$, i.e. we get a diagram

Why is this simple and transitive? If $P_{H}(Y)=∅$, then for $λ,λ_{′}∈P_{H}(Y)$ we geti.e. $λ_{′}=p∘λ$ for a unique $p$.

Why $H_{P}$ is a sheaf? GivenThen for $ϕ_{′}∈P_{H}(Y_{′})$ with $ϕ_{′}:F_{Y_{′}}→H_{Y_{′}}$ $μ$-equivariant map such that $p_{1}ϕ_{′}=p_{2}ϕ_{′}$, since sheaves satisfy descent so we get a unique $ϕ:F_{Y}→H_{Y}$ such that $f_{∗}ϕ=ϕ_{′}$.

Why $ϕ$ is $μ$-equivariant? Viz, we want commutative diagramHowever, diagram commutes fppf locally over $Y_{′}$, hence it commutes over $Y$.

Why $H_{P}≅F$ locally?

Locally we get $μ$-equivariant isomorphism $P≅μ$. So we claim $H_{P}$ is locally isomorphic to $Hom_{μ}(μ,F)$. Well, suppose we have $μα F$. Then we see $α(ζ)=α(ζ⋅1)=ζ⋅α(1)$ and hence $Hom_{μ}(μ,F)≅F$ given by $α↦α(1)$. This concludes the proof.

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注 4.2.0.8. If $F$ is an $O_{X}$-module, and $μ=Aut_{O_{X}}(F)$, then $μ$-torsor are equivalent to $O_{X}$-modules $H$ which are locally isomorphic to $F$ as $O_{X}$-modules.

例 4.2.0.9. Take $F=O_{X}$ as $O_{X}$-modules. Then we are shown that line bundles are isomorphic to $μ$-torsors, where $μ=Aut_{O_{X}}(O_{X})=O_{X}$, where $O_{X}$ is representable by $G_{m}=SpecZ[x,x_{−1}]$. Hence, we see the line bundles are the same as $G_{m}$-bundles.

Clearly we don’t have to restrict to $O_{X}$. In other word, we can take $F=O_{X}$, then $μ=GL_{n}(O_{X})$, which is representable by scheme $GL_{n}=SpecZ[x_{ij},x_{ij}:1≤i,j≤n]$. We can also define this as $M_{n}:=SpecZ[x_{ij},1≤i,j≤n]$ and then $g(x_{ij})$ is a polynomial and we look at $GL_{n}:=M_{n}\V(det)$, which is affine. In this case, we just get rank $n$ vector bundles are equivalent to $GL_{n}$-bundles.

例 4.2.0.10. We can also talk about Brauer-Sever varieties, which are locally isomorphic to $P_{n}$. Those are $PGL_{n}$-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.