The Stacks project

21.4 First cohomology and torsors

Definition 21.4.1. Let $\mathcal{C}$ be a site. Let $\mathcal{G}$ be a sheaf of (possibly non-commutative) groups on $\mathcal{C}$. A pseudo torsor, or more precisely a pseudo $\mathcal{G}$-torsor, is a sheaf of sets $\mathcal{F}$ on $\mathcal{C}$ endowed with an action $\mathcal{G} \times \mathcal{F} \to \mathcal{F}$ such that

  1. whenever $\mathcal{F}(U)$ is nonempty the action $\mathcal{G}(U) \times \mathcal{F}(U) \to \mathcal{F}(U)$ is simply transitive.

A morphism of pseudo $\mathcal{G}$-torsors $\mathcal{F} \to \mathcal{F}'$ is simply a morphism of sheaves of sets compatible with the $\mathcal{G}$-actions. A torsor, or more precisely a $\mathcal{G}$-torsor, is a pseudo $\mathcal{G}$-torsor such that in addition

  1. for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ there exists a covering $\{ U_ i \to U\} _{i \in I}$ of $U$ such that $\mathcal{F}(U_ i)$ is nonempty for all $i \in I$.

A morphism of $\mathcal{G}$-torsors is simply a morphism of pseudo $\mathcal{G}$-torsors. The trivial $\mathcal{G}$-torsor is the sheaf $\mathcal{G}$ endowed with the obvious left $\mathcal{G}$-action.

It is clear that a morphism of torsors is automatically an isomorphism.

Lemma 21.4.2. Let $\mathcal{C}$ be a site. Let $\mathcal{G}$ be a sheaf of (possibly non-commutative) groups on $\mathcal{C}$. A $\mathcal{G}$-torsor $\mathcal{F}$ is trivial if and only if $\Gamma (\mathcal{C}, \mathcal{F}) \not= \emptyset $.

Proof. Omitted. $\square$

Lemma 21.4.3. Let $\mathcal{C}$ be a site. Let $\mathcal{H}$ be an abelian sheaf on $\mathcal{C}$. There is a canonical bijection between the set of isomorphism classes of $\mathcal{H}$-torsors and $H^1(\mathcal{C}, \mathcal{H})$.

Proof. Let $\mathcal{F}$ be a $\mathcal{H}$-torsor. Consider the free abelian sheaf $\mathbf{Z}[\mathcal{F}]$ on $\mathcal{F}$. It is the sheafification of the rule which associates to $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ the collection of finite formal sums $\sum n_ i[s_ i]$ with $n_ i \in \mathbf{Z}$ and $s_ i \in \mathcal{F}(U)$. There is a natural map

\[ \sigma : \mathbf{Z}[\mathcal{F}] \longrightarrow \underline{\mathbf{Z}} \]

which to a local section $\sum n_ i[s_ i]$ associates $\sum n_ i$. The kernel of $\sigma $ is generated by sections of the form $[s] - [s']$. There is a canonical map $a : \mathop{\mathrm{Ker}}(\sigma ) \to \mathcal{H}$ which maps $[s] - [s'] \mapsto h$ where $h$ is the local section of $\mathcal{H}$ such that $h \cdot s = s'$. Consider the pushout diagram

\[ \xymatrix{ 0 \ar[r] & \mathop{\mathrm{Ker}}(\sigma ) \ar[r] \ar[d]^ a & \mathbf{Z}[\mathcal{F}] \ar[r] \ar[d] & \underline{\mathbf{Z}} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{H} \ar[r] & \mathcal{E} \ar[r] & \underline{\mathbf{Z}} \ar[r] & 0 } \]

Here $\mathcal{E}$ is the extension obtained by pushout. From the long exact cohomology sequence associated to the lower short exact sequence we obtain an element $\xi = \xi _\mathcal {F} \in H^1(\mathcal{C}, \mathcal{H})$ by applying the boundary operator to $1 \in H^0(\mathcal{C}, \underline{\mathbf{Z}})$.

Conversely, given $\xi \in H^1(\mathcal{C}, \mathcal{H})$ we can associate to $\xi $ a torsor as follows. Choose an embedding $\mathcal{H} \to \mathcal{I}$ of $\mathcal{H}$ into an injective abelian sheaf $\mathcal{I}$. We set $\mathcal{Q} = \mathcal{I}/\mathcal{H}$ so that we have a short exact sequence

\[ \xymatrix{ 0 \ar[r] & \mathcal{H} \ar[r] & \mathcal{I} \ar[r] & \mathcal{Q} \ar[r] & 0 } \]

The element $\xi $ is the image of a global section $q \in H^0(\mathcal{C}, \mathcal{Q})$ because $H^1(\mathcal{C}, \mathcal{I}) = 0$ (see Derived Categories, Lemma 13.20.4). Let $\mathcal{F} \subset \mathcal{I}$ be the subsheaf (of sets) of sections that map to $q$ in the sheaf $\mathcal{Q}$. It is easy to verify that $\mathcal{F}$ is a $\mathcal{H}$-torsor.

We omit the verification that the two constructions given above are mutually inverse. $\square$

Comments (5)

Comment #2761 by Anonymous on

Typos? On this page both and are repeatedly used for what (I think) should be the same object.

Comment #4521 by Simon on

In the last paragraph, could someone explain why it is easy to verify that is a -torsor. I don't see it, thanks!

Comment #4522 by on

The sections of the sheaf over an object of the site are the elements whose image in are equal to the restriction . Now you have to define a simply transitive action of on this set provided it is nonempty and the construction has to be compatible with restriction mappings. Finally, you have to show that if is empty, then you can find a covering such that each is nonempty. Does this help?

Comment #8676 by Gabriel on

On lemma 03AJ we have a bijection between two abelian groups. It would be nice to say that this is indeed an isomorphism. (Said otherwise, it would be nice to explain that the contracted product of torsors "is" the addition on H^1.)

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