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Tag 03AG

21.5. First cohomology and torsors

Definition 21.5.1. 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. (2)    for every $U \in \mathop{\rm Ob}\nolimits(\mathcal{C})$ 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 21.5.2. 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 21.5.3. 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 $U \in \mathop{\rm Ob}\nolimits(\mathcal{C})$ 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 $a : \mathop{\rm Ker}(\sigma) \to \mathcal{H}$ 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 $$ \xymatrix{ 0 \ar[r] & \mathop{\rm Ker}(\sigma) \ar[r] \ar[d]^a & \mathbf{Z}[\mathcal{F}] \ar[r] \ar[d] & \underline{\mathbf{Z}} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{H} \ar[r] & \mathcal{E} \ar[r] & \underline{\mathbf{Z}} \ar[r] & 0 } $$ 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. $\square$

    The code snippet corresponding to this tag is a part of the file sites-cohomology.tex and is located in lines 261–390 (see updates for more information).

    \section{First cohomology and torsors}
    \label{section-h1-torsors}
    
    \begin{definition}
    \label{definition-torsor}
    Let $\mathcal{C}$ be a site.
    Let $\mathcal{G}$ be a sheaf of (possibly non-commutative)
    groups on $\mathcal{C}$.
    A {\it pseudo torsor}, or more precisely a
    {\it 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
    \begin{enumerate}
    \item whenever $\mathcal{F}(U)$ is nonempty the action
    $\mathcal{G}(U) \times \mathcal{F}(U) \to \mathcal{F}(U)$
    is simply transitive.
    \end{enumerate}
    A {\it 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 {\it torsor}, or more precisely a
    {\it $\mathcal{G}$-torsor}, is a pseudo $\mathcal{G}$-torsor such that
    in addition
    \begin{enumerate}
    \item[(2)] for every $U \in \Ob(\mathcal{C})$
    there exists a covering $\{U_i \to U\}_{i \in I}$ of $U$
    such that $\mathcal{F}(U_i)$ is nonempty for all $i \in I$.
    \end{enumerate}
    A {\it morphism of $\mathcal{G}$-torsors} is simply a morphism of
    pseudo $\mathcal{G}$-torsors.
    The {\it trivial $\mathcal{G}$-torsor}
    is the sheaf $\mathcal{G}$ endowed with the obvious left
    $\mathcal{G}$-action.
    \end{definition}
    
    \noindent
    It is clear that a morphism of torsors is automatically an isomorphism.
    
    \begin{lemma}
    \label{lemma-trivial-torsor}
    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
    $\Gamma(\mathcal{C}, \mathcal{F}) \not = \emptyset$.
    \end{lemma}
    
    \begin{proof}
    Omitted.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-torsors-h1}
    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 $H^1(\mathcal{C}, \mathcal{H})$.
    \end{lemma}
    
    \begin{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 $U \in \Ob(\mathcal{C})$ 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
    $a : \Ker(\sigma) \to \mathcal{H}$
    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
    $$
    \xymatrix{
    0 \ar[r] &
    \Ker(\sigma) \ar[r] \ar[d]^a &
    \mathbf{Z}[\mathcal{F}] \ar[r] \ar[d] &
    \underline{\mathbf{Z}} \ar[r] \ar[d] &
    0 \\
    0 \ar[r] &
    \mathcal{H} \ar[r] &
    \mathcal{E} \ar[r] &
    \underline{\mathbf{Z}} \ar[r] &
    0
    }
    $$
    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
    $1 \in H^0(\mathcal{C}, \underline{\mathbf{Z}})$.
    
    \medskip\noindent
    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 \ref{derived-lemma-higher-derived-functors}).
    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.
    
    \medskip\noindent
    We omit the verification that the two constructions given
    above are mutually inverse.
    \end{proof}

    Comments (2)

    Comment #2761 by Anonymous on August 6, 2017 a 7:26 am UTC

    Typos? On this page both $\mathcal{G}$ and $G$ are repeatedly used for what (I think) should be the same object.

    Comment #2869 by Johan (site) on October 4, 2017 a 6:25 pm UTC

    Thanks, fixed here.

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