The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

7.16 G-sets and morphisms

Let $\varphi : G \to H$ be a homomorphism of groups. Choose (suitable) sites $\mathcal{T}_ G$ and $\mathcal{T}_ H$ as in Example 7.6.5 and Section 7.9. Let $u : \mathcal{T}_ H \to \mathcal{T}_ G$ be the functor which assigns to a $H$-set $U$ the $G$-set $U_\varphi $ which has the same underlying set but $G$ action defined by $g \cdot u = \varphi (g)u$. It is clear that $u$ commutes with finite limits and is continuous1. Applying Proposition 7.14.7 and Lemma 7.15.2 we obtain a morphism of topoi

\[ f : \mathop{\mathit{Sh}}\nolimits (\mathcal{T}_ G) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{T}_ H) \]

associated with $\varphi $. Using Proposition 7.9.1 we see that we get a pair of adjoint functors

\[ f_* : G\textit{-Sets} \longrightarrow H\textit{-Sets}, \quad f^{-1} : H\textit{-Sets} \longrightarrow G\textit{-Sets}. \]

Let's work out what are these functors in this case.

We first work out a formula for $f_*$. Recall that given a $G$-set $S$ the corresponding sheaf $\mathcal{F}_ S$ on $\mathcal{T}_ G$ is given by the rule $\mathcal{F}_ S(U) = \mathop{Mor}\nolimits _ G(U, S)$. And on the other hand, given a sheaf $\mathcal{G}$ on $\mathcal{T}_ H$ the corresponding $H$-set is given by the rule $\mathcal{G}({}_ HH)$. Hence we see that

\[ f_*S = \mathop{Mor}\nolimits _{G\textit{-Sets}}(({}_ HH)_\varphi , S). \]

If we work this out a little bit more then we get

\[ f_*S = \{ a : H \to S \mid a(gh) = ga(h) \} \]

with left $H$-action given by $(h \cdot a)(h') = a(h'h)$ for any element $a \in f_*S$.

Next, we explicitly compute $f^{-1}$. Note that since the topology on $\mathcal{T}_ G$ and $\mathcal{T}_ H$ is subcanonical, all representable presheaves are sheaves. Moreover, given an object $V$ of $\mathcal{T}_ H$ we see that $f^{-1}h_ V$ is equal to $h_{u(V)}$ (see Lemma 7.13.5). Hence we see that $f^{-1}S = S_\varphi $ for representable sheaves. Since every sheaf on $\mathcal{T}_ H$ is a coproduct of representable sheaves we conclude that this is true in general. Hence we see that for any $H$-set $T$ we have

\[ f^{-1}T = T_\varphi . \]

The adjunction between $f^{-1}$ and $f_*$ is evidenced by the formula

\[ \mathop{Mor}\nolimits _{G\textit{-Sets}}(T_\varphi , S) = \mathop{Mor}\nolimits _{H\textit{-Sets}}(T, f_*S) \]

with $f_*S$ as above. This can be proved directly. Moreover, it is then clear that $(f^{-1}, f_*)$ form an adjoint pair and that $f^{-1}$ is exact. So alternatively to the above the morphism of topoi $f : G\textit{-Sets} \to H\textit{-Sets}$ can be defined directly in this manner.

[1] Set theoretical remark: First choose $\mathcal{T}_ H$. Then choose $\mathcal{T}_ G$ to contain $u(\mathcal{T}_ H)$ and such that every covering in $\mathcal{T}_ H$ corresponds to a covering in $\mathcal{T}_ G$. This is possible by Sets, Lemmas 3.10.1, 3.10.2 and 3.11.1.

Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04D4. Beware of the difference between the letter 'O' and the digit '0'.