The Stacks project

Lemma 7.28.5. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a cocontinuous functor. Let $V$ be an object of $\mathcal{D}$. Let ${}^ u_ V\mathcal{I}$ be the category introduced in Section 7.19. We have a commutative diagram

\[ \vcenter { \xymatrix{ \, _ V^ u\mathcal{I} \ar[r]_ j \ar[d]_{u'} & \mathcal{C} \ar[d]^ u \\ \mathcal{D}/V \ar[r]^-{j_ V} & \mathcal{D} } } \quad \text{where}\quad \begin{matrix} j : (U, \psi ) \mapsto U \\ u' : (U, \psi ) \mapsto (\psi : u(U) \to V) \end{matrix} \]

Declare a family of morphisms $\{ (U_ i, \psi _ i) \to (U, \psi )\} $ of ${}^ u_ V\mathcal{I}$ to be a covering if and only if $\{ U_ i \to U\} $ is a covering in $\mathcal{C}$. Then

  1. ${}^ u_ V\mathcal{I}$ is a site,

  2. $j$ is continuous and cocontinuous,

  3. $u'$ is cocontinuous,

  4. we get a commutative diagram of topoi

    \[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits ({}^ u_ V\mathcal{I}) \ar[r]_ j \ar[d]_{f'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[d]^ f \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V) \ar[r]^-{j_ V} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) } \]

    where $f$ (resp. $f'$) corresponds to $u$ (resp. $u'$), and

  5. we have $f'_*j^{-1} = j_ V^{-1}f_*$.

Proof. Parts (1), (2), (3), and (4) are straightforward consequences of the definitions and the fact that the functor $j$ commutes with fibre products. We omit the details. To see (5) recall that $f_*$ is given by ${}_ su = {}_ pu$. Hence the value of $j_ V^{-1}f_*\mathcal{F}$ on $V'/V$ is the value of ${}_ pu\mathcal{F}$ on $V'$ which is the limit of the values of $\mathcal{F}$ on the category ${}^ u_{V'}\mathcal{I}$. Clearly, there is an equivalence of categories

\[ {}^ u_{V'}\mathcal{I} \to {}^{u'}_{V'/V}\mathcal{I} \]

Since the value of $f'_*j^{-1}\mathcal{F}$ on $V'/V$ is given by the limit of the values of $j^{-1}\mathcal{F}$ on the category ${}^{u'}_{V'/V}\mathcal{I}$ and since the values of $j^{-1}\mathcal{F}$ on objects of ${}^ u_ V\mathcal{I}$ are just the values of $\mathcal{F}$ (by Lemma 7.21.5 as $j$ is continuous and cocontinuous) we see that (5) is true. $\square$

Comments (0)

There are also:

  • 3 comment(s) on Section 7.28: Localization and morphisms

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Your email address will not be published.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 0D5R. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 7.28.5 as pdf