The Stacks project

91.9 Thickenings of ringed topoi

This section is the analogue of Section 91.3 for ringed topoi. In the following few sections we will use the following notions:

  1. A sheaf of ideals $\mathcal{I} \subset \mathcal{O}'$ on a ringed topos $(\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ is locally nilpotent if any local section of $\mathcal{I}$ is locally nilpotent.

  2. A thickening of ringed topoi is a morphism $i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ of ringed topoi such that

    1. $i_*$ is an equivalence $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$,

    2. the map $i^\sharp : \mathcal{O}' \to i_*\mathcal{O}$ is surjective, and

    3. the kernel of $i^\sharp $ is a locally nilpotent sheaf of ideals.

  3. A first order thickening of ringed topoi is a thickening $i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ of ringed topoi such that $\mathop{\mathrm{Ker}}(i^\sharp )$ has square zero.

  4. It is clear how to define morphisms of thickenings of ringed topoi, morphisms of thickenings of ringed topoi over a base ringed topos, etc.

If $i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ is a thickening of ringed topoi then we identify the underlying topoi and think of $\mathcal{O}$, $\mathcal{O}'$, and $\mathcal{I} = \mathop{\mathrm{Ker}}(i^\sharp )$ as sheaves on $\mathcal{C}$. We obtain a short exact sequence

\[ 0 \to \mathcal{I} \to \mathcal{O}' \to \mathcal{O} \to 0 \]

of $\mathcal{O}'$-modules. By Modules on Sites, Lemma 18.25.1 the category of $\mathcal{O}$-modules is equivalent to the category of $\mathcal{O}'$-modules annihilated by $\mathcal{I}$. In particular, if $i$ is a first order thickening, then $\mathcal{I}$ is a $\mathcal{O}$-module.

Situation 91.9.1. A morphism of thickenings of ringed topoi $(f, f')$ is given by a commutative diagram

91.9.1.1
\begin{equation} \label{defos-equation-morphism-thickenings-ringed-topoi} \vcenter { \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \ar[r]_ i \ar[d]_ f & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}') \ar[d]^{f'} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}), \mathcal{O}_\mathcal {B}) \ar[r]^ t & (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}'), \mathcal{O}_{\mathcal{B}'}) } } \end{equation}

of ringed topoi whose horizontal arrows are thickenings. In this situation we set $\mathcal{I} = \mathop{\mathrm{Ker}}(i^\sharp ) \subset \mathcal{O}'$ and $\mathcal{J} = \mathop{\mathrm{Ker}}(t^\sharp ) \subset \mathcal{O}_{\mathcal{B}'}$. As $f = f'$ on underlying topoi we will identify the pullback functors $f^{-1}$ and $(f')^{-1}$. Observe that $(f')^\sharp : f^{-1}\mathcal{O}_{\mathcal{B}'} \to \mathcal{O}'$ induces in particular a map $f^{-1}\mathcal{J} \to \mathcal{I}$ and therefore a map of $\mathcal{O}'$-modules

\[ (f')^*\mathcal{J} \longrightarrow \mathcal{I} \]

If $i$ and $t$ are first order thickenings, then $(f')^*\mathcal{J} = f^*\mathcal{J}$ and the map above becomes a map $f^*\mathcal{J} \to \mathcal{I}$.

Definition 91.9.2. In Situation 91.9.1 we say that $(f, f')$ is a strict morphism of thickenings if the map $(f')^*\mathcal{J} \longrightarrow \mathcal{I}$ is surjective.


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 08M6. Beware of the difference between the letter 'O' and the digit '0'.