# The Stacks Project

## Tag: 03DC

This tag has label sites-modules-lemma-exactness-pushforward-pullback and it points to

The corresponding content:

Lemma 17.14.3. Let $f : (\mathop{\textit{Sh}}\nolimits(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\mathop{\textit{Sh}}\nolimits(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi.
1. The functor $f_*$ is left exact. In fact it commutes with all limits.
2. The functor $f^*$ is right exact. In fact it commutes with all colimits.

Proof. This is true because $(f^*, f_*)$ is an adjoint pair of functors, see Lemma 17.13.2. See Categories, Section 4.23. $\square$

\begin{lemma}
\label{lemma-exactness-pushforward-pullback}
Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$
be a morphism of ringed topoi.
\begin{enumerate}
\item The functor $f_*$ is left exact. In fact it commutes with
all limits.
\item The functor $f^*$ is right exact. In fact it commutes
with all colimits.
\end{enumerate}
\end{lemma}

\begin{proof}
This is true because $(f^*, f_*)$ is an adjoint pair
of functors, see
\end{proof}


To cite this tag (see How to reference tags), use:

\cite[\href{http://stacks.math.columbia.edu/tag/03DC}{Tag 03DC}]{stacks-project}


There are no comments yet for this tag.

## Add a comment on tag 03DC

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 lower-right corner).

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 box. So in case this is tag 0321 you just have to write 0321. This captcha seems more appropriate than the usual illegible gibberish, right?