# The Stacks Project

## Tag: 01R4

This tag has label morphisms-lemma-conormal-functorial and it points to

The corresponding content:

Lemma 25.33.3. Let $$\xymatrix{ Z \ar[r]_i \ar[d]_f & X \ar[d]^g \\ Z' \ar[r]^{i'} & X' }$$ be a commutative diagram in the category of schemes. Assume $i$, $i'$ immersions. There is a canonical map of $\mathcal{O}_Z$-modules $$f^*\mathcal{C}_{Z'/X'} \longrightarrow \mathcal{C}_{Z/X}$$ characterized by the following property: For every pair of affine opens $(\mathop{\rm Spec}(R) = U \subset X, \mathop{\rm Spec}(R') = U' \subset X')$ with $f(U) \subset U'$ such that $Z \cap U = \mathop{\rm Spec}(R/I)$ and $Z' \cap U' = \mathop{\rm Spec}(R'/I')$ the induced map $$\Gamma(Z' \cap U', \mathcal{C}_{Z'/X'}) = I'/I'^2 \longrightarrow I/I^2 = \Gamma(Z \cap U, \mathcal{C}_{Z/X})$$ is the one induced by the ring map $f^\sharp : R' \to R$ which has the property $f^\sharp(I') \subset I$.

Proof. Let $\partial Z' = \overline{Z'} \setminus Z'$ and $\partial Z = \overline{Z} \setminus Z$. These are closed subsets of $X'$ and of $X$. Replacing $X'$ by $X' \setminus \partial Z'$ and $X$ by $X \setminus \big(g^{-1}(\partial Z') \cup \partial Z\big)$ we see that we may assume that $i$ and $i'$ are closed immersions.

The fact that $g \circ i$ factors through $i'$ implies that $g^*\mathcal{I}'$ maps into $\mathcal{I}$ under the canonical map $g^*\mathcal{I}' \to \mathcal{O}_X$, see Schemes, Lemmas 22.4.6 and 22.4.7. Hence we get an induced map of quasi-coherent sheaves $g^*(\mathcal{I}'/(\mathcal{I}')^2) \to \mathcal{I}/\mathcal{I}^2$. Pulling back by $i$ gives $i^*g^*(\mathcal{I}'/(\mathcal{I}')^2) \to i^*(\mathcal{I}/\mathcal{I}^2)$. Note that $i^*(\mathcal{I}/\mathcal{I}^2) = \mathcal{C}_{Z/X}$. On the other hand, $i^*g^*(\mathcal{I}'/(\mathcal{I}')^2) = f^*(i')^*(\mathcal{I}'/(\mathcal{I}')^2) = f^*\mathcal{C}_{Z'/X'}$. This gives the desired map.

Checking that the map is locally described as the given map $I'/(I')^2 \to I/I^2$ is a matter of unwinding the definitions and is omitted. Another observation is that given any $x \in i(Z)$ there do exist affine open neighbourhoods $U$, $U'$ with $f(U) \subset U'$ and $Z \cap U$ as well as $U' \cap Z'$ closed such that $x \in U$. Proof omitted. Hence the requirement of the lemma indeed characterizes the map (and could have been used to define it). $\square$

\begin{lemma}
\label{lemma-conormal-functorial}
Let
$$\xymatrix{ Z \ar[r]_i \ar[d]_f & X \ar[d]^g \\ Z' \ar[r]^{i'} & X' }$$
be a commutative diagram in the category of schemes.
Assume $i$, $i'$ immersions. There is a canonical map
of $\mathcal{O}_Z$-modules
$$f^*\mathcal{C}_{Z'/X'} \longrightarrow \mathcal{C}_{Z/X}$$
characterized by the following property: For every pair of affine opens
$(\Spec(R) = U \subset X, \Spec(R') = U' \subset X')$ with
$f(U) \subset U'$ such that
$Z \cap U = \Spec(R/I)$ and $Z' \cap U' = \Spec(R'/I')$
the induced map
$$\Gamma(Z' \cap U', \mathcal{C}_{Z'/X'}) = I'/I'^2 \longrightarrow I/I^2 = \Gamma(Z \cap U, \mathcal{C}_{Z/X})$$
is the one induced by the ring map $f^\sharp : R' \to R$ which
has the property $f^\sharp(I') \subset I$.
\end{lemma}

\begin{proof}
Let $\partial Z' = \overline{Z'} \setminus Z'$ and
$\partial Z = \overline{Z} \setminus Z$. These are closed subsets of $X'$ and
of $X$. Replacing $X'$ by $X' \setminus \partial Z'$ and $X$ by
$X \setminus \big(g^{-1}(\partial Z') \cup \partial Z\big)$ we
see that we may assume that $i$ and $i'$ are closed immersions.

\medskip\noindent
The fact that $g \circ i$ factors through $i'$ implies that
$g^*\mathcal{I}'$ maps into $\mathcal{I}$ under the canonical
map $g^*\mathcal{I}' \to \mathcal{O}_X$, see
Schemes, Lemmas
\ref{schemes-lemma-characterize-closed-subspace} and
\ref{schemes-lemma-restrict-map-to-closed}.
Hence we get an induced map of quasi-coherent sheaves
$g^*(\mathcal{I}'/(\mathcal{I}')^2) \to \mathcal{I}/\mathcal{I}^2$.
Pulling back by $i$ gives
$i^*g^*(\mathcal{I}'/(\mathcal{I}')^2) \to i^*(\mathcal{I}/\mathcal{I}^2)$.
Note that $i^*(\mathcal{I}/\mathcal{I}^2) = \mathcal{C}_{Z/X}$.
On the other hand,
$i^*g^*(\mathcal{I}'/(\mathcal{I}')^2) = f^*(i')^*(\mathcal{I}'/(\mathcal{I}')^2) = f^*\mathcal{C}_{Z'/X'}$.
This gives the desired map.

\medskip\noindent
Checking that the map is locally described as the given map
$I'/(I')^2 \to I/I^2$ is a matter of unwinding the definitions
and is omitted. Another observation is that given any
$x \in i(Z)$ there do exist affine open neighbourhoods $U$, $U'$
with $f(U) \subset U'$ and $Z \cap U$ as well as $U' \cap Z'$
closed such that $x \in U$. Proof omitted. Hence the requirement
of the lemma indeed characterizes the map (and could have been used
to define it).
\end{proof}


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

\cite[\href{http://stacks.math.columbia.edu/tag/01R4}{Tag 01R4}]{stacks-project}


## Comments (0)

There are no comments yet for this tag.

## Add a comment on tag 01R4

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 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?