Closed Immersion

In algebraic geometry, a closed immersion of schemes is a morphism of schemes $f: Z \to X$ that identifies ''Z'' as a closed subset of ''X'' such that locally, regular functions on ''Z'' can be extended to ''X''. The latter condition can be formalized by saying that $f^\#:\mathcal_X\rightarrow f_\ast\mathcal_Z$ is surjective.

An example is the inclusion map $\operatorname(R/I) \to \operatorname(R)$ induced by the canonical map $R \to R/I$.

Other characterizations

The following are equivalent:

#$f: Z \to X$ is a closed immersion.
#For every open affine $U = \operatorname(R) \subset X$, there exists an ideal $I \subset R$ such that $f^(U) = \operatorname(R/I)$ as schemes over ''U''.
#There exists an open affine covering $X = \bigcup U_j, U_j = \operatorname R_j$ and for each ''j'' there exists an ideal $I_j \subset R_j$ such that $f^(U_j) = \operatorname (R_j / I_j)$ as schemes over $U_j$.
#There is a quasi-coherent sheaf of ideals $\mathcal$ on ''X'' such that $f_\ast\mathcal_Z\cong \mathcal_X/\mathcal$ and ''f'' is an isomorphism of ''Z'' onto the global Spec of $\mathcal_X/\mathcal$ over ''X''.

Definition for locally ringed spaces

In the case of locally ringed spaces a morphism $i:Z\to X$ is a closed immersion if a similar list of criterion is satisfied

# The map $i$ is a homeomorphism of $Z$ onto its image
# The associated sheaf map $\mathcal_X \to i_*\mathcal_Z$ is surjective with kernel $\mathcal$
# The kernel $\mathcal$ is locally generated by sections as an $\mathcal_X$-module

The only varying condition is the third. It is instructive to look at a counter-example to get a feel for what the third condition yields by looking at a map which is not a closed immersion, $i:\mathbb_m\hookrightarrow \mathbb^1$ where

\mathbb_m = \text(\mathbb ,x^

If we look at the stalk of $i_*\mathcal_, _0$ at $0 \in \mathbb^1$ then there are no sections. This implies for any open subscheme $U \subset \mathbb^1$ containing $0$ the sheaf has no sections. This violates the third condition since at least one open subscheme $U$ covering $\mathbb^1$ contains $0$.

Properties

A closed immersion is finite and radicial (universally injective). In particular, a closed immersion is universally closed. A closed immersion is stable under base change and composition. The notion of a closed immersion is local in the sense that ''f'' is a closed immersion if and only if for some (equivalently every) open covering $X=\bigcup U_j$ the induced map $f:f^(U_j)\rightarrow U_j$ is a closed immersion.

If the composition $Z \to Y \to X$ is a closed immersion and $Y \to X$ is separated, then $Z \to Y$ is a closed immersion. If ''X'' is a separated ''S''-scheme, then every ''S''-section of ''X'' is a closed immersion.

If $i: Z \to X$ is a closed immersion and $\mathcal \subset \mathcal_X$ is the quasi-coherent sheaf of ideals cutting out ''Z'', then the direct image $i_*$ from the category of quasi-coherent sheaves over ''Z'' to the category of quasi-coherent sheaves over ''X'' is exact, fully faithful with the essential image consisting of $\mathcal$ such that $\mathcal \mathcal = 0$.

A flat closed immersion of finite presentation is the open immersion of an open closed subscheme.Stacks, Morphisms of schemes. Lemma 27.2

See also

* Segre embedding
*Regular embedding

Notes

References

*
*The Stacks Project
*{{Hartshorne AG

Morphisms of schemes