Interior point method

For the same-name concept in differential geometry, see immersion (mathematics).

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 regular functions on Z can be extended locally to X. The latter condition can be formalized by saying that $f^{\#}:{\mathcal{𝒪}}_X\to f_{\ast }{\mathcal{𝒪}}_Z$ is surjective.

A basic example is the inclusion map $\mathrm{Spec}(R/I)\to \mathrm{Spec}(R)$ induced by the canonical map $R\to R/I$.

Other characterizations

The following are equivalent:

1. $f:Z\to X$ is a closed immersion.
2. For every open affine $U=\mathrm{Spec}(R)\subset X$, there exists an ideal $I\subset R$ such that $f^{-1}(U)=\mathrm{Spec}(R/I)$ as schemes over U.
3. There exists an open affine covering $X=\bigcup U_j,U_j=\mathrm{Spec}{R}_j$ and for each j there exists an ideal $I_j\subset R_j$ such that $f^{-1}(U_j)=\mathrm{Spec}(R_j/I_j)$ as schemes over $U_j$.
4. There is a quasi-coherent sheaf of ideals $\mathcal{ℐ}$ on X such that $f_{\ast }{\mathcal{𝒪}}_Z\cong {\mathcal{𝒪}}_X/\mathcal{ℐ}$ and f is an isomorphism of Z onto the global Spec of ${\mathcal{𝒪}}_X/\mathcal{ℐ}$ over X.

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 $f:f^{-1}(U_j)\to U_j$ 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 $\mathcal{ℐ}\subset {\mathcal{𝒪}}_X$ 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 $\mathcal{𝒢}$ such that $\mathcal{ℐ}\mathcal{𝒢}=0$.

A flat closed immersion of finite presentation is the open immersion of an open closed subscheme.

See also

• Segre embedding

Notes

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References

• Template:EGA
• The Stacks Project
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