The Stacks project

Lemma 26.4.7. Let $f : X \to Y$ be a morphism of locally ringed spaces. Let $\mathcal{I} \subset \mathcal{O}_ Y$ be a sheaf of ideals which is locally generated by sections. Let $i : Z \to Y$ be the closed subspace associated to the sheaf of ideals $\mathcal{I}$. Let $\mathcal{J}$ be the image of the map $f^*\mathcal{I} \to f^*\mathcal{O}_ Y = \mathcal{O}_ X$. Then this ideal is locally generated by sections. Moreover, let $i' : Z' \to X$ be the associated closed subspace of $X$. There exists a unique morphism of locally ringed spaces $f' : Z' \to Z$ such that the following diagram is a commutative square of locally ringed spaces

\[ \xymatrix{ Z' \ar[d]_{f'} \ar[r]_{i'} & X \ar[d]^ f \\ Z \ar[r]^{i} & Y } \]

Moreover, this diagram is a fibre square in the category of locally ringed spaces.

Proof. The ideal $\mathcal{J}$ is locally generated by sections by Modules, Lemma 17.8.2. The rest of the lemma follows from the characterization, in Lemma 26.4.6 above, of what it means for a morphism to factor through a closed subspace. $\square$


Comments (5)

Comment #422 by Keenan Kidwell on

In the proof of 01HQ, the last word should be subspace instead of subscheme, since this lemma and the lemmas cited in the proof are about general locally ringed spaces.

Comment #582 by Anfang on

There is a typo in the second sentence. "Functions" should be changed to "sections".

Comment #8454 by on

Slogan: closed immersions of locally ringed spaces are stable under base change.

Comment #8455 by on

Here's the proof of cartesianity of the square from the statement: we need to use

Exercise. Let be a morphism of ringed spaces and let be a morphism of -modules. Then . (Hint: verify it on stalks and use that if is a commutative ring, then , for an -module and a morphism of -modules.)

Suppose we have morphisms and fitting into a commutative square which terminates in . If a factorization of this square through were to exist, then it would be unique for is monic (closed immersions of l.r.s. are monomorphisms). We see existence: We construct a map factoring through with 26.4.6: we must show that the map vanishes. But we have by 26.4.6. On the other hand, the induced map can be seen to also factor through using the monicity of .

There are also:

  • 18 comment(s) on Section 26.4: Closed immersions of locally ringed spaces

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