LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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A local morphism of stalks induces a residue-field map

Statement

Let

(f, f^{♯}) : (X, O_{X}) \to (Y, O_{Y})

be a morphism of locally ringed spaces, and let $x \in X$ . Then the local stalk map

f_{x}^{♯} : O_{Y, f (x)} ⟶ O_{X, x}

induces a field homomorphism

κ (f (x)) ⟶ κ (x)

between residue fields.

Facts & Assumptions

Given: A morphism of locally ringed spaces $(f, f^{♯}) : (X, O_{X}) \to (Y, O_{Y})$ and a point $x \in X$ .

[F1]

A local ring has a unique maximal ideal, and its residue field is the quotient by that ideal (A local ring is a nonzero commutative ring with a unique maximal ideal).

[F2]

In a morphism of locally ringed spaces, the stalk map $f_{x}^{♯} : O_{Y, f (x)} \to O_{X, x}$ sends the maximal ideal of the source into the maximal ideal of the target (Morphisms of locally ringed spaces).

Proof

technique · direct

1.1F1F2given

Let $m_{f (x)}$ and $m_{x}$ be the maximal ideals of the two stalks. By [F2], the composite $O_{Y, f (x)} \overset{f_{x}^{♯}}{\to} O_{X, x} ⟶ O_{X, x} / m_{x}$ kills $m_{f (x)}$ , so it factors through the quotient $O_{Y, f (x)} / m_{f (x)} .$

2.1F1step 1.1∎

By [F1], the source and target quotients in step 1.1 are exactly the residue fields $κ (f (x))$ and $κ (x)$ . Therefore step 1.1 is the desired field homomorphism $κ (f (x)) \to κ (x) .$

Depends on

Used by

Field-valued points and local-ring points Lemma

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Section 26.2 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, Section 6.3.1 (standard reference, not scraped)