LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Intersections of subschemes

Statement

For finitely many closed subschemes $Z_{1}, \dots, Z_{n} ↪ X$ with ideal sheaves $I_{1}, \dots, I_{n}$ , their scheme-theoretic intersection is their iterated fibre product over $X$ and is cut out by $I_{1} + \dots + I_{n}$ . For $n = 0$ the intersection and empty product over $X$ are $X$ , with zero ideal. For finitely many locally closed subschemes, restrict to the intersection of ambient opens in which they are closed and apply the same rule.

Facts & Assumptions

Given: The objects, hypotheses and conventions in the statement above.

[F1]

For $f : X \to S$ and a closed or locally closed subscheme $Z ↪ S$ , define the scheme-theoretic inverse image to be $X \times_{S} Z \to X$ . By lem-base-change-open-closed-immersions it is a closed or locally closed subscheme, respectively. For a closed ideal sheaf $I$ , the inverse-image ideal is $Im (f^{*} I \to O_{X})$ . For an open subscheme this construction is the open inverse image with its restricted sheaf. (Scheme-theoretic inverse images of subschemes)

[F2]

Let $A \to C$ be a unital ring map. For any set of variables $(t_{i})$ and any ideal $I \subseteq A [t_{i}]$ , $(A [t_{i}] / I) \otimes_{A} C ≅ C [t_{i}] / I C [t_{i}] .$ Here the extended ideal is generated by the coefficient images of all elements of $I$ . For a multiplicative subset $M \subseteq A$ , $(M^{- 1} A) \otimes_{A} C ≅ {\overline{M}}^{- 1} C .$ These are ring isomorphisms; no flatness, finite-generation or nonzero-ring hypothesis is required. (Presentations and localization under base extension)

[F3]

For $S$ -schemes $X, Y, Z$ there are natural projection-compatible isomorphisms $X \times_{S} Y ≅ Y \times_{S} X, (X \times_{S} Y) \times_{S} Z ≅ X \times_{S} (Y \times_{S} Z), X \times_{S} S ≅ X ≅ S \times_{S} X .$ Any coherence identity between these identifications holds whenever both sides induce the same ordered projections to the original factors. (Symmetry, associativity and units)

Proof

1.1

On $Spec A \subset X$ , F1 interprets pulling back $Z_{1}$ to $Z_{2}$ as the intersection. F2 gives $(A / I_{1}) \otimes_{A} (A / I_{2}) ≅ A / (I_{1} + I_{2})$ . A map to either side is exactly an $A$ -algebra map killing both ideals, so this formula respects the two projections.

givenF1F2

2.1

On principal restrictions both the quotient and ideal sum localize, so these descriptions glue. Repeating the two-ideal formula gives the finite sum; F3 identifies all bracketings and orderings. For one ideal nothing changes and for no ideals the relative terminal object is $X$ . A unit ideal gives an empty intersection, and zero ideals give unchanged factors.

F2F3step 1.1

3.1

If $Z_{i}$ is closed in an open $U_{i} \subset X$ , put $U = ⋂_{i} U_{i}$ . A test morphism factoring through every $Z_{i}$ necessarily factors through $U$ . Within $U$ the preceding closed-ideal calculation therefore represents exactly the same compatible test morphisms; composing its immersion with $U \to X$ gives the locally closed intersection. Nonreduced subschemes retain their ideal sums.

F1step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Vakil 10.2.C and H (standard reference, not scraped)