ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-26

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.

Localised Hom can fail without finite presentation of the source

Example

Fix a prime number $p$ , let $S = {p^{n} : n \in N}$ , let $M = ⨁_{n \geq 0} Z e_{n},$ and let $N = Z$ . The source $M$ is not finitely presented, and the natural map $S^{- 1} ⁣ {Hom}_{Z} (M, N) ⟶ {Hom}_{Z [1 / p]} (S^{- 1} M, S^{- 1} N)$ is not surjective.

Facts & Assumptions

Given: A prime number $p$ , the multiplicative set $S = {p^{n} : n \in N}$ , the free $Z$ -module $M = ⨁_{n \geq 0} Z e_{n}$ , and the target module $N = Z$ .

[L1]

The finite-presentation theorem gives an isomorphism only under finite-presentation hypotheses on the source (Localisation of Hom for finite and finitely presented modules).

[L2]

Localisation commutes with direct sums, so $S^{- 1} M ≅ ⨁_{n \geq 0} Z [1 / p] e_{n}$ (Localisation commutes with quotient modules and arbitrary direct sums).

[L3]

A homomorphism out of a direct sum is determined by its values on the coordinate inclusions (Universal property of a direct sum of modules, The direct sum of an indexed family of modules).

Verification

technique · direct

1.1L3algebra

The module $M$ is free on countably many generators, so it is not finitely generated and therefore not finitely presented.

1.2L2L3construct

By [L2] and [L3], there is a $Z [1 / p]$ -linear map $φ : S^{- 1} M \to Z [1 / p]$ with $φ (e_{n}) = 1 / p^{n}$ for every $n \geq 0$ .

2.1step 1.2algebra

Suppose $φ$ were in the image of the localisation-of-Hom map. Then there would be a homomorphism $f : M \to Z$ and an integer $r \geq 0$ such that $φ (e_{n}) = f (e_{n}) / p^{r}$ for every $n$ . Taking $n > r$ gives $f (e_{n}) = p^{r - n}$ , impossible in $Z$ . Therefore $φ$ is not in the image.

3.1L1step 1.1step 2.1∎

So the localisation-of-Hom map fails to be surjective for this non-finitely-presented source, exactly as warned by [L1].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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

A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., Exercise 12.26 (standard reference, not scraped)