TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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.

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Coextension of scalars is right adjoint to restriction of scalars

Statement

For a unital ring homomorphism $φ : R \to S$ , restriction of scalars

$Res : S - M o d \to R - M o d$

is left adjoint to coextension of scalars

$Coind (M) : = {Hom}_{R} (S, M) .$

Naturally in an $S$ -module $N$ and an $R$ -module $M$ ,

${Hom}_{S} (N, {Hom}_{R} (S, M)) ≅ {Hom}_{R} (Res N, M) .$

Facts & Assumptions

Given: The ring homomorphism $φ : R \to S$ , a left $S$ -module $N$ , and a left $R$ -module $M$ .

[L1]

On ${Hom}_{R} (S, M)$ the canonical left $S$ -action is $(s \cdot h) (t) = h (t s)$ (Coextension of scalars ${Hom}_{R} (S, M)$ carries its canonical left $S$ -module structure).

[F1]

Left modules and module homomorphisms form locally small categories (Left modules over a fixed ring and module homomorphisms form the large locally small category $R - M o d$ ).

[L2]

A natural family of hom-set bijections determines an adjunction (Under local smallness, transposition gives the natural hom-set bijection, and conversely).

Proof

technique · direct

1.1L1algebra

Given an $S$ -linear map $α : N \to {Hom}_{R} (S, M)$ , define $E (α) : Res N \to M$ by $E (α) (n) = α (n) (1_{S})$ . It is $R$ -linear because $α (φ (r) n) = φ (r) \cdot α (n)$ and [L1] gives $(φ (r) \cdot α (n)) (1) = α (n) (φ (r)) = r α (n) (1)$ .

1.2constructalgebra

Given an $R$ -linear map $u : Res N \to M$ , define $C (u) (n) (s) : = u (s n)$ . For $r \in R$ , $C (u) (n) (φ (r) s) = u (φ (r) s n) = r u (s n)$ , so $C (u) (n)$ is $R$ -linear.

2.1step 1.2L1

For $a, s \in S$ , $C (u) (a n) (s) = u (s a n) = C (u) (n) (s a) = (a \cdot C (u) (n)) (s)$ by [L1], so $C (u)$ is $S$ -linear.

2.2step 1.1step 1.2L1

Evaluation gives $E (C (u)) (n) = C (u) (n) (1) = u (n)$ . Conversely, $S$ -linearity of $α$ gives $C (E (α)) (n) (s) = E (α) (s n) = α (s n) (1) = (s \cdot α (n)) (1) = α (n) (s)$ .

3.1step 2.1step 2.2F1

Thus $E$ and $C$ are inverse bijections. Precomposition in $N$ and postcomposition in $M$ commute with the two displayed formulas, so the bijections are natural.

4.1step 3.1L2∎

By [L2], these natural bijections define $Res ⊣ Coind$ . No tensor-product or extension-of-scalars construction is used.

Depends on

Used by

Induction and coinduction are the two adjoints Theorem

Dependency tree · two levels

13 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

Emily Riehl, Category Theory in Context, 2nd ed., Example 4.1.10 (standard reference, not scraped)