LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

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.

The coinvariants of a right-right Hopf module form a subspace

Statement

For a right-right Hopf module $M$ over a Hopf algebra $H$ over a field $k$ , the subset $M^{co H} = {m \in M : ρ (m) = m \otimes 1_{H}}$ is a $k$ -linear subspace of $M$ .

Facts & Assumptions

Given: A field $k$ , a Hopf algebra $H$ over $k$ , and a right-right Hopf module $M$ .

[F1]

The right coaction $ρ : M \to M \otimes H$ is linear and $M^{co H}$ consists exactly of those $m$ with $ρ (m) = m \otimes 1_{H}$ (Right-right Hopf modules and coinvariants).

Proof

1.1givenF1construct

Define the linear map $T : M \to M \otimes H$ by $T (m) = ρ (m) - m \otimes 1_{H}$ . It is linear because $ρ$ and $m \mapsto m \otimes 1_{H}$ are linear, and its kernel is exactly $M^{co H}$ by [F1].

2.1step 1.1F1algebra∎

If $x, y \in \ker T$ and $a, b \in k$ , then $T (a x + b y) = a T (x) + b T (y) = 0$ . Also $T (0) = 0$ , so $\ker T$ is a $k$ -linear subspace and hence so is $M^{co H}$ .

Depends on

Right-right Hopf modules and coinvariants

Used by

Cited to discharge well-definedness by Right-right Hopf modules and coinvariants.

Dependency tree · two levels

5 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

Hans-Juergen Schneider, Lectures on Hopf Algebras, notes by Sonia Natale, Trabajos de Matematica 31/95 (standard reference, not scraped)
D. S. Drogalski, Chapter 3: Hopf Algebras, UC San Diego 207A lecture notes (standard reference, not scraped)