LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

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.

Split diagonalizable groups are dual to abelian groups

Statement

For every field $k$ , $X (D_{k} (M)) = M$ via $m \mapsto e_{m}$ , and $Hom (D_{k} (M), D_{k} (N)) = Hom (N, M)$ naturally. Moreover $k [M]$ is finitely generated as an algebra if and only if $M$ is finitely generated. For $M ≅ Z^{r} \oplus ⨁_{i} Z / n_{i} Z$ , one has $D_{k} (M) ≅ G_{m}^{r} \times \prod_{i} μ_{n_{i}}$ , where $μ_{n} = Spec k [t] / (t^{n} - 1)$ .

Facts & Assumptions

[F1]

The monomial basis, characters, and Hopf correspondence are in Diagonalizable groups and their character modules.

Proof

Given: Abelian groups $M, N$ and a field $k$ .

1.1F1algebra

If $a = \sum_{m} a_{m} e_{m}$ is group-like, coefficient comparison in $Δ (a) = a \otimes a$ gives $a_{m} a_{n} = 0$ when $m \neq n$ and $a_{m} = a_{m}^{2}$ . Over a field at most one coefficient is nonzero; the counit makes exactly one coefficient nonzero, and it is $1$ . Thus the group-like elements are precisely $e_{m}$ . A Hopf map $k [N] \to k [M]$ must send $e_{n}$ to $e_{f (n)}$ , with $f (n + n^{'}) = f (n) + f (n^{'})$ . Every such $f$ gives a Hopf map, proving both natural identifications.

2.1F1step 1.1algebra∎

Finite generators of $M$ , together with their negatives, give finite algebra generators of $k [M]$ . Conversely, take the finite union $T$ of the supports of finite algebra generators. Every product and sum has support in the submonoid generated by $T$ , so all $e_{m}$ can occur only if that monoid is $M$ ; in particular $T$ generates $M$ as a group. Direct sums become tensor products of group algebras. The algebras for $Z$ and $Z / n Z$ are respectively $k [t, t^{- 1}]$ and $k [t] / (t^{n} - 1)$ , which proves the product formula.

Depends on

Diagonalizable groups and their character modules

Used by

Tori correspond exactly to torsion-free character lattices Corollary
The multiplicative group scheme mu p is not a smooth torus Counterexample
The character lattice of a split torus Example
Multiplicative type groups split over a finite Galois extension Lemma
Multiplicative type groups and Galois character modules Theorem

Dependency tree · two levels

3 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

J. S. Milne, Algebraic Groups, corrected 2022 edition (standard reference, not scraped)
SGA 3, Expose VIII, section 1, Polo–Gille edition (standard reference, not scraped)
SGA 3, Expose X, section 1, Polo–Gille edition (standard reference, not scraped)