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Torus rational modules and affine actions are lattice gradings

Statement

Let $T = (C^{*})^{r}$ , $r \geq 0$ , with $H = C [t_{1}^{\pm 1}, \dots, t_{r}^{\pm 1}]$ and $t^{m} = \prod_{i} t_{i}^{m_{i}}$ for $m \in Z^{r}$ . Right $H$ -comodules (equivalently rational $T$ -modules) correspond to direct-sum gradings $V = ⨁_{m} V_{m}$ , with $c (v) = v \otimes t^{m}$ on $V_{m}$ and $t \cdot v = t^{m} v$ . Intertwining maps are exactly degree-preserving linear maps. For affine algebraic sets the coordinate-ring action corresponds to a grading $A = ⨁_{m} A_{m}$ with $1 \in A_{0}$ and $A_{m} A_{n} \subseteq A_{m + n}$ . Conversely every such grading of a finitely generated reduced complex algebra produces an affine algebraic $T$ -action. In the affine reconstruction assertion assume AC, used only by the published Nullstellensatz and morphism dictionary. For a function of degree $m$ , $f (t x) = t^{- m} f (x)$ ; thus function weights are opposite point-coordinate weights.

Facts & Assumptions

Given: The torus and its Laurent coordinate ring, where $Δ (t^{m}) = t^{m} \otimes t^{m}$ , $ε (t^{m}) = 1$ ; AC for affine reconstruction (The Axiom of Choice).

[F1]

Affine actions correspond to right-comodule algebra maps (Affine actions correspond to coordinate-ring coactions).

[F2]

A reduced finite-type complex algebra is $C [z_{1}, \dots, z_{n}] / I$ for radical $I$ , and the AC Nullstellensatz identifies this quotient with the coordinate algebra of $V (I)$ (The coordinate ring of a classical affine algebraic set, Classical affine algebraic sets correspond to radical ideals, and irreducible sets to prime ideals).

Proof

1.1givenalgebra

For a right comodule expand uniquely $c (v) = \sum_{m} p_{m} (v) \otimes t^{m}$ , with finitely many nonzero terms for each $v$ . Coassociativity and independence of Laurent monomials give $c (p_{m} (v)) = p_{m} (v) \otimes t^{m}$ , hence $p_{n} p_{m} = 0$ for $n \neq m$ and $p_{m}^{2} = p_{m}$ . The counit gives $v = \sum_{m} p_{m} (v)$ . Therefore $V = ⨁_{m} V_{m}$ , where $V_{m} = p_{m} (V)$ ; uniqueness follows by applying $p_{n}$ to any finite relation between homogeneous vectors. Evaluation identifies $V_{m}$ with the character eigenspace. Conversely such a direct sum defines $c$ by this finite formula, which satisfies both comodule identities. Each vector spans, together with its finitely many components, a finite-dimensional stable algebraic subspace, so the resulting action is rational.

2.1step 1.1givenalgebra

For completeness an arbitrary rational $T$ -module has such a coaction. On a finite-dimensional stable algebraic subspace, expand the regular matrix coefficients of the action in Laurent monomials; the group and identity laws give the comodule identities. On two overlapping subspaces these coactions agree on the intersection because all evaluations agree, and Laurent polynomials are determined by their values on $T$ . Thus they glue to $c$ on the union $V$ . A linear map intertwines comodules precisely when it preserves each $V_{m}$ , by coefficient comparison. Equivariant rational-module maps also intertwine comodules because their evaluations agree.

3.1F1F2step 1.1step 2.1algebra∎

For a comodule algebra, $c (a b) = c (a) c (b)$ and $c (1) = 1 \otimes 1$ imply $A_{m} A_{n} \subseteq A_{m + n}$ and $1 \in A_{0}$ . Conversely these grading laws make the coaction in step 1.1 a unital algebra map. F1 reconstructs the action on the affine set with coordinate ring $A$ . If the algebra is given abstractly, first use F2 to realize it; reducedness makes its presentation ideal radical. This is the only additional realization required, and is where AC is used. Finally evaluating the coordinate action gives $f (t^{- 1} x) = t^{m} f (x)$ , hence replacing $t$ by $t^{- 1}$ gives $f (t x) = t^{- m} f (x)$ .

Depends on

Used by

Opposite weights on the affine plane and its coordinate ring Example

Dependency tree · two levels

19 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

Michel Brion, Introduction to actions of algebraic groups (2010) (standard reference, not scraped)
Philippe Gille, Introduction to reductive group schemes over rings, full notes retrieved 2026-10-02 (standard reference, not scraped)
J. S. Milne, Algebraic Groups (2022) (standard reference, not scraped)