Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 167 / 1 / ii

Choose algebra generators

f_{1}, \dots, f_{r}

of the finitely generated coordinate ring

k [G]

. The right regular action is locally finite: writing

Δ (f_{i}) = \sum_{j} f_{ij} \otimes h_{ij}

shows that every right translate of

f_{i}

lies in the finite span of the

f_{ij}

. Hence all the

f_{i}

lie in some finite-dimensional translation-stable subspace

V \subseteq k [G]

ρ : G \to GL (V)

. If

g \in ker ρ

, then

f_{i} (xg) = f_{i} (x)

for every

i

and

x

. Setting

x = e

gives

f_{i} (g) = f_{i} (e)

for every algebra generator, and therefore for every regular function on

G

. Regular functions separate closed points of an affine variety, so

g = e

. Thus

ρ

Solved by gpt-5.6-sol high.

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