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

An affine algebraic group over $k$ is an affine variety $G$ equipped with multiplication $G\times G\to G$, inversion $G\to G$, and an identity element satisfying the group axioms, with multiplication and inversion both regular maps. Dually, the coordinate ring $k[G]$ is a commutative Hopf algebra: multiplication on $G$ induces the comultiplication $\Delta :k[G]\to k[G]\otimes k[G]$, inversion induces the antipode, and evaluation at the identity is the counit.

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Choose algebra generators $f_1,\dots ,f_r$ of the finitely generated coordinate ring $k[G]$. The right regular action is locally finite: writingshows 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 $V\subseteq k[G]$.

This gives a rational representation $\rho :G\to \mathrm{GL}(V)$. If $g\in \ker \rho$, 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 $\rho$ is a faithful representation of an affine algebraic group.

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For $G={\mathbb{G}}_a$, write $k[G]=k[t]$. Under the left-right regular representation of an affine algebraic group,so the action factors through the difference map ${\mathbb{G}}_a\times {\mathbb{G}}_a\to {\mathbb{G}}_a$. The degree filtrationis stable and has trivial one-dimensional successive quotients. The module is indecomposable; every nonzero submodule contains a nonzero translation difference of lower degree and, on iteration using suitable translations, meets the unique invariant line $k\cdot 1$.

For every finite-dimensional rational $G$-module $V$, the matrix-coefficient construction givesOn the other hand,Indeed, a nonzero finite-dimensional quotient of $k[G]$ would have a simple quotient. Every simple rational representation of the unipotent algebraic group ${\mathbb{G}}_a$ is trivial, so this would give a nonzero translation-invariant functional on $k[t]$. No such functional exists: if $c_j$ is its first nonzero value on a monomial, translating a sufficiently high monomial produces a nonzero coefficient times $c_j$, contradicting invariance.

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For $G={\mathbb{G}}_m$, write $k[G]=k[t,t^{-1}]$. Sincethe weight-space decomposition isas a $G\times G$-representation.

If $V={⨁}_r{V}_r$ is the weight decomposition of a finite-dimensional $G$-module, the matrix-coefficient argument again gives ${\mathrm{Hom}}_G(V,k[G])\cong V^{\ast }$. A map from $k[G]$ to $V$ is independently determined by the image in $V_r$ of the basis vector $t^r$; only finitely many $V_r$ are nonzero. Thusas vector spaces.

Solved by gpt-5.6-sol high.