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    Group varieties $\GG_a$ and $\GG_m$, and the group structure on $\Hom(X,G)$

    P-AGH321GROUPVAR

    Area
    Algebraic Geometry
    Topics
    Group Varieties, Morphisms, Regular Functions
    Status
    draft

    A group variety consists of a variety \(Y\) together with a morphism \(\mu: Y \times Y \to Y\) such that the set of points of \(Y\) with the operation given by \(\mu\) is a group, and such that the inverse map \(y \mapsto \inverseof{y}\) is also a morphism \(Y \to Y\).

    1. The additive group \(\GG_a\) is the variety \(\AA^1\) with the morphism \(\mu: \AA^1 \times \AA^1 \to \AA^1\) defined by \(\mu(a,b) = a+b\). Show that it is a group variety.

    2. The multiplicative group \(\GG_m\) is the variety \(\AA^1 \sm \ts{0}\) with the morphism \(\mu(a,b) = ab\). Show that it is a group variety.

    3. If \(G\) is a group variety and \(X\) is any variety, show that the set \(\Hom(X, G)\) has a natural group structure.

    4. For any variety \(X\), show that \(\Hom(X, \GG_a)\) is isomorphic to \(\mco(X)\) as a group under addition.

    5. For any variety \(X\), show that \(\Hom(X, \GG_m)\) is isomorphic to the group of units in \(\mco(X)\), under multiplication.

    Solution

    1. The variety \(\GG_a=\AA^1\) with addition is a group variety.

    The underlying set is the field \(k\), which is an abelian group under addition with identity \(0\). The multiplication map in the sense of group varieties is \[ \mu:\AA^2\to\AA^1, \qquad (a,b)\longmapsto a+b, \] whose coordinate function \(a+b\) is polynomial, hence \(\mu\) is a morphism. The inverse map is \[ \iota:\AA^1\to\AA^1, \qquad a\longmapsto-a, \] also polynomial and therefore a morphism. Thus all requirements in the definition are satisfied, proving (a).

    2. The variety \(\GG_m=\AA^1\setminus\{0\}\) with multiplication is a group variety.

    Its underlying set is \(k^\times\), a group under multiplication with identity \(1\). The multiplication map \[ \mu:\GG_m\times\GG_m\to\GG_m, \qquad (a,b)\longmapsto ab \] is the restriction of the polynomial multiplication morphism \(\AA^2\to\AA^1\) and its image avoids \(0\). Hence it is a morphism to \(\GG_m\).

    On the principal affine open \(\GG_m=D(x)\), the function \(x^{-1}\) is regular. Therefore \[ \iota:\GG_m\to\GG_m, \qquad a\longmapsto a^{-1} \] is a morphism. Thus \(\GG_m\) is a group variety, proving (b).

    3. For morphisms \(f,g:X\to G\), define\[ f*g=\mu\circ(f,g):X\to G. \]This equips \(\Hom(X,G)\) with a group structure, proving (c).

    By P-AGH316QPPRODUCT, the pair \[ (f,g):X\to G\times G, \qquad x\longmapsto(f(x),g(x)) \] is a morphism. Composing with the group multiplication morphism \(\mu:G\times G\to G\) shows that \(f*g\) is again a morphism.

    Let \(e\in G\) be the identity element. The constant map \[ e_X:X\to G, \qquad x\longmapsto e \] is a morphism and is an identity for \(*\) pointwise. If \(\iota:G\to G\) is the inverse morphism, then \[ f^{-1}=\iota\circ f \] is a morphism and is a pointwise inverse to \(f\). Associativity follows pointwise from associativity in the underlying group of \(G\): \[ ((f*g)*h)(x)=(f(x)g(x))h(x)=f(x)(g(x)h(x))=(f*(g*h))(x). \] Thus all group axioms hold.

    4. The map\[ \Phi_a:\Hom(X,\GG_a)\longrightarrow\mco(X), \qquad h\longmapsto h^*(t)=t\circ h \]is an isomorphism of additive groups.

    A morphism \(h:X\to\AA^1\) pulls the affine coordinate \(t\) back to a global regular function on \(X\), so \(\Phi_a\) is well-defined.

    Conversely, if \(f\in\mco(X)\), the map \[ h_f:X\to\AA^1, \qquad x\longmapsto f(x) \] is a morphism: locally \(f\) is a quotient of regular coordinate functions with nonvanishing denominator, which is exactly the local criterion for a morphism to affine space. Clearly \[ \Phi_a(h_f)=f, \] and a map to \(\AA^1\) is determined by its coordinate function, so this construction is inverse to \(\Phi_a\).

    Finally, for \(h_1,h_2:X\to\GG_a\), \[ \Phi_a(h_1*h_2)(x)=h_1(x)+h_2(x) =\Phi_a(h_1)(x)+\Phi_a(h_2)(x). \] Hence \(\Phi_a\) is a group isomorphism, proving (d).

    5. The map\[ \Phi_m:\Hom(X,\GG_m)\longrightarrow\mco(X)^\times, \qquad h\longmapsto t\circ h \]is an isomorphism of multiplicative groups.

    If \(h:X\to\GG_m\) is a morphism, then \[ u=t\circ h \] is a global regular function which never vanishes. Composing \(h\) with the inversion morphism on \(\GG_m\) shows that \[ u^{-1}=t\circ\iota\circ h \] is also globally regular. Thus \(u\in\mco(X)^\times\).

    Conversely, let \(u\in\mco(X)^\times\) and let \(v\in\mco(X)\) satisfy \(uv=1\). Then \(u(x)\ne0\) for every \(x\in X\), so the morphism \[ x\longmapsto u(x) \] to \(\AA^1\) factors set-theoretically through the open subvariety \(\GG_m\). By the subvariety restriction criterion, it is a morphism \(X\to\GG_m\). These constructions are inverse.

    Pointwise multiplication gives \[ \Phi_m(h_1*h_2)=\Phi_m(h_1)\Phi_m(h_2), \] so \(\Phi_m\) is a group isomorphism, proving (e).

    6. Q.E.D.

    Steps <1>1–<1>5 prove (a)–(e), respectively.

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