A subset of a topological space is locally closed if it is an open subset of its closure, equivalently if it is the intersection of an open set with a closed set.
If \(X\) is a quasi-affine or quasi-projective variety and \(Y \subseteq X\) is an irreducible locally closed subset, then \(Y\) is itself quasi-affine, respectively quasi-projective, by virtue of being a locally closed subset of the same affine or projective space. This is the induced structure on \(Y\), and \(Y\) is called a subvariety of \(X\).
Now let \(\phi: X \to Y\) be a morphism, and let \(X' \subseteq X\) and \(Y' \subseteq Y\) be irreducible locally closed subsets with \(\phi(X') \subseteq Y'\). Show that \(\ro{\phi}{X'} : X' \to Y'\) is a morphism.
Solution
Write \[ \phi'=\ro{\phi}{X'}:X'\longrightarrow Y'. \]
1. The map \(\phi'\) is continuous for the induced topologies on \(X'\) and \(Y'\).
Let \(C\subseteq Y'\) be closed. Because \(Y'\) has the subspace topology from \(Y\), there is a closed subset \(D\subseteq Y\) with \[ C=Y'\cap D. \] Since \(\phi(X')\subseteq Y'\), \[ (\phi')^{-1}(C)=X'\cap\phi^{-1}(D). \] The set \(\phi^{-1}(D)\) is closed in \(X\) because \(\phi\) is continuous, so its intersection with \(X'\) is closed in \(X'\). Thus \(\phi'\) is continuous.
2. A regular function on an open subset of \(Y'\) is, near each point, the restriction of a regular function on an open subset of \(Y\).
Fix an open subset \(V'\subseteq Y'\), a regular function \(f\) on \(V'\), and a point \(Q\in V'\). By the definition of a regular function on the induced quasi-affine or quasi-projective structure, after shrinking around \(Q\) in \(V'\), the function \(f\) is represented by a quotient \[ \frac{g}{h} \] of ambient polynomial functions, with \(h\) nonzero there; in the projective case \(g\) and \(h\) are homogeneous of the same degree. The same quotient is regular on the ambient open subset of \(Y\) where \(h\ne0\). Intersecting that open subset with a sufficiently small ambient open neighborhood whose intersection with \(Y'\) lies in \(V'\) gives an open neighborhood \(V\subseteq Y\) of \(Q\) and a regular function \(\widetilde f\) on \(V\) whose restriction to a neighborhood of \(Q\) in \(Y'\) equals \(f\).
3. The pullback of every regular function by \(\phi'\) is regular.
Let \(V'\subseteq Y'\) be open, let \(f\in\mco(V')\), and fix \[ P\in(\phi')^{-1}(V'). \] Put \(Q=\phi'(P)\). By step 2, after shrinking around \(Q\), choose an open neighborhood \(V\subseteq Y\) and a regular function \(\widetilde f\) on \(V\) whose restriction to \(Y'\) agrees with \(f\) near \(Q\).
Because \(\phi\) is a morphism, \[ \widetilde f\circ\phi \] is regular on \(\phi^{-1}(V)\). Restricting this regular function to the induced subvariety \(X'\) gives a regular function near \(P\), and there it equals \[ f\circ\phi'. \] Thus \(f\circ\phi'\) is regular in a neighborhood of every point of \((\phi')^{-1}(V')\). Regularity is local, so it is regular on all of \((\phi')^{-1}(V')\).
4. Q.E.D.
Step 1 proves continuity, and step 3 proves the regular-function pullback condition. These are precisely the defining conditions for \(\phi':X'\to Y'\) to be a morphism.
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P-AGH321GROUPVARGroup varieties $\GG_a$ and $\GG_m$, and the group structure on $\Hom(X,G)$uses