Fulton 2.4

2.17. Let $V=V(Y^2-X^2(X+1))\subset\mathbb{A}^2,$ and $\overline{X},\overline{Y}$ the residues of $X,Y$ in $\Gamma(V)$ ; let $z=\overline{X}/\overline{Y}\in k(V).$ Find the pole sets of $z$ and $z^2.$

The pole set of $z$ is $\{(0,0)\}.$

First observe that $z=\overline{Y}/\overline{X}=\overline{Y}/^2\overline{XY}=\overline{X(X+1)}/\overline{Y}.$ Therefore $z$ is clearly defined at all points $(x,y)$ where $x\ne 0$ or $y\ne 0.$ It remains to show that $z$ is not defined at $(0,0).$

Suppose $\overline{F}/\overline{G}=\overline{Y}/\overline{X}$ for some $F,G\in k[X,Y].$ Then $\overline{YG}=\overline{XF}$ and $YG-XF\in (Y^2-X^2(X+1)).$ So $YG-XF=H(Y^2-X^2(X+1))$ for some $H\in k[X,Y].$ Hence $Y(G-HY)=X(F-HX(X+1)).$ So $G-HY\in (X).$ Therefore $0=G(0,0)-(HY)(0,0)=G(0,0).$

The pole set of $z^2$ is empty.

Observe that $z=\overline{Y}^2/\overline{X}^2=\overline{X^2(X+1)}/\overline{X}^2=\overline{X+1}.$

2.18. Let $\mathcal{O}_{P}(V)$ be the local ring of a variety $V$ at a point $P.$ Show that there is a natural one-to-one correspondence between the prie ideals in $\mathcal{O}_{P}(V)$ and the subvarieties of $V$ that pass through $P.$

It suffices to show that there is a one-to-one correspondence between prime ideals in $\mathcal{O}_{P}(V)$ and prime ideals in $\Gamma(V)$ containing in $I(P).$

Note from Proposition 3 that an ideal $I\subset\mathcal{O}_{P}(V)$ is generated by $I\cap\Gamma(V).$ Thus if $I$ is prime, then $J=I(I\cap\Gamma(V))$ is a prime ideal in $\Gamma(V).$

Suppose $J\subset\Gamma(V)$ is a prime ideal, $I(P)\subset J.$ Note that $I(J),$ the ideal generated by $J$ in $\mathcal{O}_{P}(V),$ is a proper ideal since $J$ does not contain any units. Suppose $a/b,c/d\in\mathcal{O}_{P}(V)$ such that $(ac)/(bd)\in I(J).$ Since $1/b,1/d,1/(bd)$ are units, they are not in $I(J).$ Thus $ac\in I(J)$ and therefore $ac\in J.$ So $a$ or $c$ must be in $J\subset I(J).$ Thus $I(J)$ is prime.

2.19. Let $f$ be a rational function on a variety $V.$ Let $U=\{P\in V \ | \ f \text{ is defined at } P\}.$ Then $f$ defines a function $U$ to $k.$ Show that this function determines $f$ uniquely.

Suppose $f=a/b=c/d,$ $a,b,c,d\in\Gamma(V).$ Observe that for all $P\in U,$ $f(P),$ $f(P)b(P)d(P)=a(P)d(P)=c(P)b(P).$ Moreover, for all $P\in V\setminus U,$ $0=a(P)d(P)=c(P)b(P).$ Therefore $ad=cb.$ So $a/b=c/d.$

2.20. Let $V$ and $f$ be as in the example given in this section.

Show that the pole set of $f$ is exactly $\{(x,y,z,w) \ | \ y=0 \text{ and } w=0\}.$

Clearly $f$ is defined everywhere $y\ne 0$ or $w\ne 0.$ It remains to show that $f$ is not defined when $y=w=0.$

Suppose $f=\overline{F}/\overline{G}.$ Then $\overline{XG}=\overline{YF}$ and $XG-YF\in(XW-YZ).$ Thus $XG-YF=H(XW-YZ)$ and $X(G-HW)=Y(F-HZ),$ $H\in k[X,Y,Z,W].$ So $G-HW\in (Y).$ Let $P=(x,0,z,0),$ $x,z\in k.$ Observe that $G(P)-(HW)(P)=0.$ Since $(HW)(P)=H(P)(0)=0,$ $G(P)=0.$ Thus $f$ is not defined at $P$ for any choice of representative.

Show that it is impossible to write $f=a/b$ where $a,b\in\Gamma(V),$ and $b(P)\ne 0$ for every $P$ where $f$ is defined.

Suppose $f=\overline{F}/\overline{G}$ such that $\overline{F}/\overline{G}$ is defined at all points $f$ is defiend. By the same argument above, $G-HW\in (Y).$ So if $Y=0,$ $G=HW.$ If $W=0,$ then $G=AY,$ $A\in k[X,Y,Z,W].$ So $G=AY+HW.$

Since $\overline{F}/\overline{G}$ is defined at all points $f$ is defined, $V(G)\cap V$ is the pole set of $f.$ Thus $I(V(G)\cap V)=(Y,W),$ since $(Y,W)$ is the ideal of the pole set. However, for any choice of $A,H,$ observe that $I(V(G)\cap V)=(AY+HW, XW-YZ)\ne (Y,W),$ a contradiction.

2.21.* Let $\varphi:V\to W$ be a polynomial map of affine varieties, $\widetilde{\varphi}:\Gamma(W)\to\Gamma(V)$ the induced map on coordinate rings. Suppose $P\in V,$ $\varphi(P)=Q.$ Show that $\widetilde{\varphi}$ extends uniquely to a ring homomorphism (also writter $\widetilde{\varphi}$ ) from $\mathcal{O}_{Q}(W)$ to $\mathcal{O}_{P}(V).$ (Note that $\widetilde{\varphi}$ may not extend to all of $k(W).$ ) Show that $\widetilde{\varphi}(\mathfrak{m}_{Q}(W))\subset\mathfrak{m}_{P}(V).$

Suppose $\psi:\mathcal{O}_{Q}(W)\to\mathcal{O}_{P}(V)$ is a homomorphism such that $\psi(a)=\widetilde{\varphi}(a)$ for any $a\in\Gamma(W).$ If $a$ is a unit in $\mathcal{O}_{Q}(W)$ then $\psi(a^{-1})=\psi(a)^{-1}=\widetilde{\varphi}(a)^{-1}.$ So for any $a/b\in\mathcal{O}_{Q}(W),$ $\psi(a/b)=\widetilde{\varphi}(a)/\widetilde{\varphi}(b).$ Therefore $\widetilde{\varphi}$ extends uniquely to a homomorphism $\mathcal{O}_{Q}(W)\to\mathcal{O}_{P}(V).$

Suppose $f\in\mathfrak{m}_{Q}(W).$ Then $\widetilde{\varphi}(f)(P)=f(Q)=0.$ So $\widetilde{\varphi}(\mathfrak{m}_{Q}(W))\subset\mathfrak{m}_{P}(V).$

2.22.* Let $T:\mathbb{A}^n\to\mathbb{A}^n$ be an affine change of coordinates, $T(P)=Q.$ Show that $\widetilde{T}:\mathcal{O}_{Q}(\mathbb{A}^n)\to\mathcal{O}_{P}(\mathbb{A}^n)$ is an isomorphism. Show that $\widetilde{T}$ induces an isomorphism $\mathcal{O}_{Q}(V)\to \mathcal{O}_{P}(V)$ if $P\in V^T,$ for $V$ a subvariety of $\mathbb{A}^n.$

Since $\widetilde{T}:\Gamma(\mathbb{A}^n)\to\Gamma(\mathbb{A}^n)$ is an isomorphism, it extends uniquely to an isomorphism $\widetilde{T}:\mathcal{O}_{Q}(\mathbb{A}^n)\to\mathcal{O}_{P}(\mathbb{A}^n),$ by Problem 2.21.

Let $\pi_V:\mathcal{O}_{Q}(\mathbb{A}^n)\to\mathcal{O}_{Q}(V)$ and $\pi_{V^T}:\mathcal{O}_{P}(\mathbb{A}^n)\to \mathcal{O}_{P}(V^T)$ be the natural homomorphisms (see Lemma 2.13). Observe that $\ker(\pi_{V^T}\circ\widetilde{T})\subset\mathcal{O}_{Q}(\mathbb{A}^n)$ is equal to the ideal generated by $\ker(\pi_{V^T}\circ\widetilde{T})\cap\Gamma(\mathbb{A}^n),$ similarly for $\ker\pi_V.$ Thus since $\ker\pi_V\cap\Gamma(\mathbb{A}^n)=\ker(\pi_{V^T}\circ \widetilde{T})\cap\Gamma(\mathbb{A}^n),$ $\ker\pi_V=\ker(\pi_{V^T}\circ\widetilde{T}).$ So $\pi_{V^T}\circ\widetilde{T}$ descends, by Lemma 1.1, to an isomorphism $\mathcal{O}_{Q}(V)\to\mathcal{O}_{P}(V^T).$