Fulton 2.1

2.1.* Show the map that associates each $F\in k[X_1,\ldots,X_n]$ to a polynomial function in $\mathcal{F}(V,k)$ is a ring homomorphism with kernel $I(V).$

Clearly the map is a ring homorphism. Suppose $F\in k[X_1,\ldots,X_n]$ such that $F$ is the zero function in $\mathcal{F}(V,k).$ Then $F(P)=0$ for all $P\in V.$ Thus $F\in I(V).$ Similarly, if $F\in I(V)$ then $F$ is the zero function.

2.2.* Let $V\subset\mathbb{A}^n$ be a variety. A subvariety of $V$ is a variety $W\subset\mathbb{A}^n$ that is contained in $V.$ Show that there is a natural one-to-one correspondence between algebraic subsets (resp. subvarieties, resp. points) of $V$ and radical ideals (resp. prime ideals, resp. maximal ideals) of $\Gamma(V).$

Recall there is a one to one correspondence between radical, prime, and maximal ideals $J/I(V)$ in $\Gamma(V)$ and radical, prime, and maximal ideals $J$ in $k[X_1,\ldots,X_n]$ such that $I(V)\subset J.$ Observe that that $W$ is an algebraic subset, subvariety, or point of $V,$ if and only if $I(W)$ is a radical, prime, or maximal ideal in $k[X_1,\ldots,X_n]$ with $I(V)\subset I(W).$

2.3.* Let $W$ be a subvariety of a variety $V,$ and let $I_V(W)$ be the ideal of $\Gamma(V)$ corresponding to $W.$

(a) Show that every polynomial function on $V$ restricts to a polynomial function on $W.$

Suppose $f\in\mathcal{F}(V,k)$ is a polynomial function. Then there exists $F\in k[X_1,\ldots,X_n]$ such that $f(a_1,\ldots,a_n)=F(a_1,\ldots,a_n)$ for all $(a_1,\ldots,a_n)\in V.$ Since $W\subset V,$ $f|_W\in\mathcal{F}(W,k)$ is polynomial as well. In terms of coordinate rings, this restriction map defines a homomorphism $\varphi:\Gamma(V)\to\Gamma(W)$ where $\varphi(F+I(V))=F+I(W).$

(b) Show the map $\varphi:\Gamma(V)\to\Gamma(W)$ defined in (a) is surjective with kernel $I_V(W),$ so $\Gamma(W)\cong\Gamma(V)/I_V(W).$

Let $\pi_V:k[X_1,\ldots,X_n]\to \Gamma(V)$ and $\pi_W:k[X_1,\ldots,X_n]\to \Gamma(W)$ be the natural homomorphisms. Since $\pi_W,\pi_V$ are surjective and $\ker\pi_V\subset\ker\pi_W,$ by descending to the quotient (Lemma 1.1) there exists a surjective homomorphism $\widetilde{\pi_W}:\Gamma(V)\to\Gamma(W)$ with $\ker\widetilde{\pi_W}=\pi_W(I(V))=I_V(W).$ Moreover, $\widetilde{\pi_W}(F+I(V))=\pi_W(F)=F+I(W)=\varphi(F+I(V)).$ So $\widetilde{\pi_W}=\varphi.$

2.4.* Let $V\subset\mathbb{A}^n$ be a nonempty variety. Show the following are equivalent: (1) $V$ is a point; (2) $\Gamma(V)=k$ ; (3) $\dim_k\Gamma(V)<\infty.$

(1) $\Leftrightarrow$ (2). Suppose $V=\{(a_1,\ldots,a_n)\}\subset\mathbb{A}^n.$ Then $I(V)=(X_1-a_1,\ldots,X_n-a_n).$ Therefore $\Gamma(V)=k.$ Conversely, suppose $\Gamma(V)=k.$ Then $I(V)$ is maximal. Since $k$ is algebraically closed, $I(V)=(X-a_1,\ldots,X-a_n).$ Thus $V$ is a point.

(2) $\Leftrightarrow$ (3). Clearly if $\Gamma(V)=k,$ then $\dim_k\Gamma(V)=1.$ Conversely, suppose $\dim_k\Gamma(V)<\infty.$ Let $L$ be the field of fractions for $\Gamma(V).$ By Lemma 2.1, $L$ is finite field extension of $k.$ Since $k$ is algebraically closed, by Problem 1.48, $L=k.$ Thus $\Gamma(V)=L=k.$

2.5. Let $F$ be an irreducible polynomial in $k[X,Y],$ and suppose $F$ is monic in $Y$ : $F=Y^n+a_1(X)Y^{n-1}+\ldots+a_n(X),$ with $n>0.$ Let $V=V(F)\subset\mathbb{A}^2.$ Show that the natural homomorphism from $k[X]$ to $\Gamma(V)=k[X,Y]/I(F)$ is one-to-one, so that $k[X]$ may be regarded as a subring of $\Gamma(V)$ ; show that the residues $\overline{1},\overline{Y},\ldots, \overline{Y}^n-1$ generate $\Gamma(V)$ over $k[X]$ as a module.

Observe that if $n=0,$ then $F(X,Y)=1$ is constant, $(F)=k[X,Y],$ and $k[X,Y]/(F)$ is the trivial ring. Thus suppose $n\ge 1.$

Note that the natural homomorphism $k[X]\to\Gamma(V)$ maps $G\mapsto G+(F).$ Suppose $G,H\in k[X]$ such that $H+(F)=G+(F),$ that is, $H-G\in (F).$ Observe that $F\nmid H-G,$ since $n\ge 1,$ so $G-H=0$ and $G=H.$ So the map is one-to-one.

Let $G+(F)\in\Gamma(V).$ Observe we have $Y^n+(F)=-a_1(X)Y^{n-1}-\ldots-a_0(X)+(F).$ Hence for any $m>0,$ $Y^m+(F)=\sum_{i=0}^{n-1}b_{m,i}(X)Y^i$ for some $b_{m,i}\in k[X].$ Thus $\overline{1},\overline{Y},\ldots, \overline{Y}^{n-1}$ generate $\Gamma(V)$ over $k[X]$ as a module.