Fulton 2.8

2.39.* Prove the following relations among ideals $I_i, J$ in a ring $R$ :

(a) $(I_1+I_2)J=I_1J+I_2J.$

Suppose $x=(i_1+i_2)j\in (I_1+I_2)J.$ Then $x=i_1j+i_2j\in I_1J+I_2J.$ Suppose $x=i_1j_1+i_2j_2\in I_1J+I_2J.$ Then $i_1j_1,i_2j_2\in (I_1+I_2)J$ and thus $x\in (I_1+I_2)J.$ So $(I_1+I_2)J=I_1J+I_2J.$

(b) $(I_1\ldots I_N)^n=I_1^n\ldots I_N^n.$

Observe that $(a_1\ldots a_N)^n=a_1^n\ldots a_N^n$ where $a_k\in I_k.$ Thus $(I_1\ldots I_N)^n=I_1^n\ldots I_N^n.$

2.40.* (a) Suppose $I,J$ are comaximal ideals in $R.$ Show that $I+J^2=R.$ Show that $I^m$ and $J^n$ are comaximal for all $m,n.$

Let $a\in I, b\in J$ such that $a+b=1.$ Then $ab\in I, b^2\in J^2$ and $ab+b^2=b.$ Thus $a,b\in I+J^2$ and therefore $1\in I+J^2.$

By induction, the above implies that $I^m$ and $J^n$ are comaximal for all $m,n.$

(b) Suppose $I_1,\ldots,I_N$ are ideals in $R,$ and $I_i$ and $J_i=\cap_{j\ne i}I_j$ are comaximal for all $i.$ Show that $I_1^n\cap\ldots\cap I_N^n=(I_1\ldots I_N)^n= (I_1\cap\ldots\cap I_N)^n$ for all $n.$

This follows from part (a) and the fact that $I\cap J=IJ$ when $I,J$ are comaximal.

2.41.* Let $I,J$ be ideals in $R.$ Suppose $I$ is finitely generated and $I\subset\text{Rad}(J).$ Show that $I^n\subset J$ for some $n.$

Let $I=(a_1,\ldots,a_N).$ Since $I\subset\text{Rad}(J)$ for each $i,$ $a_i^{n_i}\in J$ for some $n_i\ge 1.$ If we take $n=N\cdot\max\left\{ n_i \, | \, 1\le i\le N \right\}$ then, by the multinomial theorem and pigeon hole principle, $I^n=((a_1)+\ldots+(a_N))^n\subset (a_1)^{n_1}+\ldots+(a_N)^{n_N}\subset J.$

2.42.* (a) Let $I\subset J$ be ideals in a ring $R.$ Show that there is a natural ring homomorphism from $R/I$ onto $R/J.$

This follows from applying Lemma 1.1 to the natural maps $R\to R/I,$ $R\to R/J.$

(b) Let $I$ be an ideal in a ring $R,$ $R$ a subring of a ring $S.$ Show that there is a natural ring homomorphism from $R/I$ to $S/IS.$

Define $\varphi:R\to S/IS$ by $r\mapsto r+IS.$ Note that this is a well defined homomorphism since $I\subset IS.$ Moreover, $\varphi(I)=\{0+IS\}\subset S/IS.$ Thus we may apply Lemma 1.1 to the natural map $R\to R/I$ and the map $\varphi:R\to S/IS$ and yield the desired result.

2.43.* Let $P=(0,\ldots,0)\in\mathbb{A}^n,$ $\mathcal{O}=\mathcal{O}_{P}(\mathbb{A}^n),$ $\mathfrak{m}=\mathfrak{m}_{P}(\mathbb{A}^n).$ Let $I\subset k[X_1,\ldots,X_n]$ be the ideal generated by $X_1,\ldots,X_n.$ Show that $I\mathcal{O}=\mathfrak{m},$ so $I^r\mathcal{O}=\mathfrak{m}^r$ for all $r.$

Observe that if $f=F/G\in\mathcal{O},$ then $f\in\mathfrak{m},$ that is $f(P)=0,$ if and only if $F(P)=0.$ Moreover, $F(P)=0$ if and only if $F\in I(P)=I.$ So $\mathfrak{m}=I\mathcal{O}.$

2.44.* Let $V$ be a variety in $\mathbb{A}^n,$ $I=I(V)\subset k[X_1,\ldots,X_n],$ $P\in V,$ and let $J$ be an ideal of $k[X_1,\ldots,X_n]$ that contains $I.$ Let $J&39;$ be the image of $J$ in $\Gamma(V).$ Show that there is a homomoprhism $\varphi$ from $\mathcal{O}_{P}(\mathbb{A}^n)/J\mathcal{O}_{P}(\mathbb{A}^n)$ to $\mathcal{O}_{P}(V)/J&39;\mathcal{O}_{P}(V),$ and that $\varphi$ is an isomorphism. In particular, $\mathcal{O}_{P}(\mathbb{A}^n)/I\mathcal{O}_{P}(\mathbb{A}^n)$ is isomorphic to $\mathcal{O}_{P}(V).$

Observe that by Lemma 2.13 the homomorphism $\varphi:\Gamma(\mathbb{A}^n)\to\Gamma(V)$ extends naturally to a homomorphism $\varphi&39;:\mathcal{O}_{P}(\mathbb{A}^n)\to\mathcal{O}_{P}(V).$ Moreover, note that $J&39;=\varphi(J),$ thus $\varphi&39;(J\mathcal{O}_{P}(\mathbb{A}^n))=J&39;\mathcal{O}_{P}(V).$ Let $\pi:\mathcal{O}_{P}(\mathbb{A}^n)\to J\mathcal{O}_{P}(\mathbb{A}^n)$ and $\pi_V:\mathcal{O}_{P}(V)\to \mathcal{O}_{P}(V)/J&39;\mathcal{O}_{P}(V)$ be the natural homomorphisms. $\xymatrix{ \mathcal{O}_{P}(\mathbb{A}^n) \ar[r]^{\varphi&39;} \ar[d]^{\pi} & \mathcal{O}_{P}(V) \ar[d]^{\pi_V}\\ \mathcal{O}_{P}(\mathbb{A}^n)/J\mathcal{O}_{P}(\mathbb{A}^n) \ar[r] & \mathcal{O}_{P}(V)/J&39;\mathcal{O}_{P}(V) }$ Observe that $\ker\pi=J\mathcal{O}_{P}(\mathbb{A}^n)=\varphi&39;^{-1}(J&39;\mathcal{O}_{P}(V))=\ker\pi_V\circ\varphi&39;.$ Thus by Lemma 1.1, the homomorphism $\pi_V\circ\varphi&39;$ induces an isomorphism $\mathcal{O}_{P}(\mathbb{A}^n)/J\mathcal{O}_{P}(\mathbb{A}^n)\to\mathcal{O}_{P}(V)/J&39;\mathcal{O}_{P}(V).$

2.45.* Show that ideals $I,J\subset k[X_1,\ldots,X_n]$ ( $k$ algebraically closed) are comaximal if and only if $V(I)\cap V(J)=\emptyset.$

Note that $V(I)\cap V(J)=V(I+J).$ By the weak Nullstellensatz, $V(I+J)=\emptyset$ if and only if $I+J=k[X_1,\ldots,X_n].$ Thus $I+J$ are comaximal if and only if $V(I)\cap V(J)=\emptyset.$

2.46.* Let $I=(X,Y)\subset k[X,Y].$ Show that $\dim_k(k[X,Y]/I^n)= 1+2+\ldots+n=\frac{n(n+1)}{2}.$

Observe that the residues of monomials of degree $d<n$ span $k[X,Y]/I^n$ and are linearly independent over $k.$ By Problem 2.35(a) there are $d+1$ monomials of degree $d.$ Therefore $k[X,Y]/I^n$ is of dimension $1+2+\ldots+n=\frac{n(n+1)}{2}.$