Homework for Introduction to Algebraic Geometry

Unless otherwise stated a ring would mean a commutative ring with identity. If unspecified, $k$ denotes a field and $R$ a ring.

Class 1

(1) Let $R$ be a nonzero commutative ring with 1. Let $J \subset I$ be ideals of $R$. Let $\bar I$ denote the ideal $I / J$ of $\bar R=R/J$. Show that $\bar R/\bar I$ is isomorphic to $R/I$.

(2) Let $R$ be a nonzero commutative ring with 1. Show that $R[x,y]/(x-f(y))$ is isomorphic to $R[y]$, where $R[x,y]$ is a polynomial ring over $R$ in two variables and $f(y)\in R[y]\subset R[x,y]$.

(3) For $a,b\in \mathbb{R}$, show that $(x-a,y-b)$ is a maximal ideal of $\mathbb{R}[x,y]$. Try this by direct computation.

(4) Show that $(x^2+y^2-1)$ is a prime ideal in $\mathbb{R}[x,y]$.

(5) Let $R$ be an integral domain and $x\in R$. Show that if $(x)$ is a prime ideal of $R$ then $x$ is irreducible. In class it was shown that the converse is true if $R$ is a UFD. Is the converse true in general?

Class 2

(6) Give an example of a reduced ring which is not an integral domain.

(7) For a commutative ring $R$ and an ideal $I$ of $R$, prove that $\sqrt{I}$ is the intersection of all prime ideals containing $I$.

(8) Let $I$ and $J$ be ideals of $k[X_1,\ldots,X_n]$. Show that $Z(IJ)=Z(I\cap J)= Z(I)\cup Z(J)$.

Class 3

(9) Let $\{I_l\}_l$ be a collection of ideals in $k[X_1,\ldots,X_n]$. Show that $\cap_l Z(I_l)=Z(I)$ where $I=(\cup I_l)$.

(10) Let $(X,\mathcal{O}_X)$ be an affine variety. Show that $C\subset X$ is closed iff there exist an ideal $I$ of $\mathcal{O}_X$ such that $C=Z(I):=\{p\in X: f(p)=0 \forall f\in I\}$.

Class 4

(11) List all the ideals of $\mathbb{Z}_{(p)}$ and the prime ideals of $\mathbb{Z}[\frac{1}{p}]$, where $\mathbb{Z}[\frac{1}{p}]=S^{-1}\mathbb{Z}$ for $S=\{1, p, p^2, \dots \}$.

(12) Let $R$ be a ring and $S$ a multiplicative subset. Let $J$ be an ideal of $S^{-1}R$. The subset $\phi^{-1}(J)$ of $R$ is an ideal of $R$ and is (sometimes) denoted by $J\cap R$, where $\phi:R\to S^{1-}R$ is the natural localization map. Show that $J=(J\cap R)S^{-1}R$.

(13) Let $q:R\to \bar R$ be surjective ring homomorphism and $S$ be a multiplicative subset of $R$ then $q(S)$ is a multiplicative subset of $\bar R$ and $q(S)^{-1}\bar R$ is isomorphic to $S^{-1}R/\text{ker}(q)S^{-1}R$.

Class 5

(14) Show that $R[[x_1, x_2,\ldots, x_n]]$ is noetherian if $R$ is a noetherian ring.

(15) Find a non-noetherian subring of the polynomial ring $k[x,y]$ where $k$ is a field.

(16) Give a description of all $k[x,y]$-module in terms of $k$-vector spaces.

Class 6

(16.5) Let $R$ be a ring and $f:M\to N$ be an $R$-module homomorphism. Show that:
(i) $f$ is injective iff $0\to M\to N$ is an exact sequence.
(ii) $f$ is surjective iff $M\to N \to 0$ is an exact sequence.
(iii) $0\to \text{ker}(f)\to M\to N \to \text{coker}(f)\to 0$ is an exact sequence.

(17) Let $0\to A\to B\to C\to 0$ be a short exact sequence of $R$-modules and $N$ an $R$-module. Show that the following sequences are exact: $$0\to Hom_R(C,N)\to Hom_R(B,N)\to Hom_R(A,N)$$ $$0\to Hom_R(N,A)\to Hom_R(N,B)\to Hom_R(N,C)$$ For this reason Hom is said to be a left exact "functor".

(18) For a field $k$ show that $S^{-1}[k[x,y]/(xy)]$ is isomorphic to $k[x]_{(x)}$ where $S=(k[x,y]/(xy))\setminus (\bar x,\bar y-1)$.

(19) Let $R$ be a ring, $I$ an ideal of $R$ and $S$ a multiplicative subset of $R$. Show that $0\to I \to R \to R/I\to 0$ is an exact sequence of $R$-modules. Use this to prove (13).

Class 7

(20) Let $B$ be a fnitely generated $A$-algebra and $C$ be a finitely generated $B$ algebra then $C$ is a finitely generated $A$-algebra.

(21) Let $\mathbb{Z} \subset A$ be integral domains. If $A$ is a finite $Z$-module then show that there exist $\alpha_1,\ldots,\alpha_n\in \bar{\mathbb{Q}}$such that $A$ is isomorphic to the subring $\mathbb{Z}[\alpha_1,\ldots, \alpha_n]$ of $\bar{\mathbb{Q}}$.

(22) Let $B$ be an integral domain and $A$ a subring of $B$. Let $K$ be the field of fractions of $A$. Show that if $\alpha \in B$ is integral over $A$ then every $K$-Galois conjugate of $\alpha$ (which lies in an algebraic closure of the quotient field of $B$) is integral over $A$.

Class 8

(23) Show that $\mathbb{Z}[i]$ is the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[i]$.

(24) Let $A\subset B$ be rings and $S\subset A$ be a multiplicative set. Let $C$ be the integral closure of $A$ in $B$. Show that $S^{-1}C$ is the integral closure of $S^{-1}A$ in $S^{-1}B$.

(25) Show that the ring $k[X,Y]/(Y^2-X^2-X^3)$ is not a normal domain.

(26) Recall that an $R$-module is called noetherian if it satisfies the ascending chain condition, i.e. there is no infinite strictly increasing chian of submodules. Show that a finitely generated $R$-module is noetherian if $R$ is noetherian.

Class 9

(27) Let $X\subset \mathbb{A}^n$ and $Y\subset \mathbb{A}^m$ be affine varieties over an algebraically closed field $k$. Let $f_1, \ldots ,f_m \in k[X_1,\ldots,X_n]$ be polynomials such that for all $\underline a=(a_1,\ldots,a_n)\in X$ the $m$-tuple $(f_1(\underline a),\ldots, f_m(\underline a))\in Y$ then the map $F:X\to Y$ given by $F(\underline a)=(f_1(\underline a),\ldots, f_m(\underline a))$ is a morphism of affine varieties.
It was shown in class that $F$ defines a ring homomorphism $\Phi:k[Y_1,\ldots, Y_m]\to k[X_1,\ldots, X_n]$ by $\Phi(g(Y_1,\ldots,Y_n))=g(f_1,\ldots,f_m)$ such that if $g\in I(Y)\subset k[Y_1,\ldots Y_m]$ then $\Phi(g)\in I(X)\subset k[X_1,\ldots, X_n]$. Hence $\Phi$ induces a ring homomorphism $\phi: k[Y_1,\ldots,Y_m]/I(Y)\to k[X_1,\ldots X_m]/I(X)$. Conversely given a $k$-algebra homomorphism from $\mathcal{O}_Y\to \mathcal{O}_X$, it defines a morphism from $X\to Y$. Show that these two operations are inverse to each other.

(28) For the following $\mathbb{C}$-algebra homomorphism. Describe the corresponding varieties and the morphism between them.
(i) The map $\phi^{\#}:\mathbb{C}[y]\to \mathbb{C}[x,y]/(y^2-x^3-x)$ which sends $y$ to $\bar y$ the image of $y$ in the co-domain.
(ii) The inclusion $\mathbb{C}[y]\to \mathbb{C}[x,y]$.
(iii) The inclusion $\mathbb{C}[x] \to \mathbb{C}[x,x^{-1}, (x-1)^{-1}]$.

(29) Consider the space curve $C\subset \mathbb{A}^3$ given by the image of the morphism $\phi: \mathbb{A}^1\to \mathbb{A}^3$ which send $t\to (t,t^2,t^3)$. Compute the co-ordinate ring $\mathcal{O}_C$ and the ring homomorphism $\phi^{\#}:\mathcal{O}_C\to \mathcal{O}_{\mathbb{A}^1}$.

Class 10

(30) (Prime avoidance lemma) Let $I_1, I_2, \ldots, I_n$ be prime ideals and $J$ be an ideal of $R$, such that $J \subset \bigcup_j I_j$. Then $J$ is contained $I_j$ for some $j$.

(31) Let $\phi:X\to Y$ be a surjective morphism of varieties. Show that the induced map of $k$-algebra $\phi^{\#}:\mathcal{O}_Y\to \mathcal{O}_X$ is injective. Also show that the converse is false.

(32) (Gauss Lemma) Let $A$ be a UFD and $K$ its field of fractions. Let $f(x)$ be a monic polynomial in $A[x]$. Show that $f(x)$ is irreducible in $K[x]$ if and only if it is irreducible in $A[x]$.

Class 11

(33) Let $R$ be a ring and $M$ an $R$-module. Show that for any integer $n>0$, $R^n\otimes_R M\cong M^n$ as $R$-modules.

(34) Let $R$ be a ring , $I$ an $R$-ideal and $M$ an R-module. Show that $R/I \otimes_R M \cong M/IM$.

(35) Let $R$ be a ring.
(i) Show that if $R^n\cong R^m$ as $R$-modules then $n=m$.
(ii) If there exist a $R$-linear surjection $R^n\to R^m$ then $n\ge m$.

Class 12

(36) Let $A$ and $B$ be rings. Let $M$ be an $A$-module and $N$ a $(A,B)$-bimodule. Show that $M\otimes_A N$ has a natural $B$-module structure.

(37) Let $X$ and $Y$ irreducible algebraic set. Then $X\times Y$ is irreducible. (Hint: Suppose $X\times Y= V_1 \cup V_2$, $V_i$ closed subsets of $X\times Y$. Let $U_i=\{y\in Y: V_i\cap X\times\{y\}=X\times \{y\}\}$. Note $X\times \{y\}=(V_1\cap X\times\{y\}) \cup (V_2\cap X\times\{y\})$. Use $X$ irreducible to show that $U_1\cup U_2=Y$. Also $V_i\supset X\times U_i$ and hence $V_i\supset X\times \bar U_i$. But $Y$ is irreducible, hence $\bar U_1=Y$ or $\bar U_2=Y$.

Class 13


Class 14

(38) (Power of prime ideals need not be primary): Let $R=k[x,y,z]/(z^2-xy)$. Show that $P=(\bar x, \bar z)$ is a prime ideal of $R$. Show that $P^2$ is not primary. Also show $\sqrt{P^2}=P$. (Hint: First show $\bar x\bar y \in P^2$)

(39) Let $R$ be a ring and $I$ an ideal in $R$. Let $R[X]$ be a polynomial ring over $R$ and $I[X]$ the ideal generated by $I$ in $R[X]$. Show that if $I$ is primary then $I[X]$ is primary. (Hint: Show that zero divisors in $R[X]/I[X]\cong (R/I)[X]$ and zerodivisors in $(R/I)[X]$ are nilpotents.)

(40) Let $k$ be a field and $k[X_1,\ldots, X_n]$ be a polynomial ring. Show that every power of prime ideals $(X_1,\ldots, X_i)$ is a primary ideal for $1\le i \le n$. (Hint: Use (39))

Class 15

(41) (Minimal primary decomposition may not be unique) In the ring $k[X,Y]$ show that $(X^2,XY)=(X)\cap (X,Y)^2=(X)\cap (X^2,Y)$.

(42) Let $R=k[X,Y,Z]$ and $I=(X^2,XY,YZ,ZX)$. Determine the minimal prime ideals of $I$. Show that $(X,Y,Z)$ is an embedded prime of $I$.

(43) Let $R$ be a ring and $S$ a multiplicative subset of $R$. Let $Q$ be a primary ideal of $R$ disjoint with $S$ then show that $S^{-1}Q$ is a primary ideal of $S^{-1}R$.

Class 16

(38) Let $A\subset B$ be integral extension. Show that dimension of $A$ and $B$ are same.

(39) Find an example of a ring extension $A \subset B$, a prime ideal $P$ in $A$ and prime ideals $Q_1\subset Q_2$ in $B$ such that $Q_1\cap A=Q_2\cap A=P$.

(40) Find an example of a ring extension $A \subset B$ such that the induced map from Spec$(B)$ to Spec$(A)$ is surjective (i.e. lying over holds) but going up property for the extension fails.

Class 17

(41) Let $A$ be a ring and $S$ a multiplicative subset. Show that going down theorem holds for the following ring homomorphisms.
(i) $A\subset A[X]$ where $A[X]$ is the polynomial ring over $A$.
(ii) $A \to S^{-1}A$ where $S$ is a multiplicative subset of $A$.

(42) Let $X$ be an irreducible affine variety and $p\in X$ a point. Show that R_p=$\{f \in k(X): f \text{ is regular at } p\}$ is a local ring isomorphic to $k[X]_P$ where $k[X]=\mathcal{O}_X$ is the co-ordinate ring of $X$ and $P$ is the maximal ideal $\{f\in k[X]: f(p)=0\}$.

(43) Let $f:X\dashrightarrow Y$ and $g:Y\dashrightarrow Z$ be rational maps between varieties. Show that if $f$ is dominant then there exist a rational map $h:X \dashrightarrow Z$ which agrees with $g\circ f$ on an open subset of $X$. Show that if f is not dominant then the above conclusion may fail.

Class 18

(44) Show that Zariski topology on an irreducible affine curve is same as the cofinite topology. Deduce that any two affine curves are homeomorphic.

(45) Show that every nonempty open subset of an irreducible affine curve, whose coordinate ring is a PID, is an affine curve. (The result is true without PID hypothesis on the coordinate ring).

(46) Let $U$ be a nonempty open subset of an affine variety $X$. Note that $\mathcal{O}_X\subset \mathcal{O}(U) \subset k(X)$. Show that if $\Phi:U\to Y$ is an isomorphim of $U$ to an affine variety $Y$ then $\Phi$ induces an isomorphism $\mathcal{O}_Y\to \mathcal{O}(U)$. (Hint: $\Phi$ is a restrcition of a rational map $\tilde \Phi:X\dashrightarrow Y$ and $\Phi$ is an isomorphism means there is a morphism from $\Psi:Y\to X$ whose image lie in $U$ such that $\Phi\circ \Psi=id_Y$ and $\Psi\circ\Phi=id_U$. Show that $\tilde \Phi^{\#}:\mathcal{O}_Y\to k(X)$ is injective and surjects onto $\mathcal{O}(U))$

Class 19

(47) Let $k$ be an infinite field. Show that $f\in k[X_0,\ldots, X_n]$ is homogenous iff there exist $d\ge 0$ such that $f(\lambda a_0,\ldots, \lambda a_n)=\lambda^d f(a_0,\ldots,a_n)$ for all $\lambda, a_0, a_1, \ldots, a_n \in k$.

(48) Show that proper Zariski closed subset of $\mathbb{P}^1$ are finite set of points.

(49) Let $X$ be an algebraic subset of $\mathbb{P}^n$. Show that $I=\{ f \in k[X_0,\ldots, X_n] : f(a_0,\ldots,a_n)=0 \forall (a_0,\dots,a_n)\in \mathbb{A}^{n+1} \text{ with } [a_0,\ldots, a_n]\in X\}$ is a homogenous ideal and equals $I(X)$.

(50) Consider the algebraic set $X$ in $\mathbb{P}^2$ defined by the homogenous polynomial $X_0^2+X_1^2-X_2^2$. As we saw in the class, $X= \{[\frac{X_0}{X_2},\frac{X_1}{X_2},1]: (\frac{X_0}{X_2})^2+(\frac{X_1}{X_2})^2-1=0\} \sqcup \{[1,\pm i,0]\}$ (view it as $(X_2\ne 0)\sqcup (X_2=0)$).
Also $X=\{[1,\frac{X_1}{X_0},\frac{X_2}{X_0}]: 1+(\frac{X_1}{X_0})^2-(\frac{X_2}{X_0})^2=0\} \sqcup \{[0,\pm 1,1]\}$.
View $Y=X\setminus \{[1,\pm i,0], [0,\pm 1,1]\}$ as (1) circle minus two points and (2) hyperbola minus two points.
Show that the affine co-ordinate ring of $Y$ viewed as in (1) is $A=k[x,y,x^{-1}]/(x^2+y^2-1)$; and as in (2) is $B=k[u,v,v^{-1}]/(1+u^2-v^2)$. Show that $A$ and $B$ are isomorphic directly by giving explicit ring homomorphism. (Hint: $x=X_0/X_2, y=X_1/X_2, u=X_1/X_0, v=X_2/X_0$).

Class 20

(51) Let $S\subset \mathbb{P}^n$, show that $V(I(S))$ is the Zariski closure of $S$.

(52) Show that the map $\Phi_0:\mathbb{A}^n\to \mathbb{P}^n$ is homemorphism onto its image. The topology on a subset of $\mathbb{P}^n$ is the subspace topology inherited from the Zariski topology and the topology on $\mathbb{A}^n$ is the Zariski topology.

(53) Let $X\subset \mathbb{P}^n$ be a projective algebraic subset. Recall that the cone of $X$, $CX=\{(a_0,a_1,\ldots,a_n): [a_0,\ldots,a_n]\in X\} \cup \{origin\}\subset \mathbb{A}^{n+1}$. Show that $X$ is irreducible iff $CX$ is irreducible.

Class 21

(53) Let $X$ be an irreducible algebraic subset of $\mathbb{P}^n$ and $P\in X$ be a point. Show that the set of rational functions on $X$ which are regular at $P$ is a local ring.