Homework for Introduction to Algebraic Geometry

Unless otherwise stated a ring would mean a commutative ring with identity.

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. Show 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 a 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) For an affine algebraic set $X$ in $\mathbb{A}^n_k$, show that $Z(I(X))=X$.

Class 4

(9) Let $I$ denote the defining ideal of an algebraic set $X$ in $\mathbb{A}^n_k$ where $k$ is an algebraically closed field. Recall that the cordinate ring of $X$ is $R=k[x_1,\ldots,x_n]/I$. Show that $R$ is a reduced ring.

(10) Let $S$ be multiplicative subset of a ring $R$. Show that the construction $(S^{-1}R,+,.)$ done in the class is a ring and the map $\phi:R\to S^{-1}R$ sending $r$ to the conjugacy class $\frac{r}{1}$ is a ring homomorphism.

(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 $S=\{1, p, p^2, \dots \}$.

Class 5

(12) State and prove the universal property of the localization of a ring $R$ with respect to a multiplicative subset $S$.

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

(14) 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.

(15) Let $R$ be a ring and $M$ be an $R$-module. Show that $Hom_R(R,M)$ is isomorphic to $M$ as an $R$-module.

(16) 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".

Class 6

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

(18) 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 $S^{-1}R/S^{-1}\text{ker}(q)$.

(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 (18).

Class 7

(20) Let $R$ be a ring and $A$, $B$ and $C$ be $R$-modules. Show that $A\otimes_R (B \otimes_R C)$ is isomorphic to $(A \otimes_R B)\otimes_R C$ as $R$-modules.

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

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

Class 8

(23) 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.

(24) 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 $n\ge m$.
(iii) If $f:R^n\to R^m$ is an $R$-linear injection, show that $n\le m$.

Class 9

(25) (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 $A[x]$.

(26) State Hom-Tensor duality. Use it and left exactness of $Hom$ to show that $\otimes$ is right exact. (Hint: First show that if $A\to B\to C\to 0$ is a complex of $R$-modules such that $0\to Hom(C,N)\to Hom(B,N)\to Hom(A,N)$ is exact for all $R$-module $N$ then the complex $A\to B\to C\to 0$ is exact.)

(27) 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 $\bar{\mathbb{Q}}$.

Class 10

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

(29) 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 13

(30) Let $R$ be a ring and $P$ a projective $R$-module, i.e. for any surjective $R$-module homomorphism $f:A \to B$ and an $R$-module homomorphism $g:P\to B$ there exist an $R$-module homomorphism $\tilde g:P\to A$ such that $g=f\circ \tilde g$. Show that for any surjective $R$-module homomorphism $f:F\to P$ where $F$ is a free $R$-module, $F$ is isomorphic to $P\oplus ker(f)$.

(31) 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.
Step 1: Define a ring homomorphism $\Phi:k[Y_1,\ldots, Y_m]\to k[X_1,\ldots, X_n]$ by $\Phi(g(Y_1,\ldots,Y_n))=g(f_1,\ldots,f_m)$. Show 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 $\phi: k[Y_1,\ldots,Y_m]/I(Y)\to k[X_1,\ldots X_m]/I(X)$.
Step 2: Let $\underline a \in X$ and $b_i=f_i(\underline a)$ then show that $\Phi^{-1}((X_1-a_1,\ldots,X_n-a_n))=(Y_1-b_1,\ldots,Y_m-b_m)$. (Hint: Note that the polynomials $f_i-b_i \in (X_1-a_1,\dots,x_n-a_n)$).

Class 14

(32) Let $k$ be a field and $R=k[X]/(X(X-1))$. Show that the $R$-module $M=R/(\bar X)$ is projective but not free.

(33) Let $M$ be a projective $R$-module and $S$ is a multiplicative subset of $R$ then $S^{-1}M$ is a projective $S^{-1}R$-module.

Class 16

(34) 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.)

(35) 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 (34))

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

(37) 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 17

(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 18

(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 $A$.
(iii) (optional) More generally, $A \to B$ is a flat homomorphism.

(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 $P$ is the maximal ideal $\{f\in k[X]: f(p)=0\}$.

(43) 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 $t\to (t,t^2,t^3)$. Show that $\phi$ induces an isomorphism $\mathbb{A}^1 \to C$.

Class 19

(44) Show that Zariski topology on an irreducible affine curve is same as the cofinite topology.

(45) Show that every Zariski 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).

Class 20

(46) 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 21

(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. Obtain that the affine co-ordinate ring of $Y$ in (1) is $A=k[x,y,x^{-1}]/(x^2+y^2-1)$ and 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 22

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

(52) Let $X$ be an irreducible algebraic subset of $\PP^n$ 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.

(53) Compute the field of rational functions of $\PP^n$.