# Appendix: Course Exercises ## Problem Set 1 :::{.exercise title="Gathmann 1.19"} Prove that every affine variety $X\subset \AA^n/k$ consisting of only finitely many points can be written as the zero locus of $n$ polynomials. > Hint: Use interpolation. It is useful to assume at first that all points in $X$ have different $x_1\dash$coordinates. ::: :::{.exercise title="Gathmann 1.21"} Determine $\sqrt{I}$ for \[ I\da \gens{x_1^3 - x_2^6,\, x_1 x_2 - x_2^3} \normal \CC[x_1, x_2] .\] ::: :::{.exercise title="Gathmann 1.22"} Let $X\subset \AA^3/k$ be the union of the three coordinate axes. Compute generators for the ideal $I(X)$ and show that it can not be generated by fewer than 3 elements. ::: :::{.exercise title="Gathmann 1.23: Relative Nullstellensatz"} Let $Y\subset \AA^n/k$ be an affine variety and define $A(Y)$ by the quotient \[ \pi: k[x_1,\cdots, x_n] \to A(Y) \da k[x_1, \cdots, x_n]/I(Y) .\] a. Show that $V_Y(J) = V(\pi^{-1}(J))$ for every $J\normal A(Y)$. b. Show that $\pi^{-1} (I_Y(X)) = I(X)$ for every affine subvariety $X\subseteq Y$. c. Using the fact that $I(V(J)) \subset \sqrt{J}$ for every $J\normal k[x_1, \cdots, x_n]$, deduce that $I_Y(V_Y(J)) \subset \sqrt{J}$ for every $J\normal A(Y)$. Conclude that there is an inclusion-reversing bijection \[ \correspond{\text{Affine subvarieties}\\ \text{of } Y} \iff \correspond{\text{Radical ideals} \\ \text{in } A(Y)} .\] ::: :::{.exercise title="Extra"} Let $J \normal k[x_1, \cdots, x_n]$ be an ideal, and find a counterexample to $I(V(J)) =\sqrt{J}$ when $k$ is not algebraically closed. ::: ## Problem Set 2 :::{.exercise title="Gathmann 2.17"} Find the irreducible components of \[ X = V(x - yz, xz - y^2) \subset \AA^3/\CC .\] ::: :::{.exercise title="Gathmann 2.18"} Let $X\subset \AA^n$ be an arbitrary subset and show that \[ V(I(X)) = \bar{X} .\] ::: :::{.exercise title="Gathmann 2.21"} Let $\ts{U_i}_{i\in I} \covers X$ be an open cover of a topological space with $U_i \intersect U_j \neq \emptyset$ for every $i, j$. a. Show that if $U_i$ is connected for every $i$ then $X$ is connected. b. Show that if $U_i$ is irreducible for every $i$ then $X$ is irreducible. ::: :::{.exercise title="Gathmann 2.22"} Let $f:X\to Y$ be a continuous map of topological spaces. a. Show that if $X$ is connected then $f(X)$ is connected. b. Show that if $X$ is irreducible then $f(X)$ is irreducible. ::: :::{.exercise title="Gathmann 2.23"} Let $X$ be an affine variety. a. Show that if $Y_1, Y_2 \subset X$ are subvarieties then \[ I(\bar{Y_1\sm Y_2}) = I(Y_1): I(Y_2) .\] b. If $J_1, J_2 \normal A(X)$ are radical, then \[ \bar{V(J_1) \sm V(J_2)} = V(J_1: J_2) .\] ::: :::{.exercise title="Gathmann 2.24"} Let $X \subset \AA^n,\, Y\subset \AA^m$ be irreducible affine varieties, and show that $X\cross Y\subset \AA^{n+m}$ is irreducible. ::: ## Problem Set 3 :::{.exercise title="Gathmann 2.33"} Define \[ X \da \ts{M \in \mat(2\times 3, k) \st \rk M \leq 1} \subseteq \AA^6/k .\] Show that $X$ is an irreducible variety, and find its dimension. ::: :::{.exercise title="Gathmann 2.34"} Let $X$ be a topological space, and show a. If $\ts{U_i}_{i\in I} \covers X$, then $\dim X = \sup_{i\in I} \dim U_i$. b. If $X$ is an irreducible affine variety and $U\subset X$ is a nonempty subset, then $\dim X = \dim U$. Does this hold for any irreducible topological space? ::: :::{.exercise title="Gathmann 2.36"} Prove the following: a. Every noetherian topological space is compact. In particular, every open subset of an affine variety is compact in the Zariski topology. b. A complex affine variety of dimension at least 1 is never compact in the classical topology. ::: :::{.exercise title="Gathmann 2.40"} Let \[ R = k[x_1, x_2, x_3, x_4] / \gens{x_1 x_4 - x_2 x_3} \] and show the following: a. $R$ is an integral domain of dimension 3. b. $x_1, \cdots, x_4$ are irreducible but not prime in $R$, and thus $R$ is not a UFD. c. $x_1 x_4$ and $x_2 x_3$ are two decompositions of the same element in $R$ which are nonassociate. d. $\gens{x_1, x_2}$ is a prime ideal of codimension 1 in $R$ that is not principal. ::: :::{.exercise title="Problem 5"} Consider a set $U$ in the complement of $(0, 0) \in \AA^2$. Prove that any regular function on $U$ extends to a regular function on all of $\AA^2$. ::: ## Problem Set 4 :::{.exercise title="Gathmann 3.20"} Let $X\subset \AA^n$be an affine variety and $a\in X$. Show that \[ \OO_{X, a} = \OO_{\AA^n, a} / I(X) \OO_{A^n,a} ,\] where $I(X) \OO_{\AA^n, a}$ denotes the ideal in $\OO_{\AA^n, a}$ generated by all quotients $f/1$ for $f\in I(X)$. ::: :::{.exercise title="Gathmann 3.21"} Let $a\in \RR$, and consider sheaves $\mathcal{F}$ on $\RR$ with the standard topology: 1. $\mathcal{F} \da$ the sheaf of continuous functions 2. $\mathcal{F} \da$ the sheaf of locally polynomial functions. For which is the stalk $\mathcal{F}_a$ a local ring? > Recall that a local ring has precisely one maximal ideal. ::: :::{.exercise title="Gathmann 3.22"} Let $\phi, \psi \in \mathcal{F}(U)$ be two sections of some sheaf $\mathcal{F}$ on an open $U\subseteq X$ and show that a. If $\phi, \psi$ agree on all stalks, so $\bar{(U, \phi)} = \bar{(U, \psi)} \in \mathcal{F}_a$ for all $a\in U$, then $\phi$ and $\psi$ are equal. b. If $\mathcal{F} \da \OO_X$ is the sheaf of regular functions on some irreducible affine variety $X$, then if $\psi = \phi$ on one stalk $\mathcal{F}_a$, then $\phi = \psi$ everywhere. c. For a general sheaf $\mathcal{F}$ on $X$, (b) is false. ::: :::{.exercise title="Gathmann 3.23: Geometry of a Certain Localization"} Let $Y\subset X$ be a nonempty and irreducible subvariety of an affine variety $X$, and show that the stalk $\OO_{X, Y}$ of $\OO_X$ at $Y$ is a $k\dash$algebra which is isomorphic to the localization $A(X)_{I(Y)}$. ::: :::{.exercise title="Gathmann 3.24"} Let $\mathcal{F}$ be a sheaf on $X$ a topological space and $a\in X$. Show that the stalk $\mathcal{F}_a$ is a *local object*, i.e. if $U\subset X$ is an open neighborhood of $a$, then $\mathcal{F}_a$ is isomorphic to the stalk of $\ro{ \mathcal{F} }{U}$ at $a$ on $U$ viewed as a topological space. ::: ## Problem Set 5 :::{.exercise title="Gathmann 4.13"} Let $f:X\to Y$ be a morphism of affine varieties and $f^*: A(Y) \to A(X)$ the induced map on coordinate rings. Determine if the following statements are true or false: a. $f$ is surjective $\iff f^*$ is injective. b. $f$ is injective $\iff f^*$ is surjective. c. If $f:\AA^1\to\AA^1$ is an isomorphism, then $f$ is *affine linear*, i.e. $f(x) = ax+b$ for some $a, b\in k$. d. If $f:\AA^2\to\AA^2$ is an isomorphism, then $f$ is *affine linear*, i.e. $f(x) = Ax+b$ for some $a \in \Mat(2\times 2, k)$ and $b\in k^2$. ::: :::{.exercise title="Gathmann 4.19"} Which of the following are isomorphic as ringed spaces over $\CC$? (a) $\mathbb{A}^{1} \backslash\{1\}$ (b) $V\left(x_{1}^{2}+x_{2}^{2}\right) \subset \mathbb{A}^{2}$ (c) $V\left(x_{2}-x_{1}^{2}, x_{3}-x_{1}^{3}\right) \backslash\{0\} \subset \mathbb{A}^{3}$ (d) $V\left(x_{1} x_{2}\right) \subset \mathbb{A}^{2}$ (e) $V\left(x_{2}^{2}-x_{1}^{3}-x_{1}^{2}\right) \subset \mathbb{A}^{2}$ (f) $V\left(x_{1}^{2}-x_{2}^{2}-1\right) \subset \mathbb{A}^{2}$ ::: :::{.exercise title="Gathmann 5.7"} Show that a. Every morphism $f:\AA^1\smz \to \PP^1$ can be extended to a morphism $\hat f: \AA^1 \to \PP^1$. b. Not every morphism $f:\AA^2\smz \to \PP^1$ can be extended to a morphism $\hat f: \AA^2 \to \PP^1$. c. Every morphism $\PP^1\to \AA^1$ is constant. ::: :::{.exercise title="Gathmann 5.8"} Show that a. Every isomorphism $f:\PP^1\to \PP^1$ is of the form \[ f(x) = {ax+b \over cx+d} && a,b,c,d\in k .\] where $x$ is an affine coordinate on $\AA^1\subset \PP^1$. b. Given three distinct points $a_i \in \PP^1$ and three distinct points $b_i \in \PP^1$, there is a unique isomorphism $f:\PP^1 \to \PP^1$ such that $f(a_i) = b_i$ for all $i$. ::: :::{.proposition title="?"} There is a bijection \[ \ts{ \text { morphisms } X \rightarrow Y } &\stackrel{1: 1}{\leftrightarrow} \ts{ K\dash\text{algebra morphisms } \mathscr{O}_{Y}(Y) \rightarrow \mathscr{O}_{X}(X) } \\ f &\mapsto f^{*} \] ::: :::{.exercise title="Gathmann 5.9"} Does the above bijection hold if a. $X$ is an arbitrary prevariety but $Y$ is still affine? a. $Y$ is an arbitrary prevariety but $X$ is still affine? :::