Appendix: Course Exercises

Prove that every affine variety \(X\subset {\mathbf{A}}^n/k\) 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 \(x_1{\hbox{-}}\)coordinates.

Determine \(\sqrt{I}\) for \begin{align*} I\coloneqq\left\langle{x_1^3 - x_2^6,\, x_1 x_2 - x_2^3}\right\rangle {~\trianglelefteq~}{\mathbf{C}}[x_1, x_2] .\end{align*}

Let \(X\subset {\mathbf{A}}^3/k\) 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.

Let \(Y\subset {\mathbf{A}}^n/k\) be an affine variety and define \(A(Y)\) by the quotient \begin{align*} \pi: k[x_1,\cdots, x_n] \to A(Y) \coloneqq k[x_1, \cdots, x_n]/I(Y) .\end{align*}

• Show that \(V_Y(J) = V(\pi^{-1}(J))\) for every \(J{~\trianglelefteq~}A(Y)\).

• Show that \(\pi^{-1} (I_Y(X)) = I(X)\) for every affine subvariety \(X\subseteq Y\).

• Using the fact that \(I(V(J)) \subset \sqrt{J}\) for every \(J{~\trianglelefteq~}k[x_1, \cdots, x_n]\), deduce that \(I_Y(V_Y(J)) \subset \sqrt{J}\) for every \(J{~\trianglelefteq~}A(Y)\).

Conclude that there is an inclusion-reversing bijection \begin{align*} \left\{{\substack{\text{Affine subvarieties}\\ \text{of } Y}}\right\} \iff \left\{{\substack{\text{Radical ideals} \\ \text{in } A(Y)}}\right\} .\end{align*}

Let \(J {~\trianglelefteq~}k[x_1, \cdots, x_n]\) be an ideal, and find a counterexample to \(I(V(J)) =\sqrt{J}\) when \(k\) is not algebraically closed.

Find the irreducible components of \begin{align*} X = V(x - yz, xz - y^2) \subset {\mathbf{A}}^3/{\mathbf{C}} .\end{align*}

Let \(X\subset {\mathbf{A}}^n\) be an arbitrary subset and show that \begin{align*} V(I(X)) = \mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu .\end{align*}

Let \(\left\{{U_i}\right\}_{i\in I} \rightrightarrows X\) be an open cover of a topological space with \(U_i \cap U_j \neq \emptyset\) for every \(i, j\).

• Show that if \(U_i\) is connected for every \(i\) then \(X\) is connected.

• Show that if \(U_i\) is irreducible for every \(i\) then \(X\) is irreducible.

Let \(f:X\to Y\) be a continuous map of topological spaces.

• Show that if \(X\) is connected then \(f(X)\) is connected.

• Show that if \(X\) is irreducible then \(f(X)\) is irreducible.

Let \(X\) be an affine variety.

• Show that if \(Y_1, Y_2 \subset X\) are subvarieties then \begin{align*} I(\mkern 1.5mu\overline{\mkern-1.5muY_1\setminus Y_2\mkern-1.5mu}\mkern 1.5mu) = I(Y_1): I(Y_2) .\end{align*}

• If \(J_1, J_2 {~\trianglelefteq~}A(X)\) are radical, then \begin{align*} \mkern 1.5mu\overline{\mkern-1.5muV(J_1) \setminus V(J_2)\mkern-1.5mu}\mkern 1.5mu = V(J_1: J_2) .\end{align*}

Let \(X \subset {\mathbf{A}}^n,\, Y\subset {\mathbf{A}}^m\) be irreducible affine varieties, and show that \(X\times Y\subset {\mathbf{A}}^{n+m}\) is irreducible.

Define \begin{align*} X \coloneqq\left\{{M \in \operatorname{Mat}(2\times 3, k) {~\mathrel{\Big\vert}~}{\operatorname{rank}}M \leq 1}\right\} \subseteq {\mathbf{A}}^6/k .\end{align*}

Show that \(X\) is an irreducible variety, and find its dimension.

Let \(X\) be a topological space, and show

• If \(\left\{{U_i}\right\}_{i\in I} \rightrightarrows X\), then \(\dim X = \sup_{i\in I} \dim U_i\).

• 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?

Prove the following:

• Every noetherian topological space is compact. In particular, every open subset of an affine variety is compact in the Zariski topology.

• A complex affine variety of dimension at least 1 is never compact in the classical topology.

Let \begin{align*} R = k[x_1, x_2, x_3, x_4] / \left\langle{x_1 x_4 - x_2 x_3}\right\rangle \end{align*} and show the following:

• \(R\) is an integral domain of dimension 3.

• \(x_1, \cdots, x_4\) are irreducible but not prime in \(R\), and thus \(R\) is not a UFD.

• \(x_1 x_4\) and \(x_2 x_3\) are two decompositions of the same element in \(R\) which are nonassociate.

• \(\left\langle{x_1, x_2}\right\rangle\) is a prime ideal of codimension 1 in \(R\) that is not principal.

Consider a set \(U\) in the complement of \((0, 0) \in {\mathbf{A}}^2\). Prove that any regular function on \(U\) extends to a regular function on all of \({\mathbf{A}}^2\).

Let \(X\subset {\mathbf{A}}^n\)be an affine variety and \(a\in X\). Show that \begin{align*} {\mathcal{O}}_{X, a} = {\mathcal{O}}_{{\mathbf{A}}^n, a} / I(X) {\mathcal{O}}_{A^n,a} ,\end{align*} where \(I(X) {\mathcal{O}}_{{\mathbf{A}}^n, a}\) denotes the ideal in \({\mathcal{O}}_{{\mathbf{A}}^n, a}\) generated by all quotients \(f/1\) for \(f\in I(X)\).

Let \(a\in {\mathbb{R}}\), and consider sheaves \(\mathcal{F}\) on \({\mathbb{R}}\) with the standard topology:

• \(\mathcal{F} \coloneqq\) the sheaf of continuous functions
• \(\mathcal{F} \coloneqq\) 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.

Let \(\phi, \psi \in \mathcal{F}(U)\) be two sections of some sheaf \(\mathcal{F}\) on an open \(U\subseteq X\) and show that

• If \(\phi, \psi\) agree on all stalks, so \(\mkern 1.5mu\overline{\mkern-1.5mu(U, \phi)\mkern-1.5mu}\mkern 1.5mu = \mkern 1.5mu\overline{\mkern-1.5mu(U, \psi)\mkern-1.5mu}\mkern 1.5mu \in \mathcal{F}_a\) for all \(a\in U\), then \(\phi\) and \(\psi\) are equal.

• If \(\mathcal{F} \coloneqq{\mathcal{O}}_X\) 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.

• For a general sheaf \(\mathcal{F}\) on \(X\), (b) is false.

Let \(Y\subset X\) be a nonempty and irreducible subvariety of an affine variety \(X\), and show that the stalk \({\mathcal{O}}_{X, Y}\) of \({\mathcal{O}}_X\) at \(Y\) is a \(k{\hbox{-}}\)algebra which is isomorphic to the localization \(A(X)_{I(Y)}\).

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 \({ \left.{{ \mathcal{F} }} \right|_{{U}} }\) at \(a\) on \(U\) viewed as a topological space.

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:

• \(f\) is surjective \(\iff f^*\) is injective.

• \(f\) is injective \(\iff f^*\) is surjective.

• If \(f:{\mathbf{A}}^1\to{\mathbf{A}}^1\) is an isomorphism, then \(f\) is affine linear, i.e. \(f(x) = ax+b\) for some \(a, b\in k\).

• If \(f:{\mathbf{A}}^2\to{\mathbf{A}}^2\) is an isomorphism, then \(f\) is affine linear, i.e. \(f(x) = Ax+b\) for some \(a \in \operatorname{Mat}(2\times 2, k)\) and \(b\in k^2\).

Which of the following are isomorphic as ringed spaces over \({\mathbf{C}}\)?

• \(\mathbb{A}^{1} \backslash\{1\}\)

• \(V\left(x_{1}^{2}+x_{2}^{2}\right) \subset \mathbb{A}^{2}\)

• \(V\left(x_{2}-x_{1}^{2}, x_{3}-x_{1}^{3}\right) \backslash\{0\} \subset \mathbb{A}^{3}\)

• \(V\left(x_{1} x_{2}\right) \subset \mathbb{A}^{2}\)

• \(V\left(x_{2}^{2}-x_{1}^{3}-x_{1}^{2}\right) \subset \mathbb{A}^{2}\)

• \(V\left(x_{1}^{2}-x_{2}^{2}-1\right) \subset \mathbb{A}^{2}\)

Show that

• Every morphism \(f:{\mathbf{A}}^1\setminus\left\{{0}\right\}\to {\mathbf{P}}^1\) can be extended to a morphism \(\widehat{f}: {\mathbf{A}}^1 \to {\mathbf{P}}^1\).

• Not every morphism \(f:{\mathbf{A}}^2\setminus\left\{{0}\right\}\to {\mathbf{P}}^1\) can be extended to a morphism \(\widehat{f}: {\mathbf{A}}^2 \to {\mathbf{P}}^1\).

• Every morphism \({\mathbf{P}}^1\to {\mathbf{A}}^1\) is constant.

Show that

• Every isomorphism \(f:{\mathbf{P}}^1\to {\mathbf{P}}^1\) is of the form \begin{align*} f(x) = {ax+b \over cx+d} && a,b,c,d\in k .\end{align*} where \(x\) is an affine coordinate on \({\mathbf{A}}^1\subset {\mathbf{P}}^1\).

• Given three distinct points \(a_i \in {\mathbf{P}}^1\) and three distinct points \(b_i \in {\mathbf{P}}^1\), there is a unique isomorphism \(f:{\mathbf{P}}^1 \to {\mathbf{P}}^1\) such that \(f(a_i) = b_i\) for all \(i\).

Does the above bijection hold if

• \(X\) is an arbitrary prevariety but \(Y\) is still affine?
• \(Y\) is an arbitrary prevariety but \(X\) is still affine?