189-571B: Higher Algebra II

Assignment 3. Due: Monday, March 11.

This assignment is meant to help you study for the midterm exam which is scheduled on March 13. The Monday deadline will allow us to discuss any questions you might have about it, ahead of the midterm exam.

1. Let $V\subset {\mathbb A}^n$ be a variety over a field $k$, corresponding to an ideal $I$ of $k[x_1,\ldots,x_n]$. Show that the maximal ideals of $k[x_1,\ldots, x_n]$ containing $I$ are in natural bijection with the Galois orbits of points over the algebraic closure. (The absolute Galois group of $k$ acts naturally and continuously on $V(\bar k)$, and a Galois orbit is just a -necessarily finite- orbit for this action.)

2. Explain whether the following assertions are true or false. Here $\bar k$ denotes the algebraic closure of a field $k$.

(a) If $I$ is a maximal ideal of $k[x_1,\ldots, x_n]$, then it generates a maximal ideal of $\bar k[x_1,\ldots, x_n]$.

(b) If $I$ is a prime ideal of $k[x_1,\ldots, x_n]$, then it generates a prime ideal of $\bar k[x_1,\ldots, x_n]$.

(c) If $I$ is a maximal ideal of $k[x_1,\ldots, x_n]$, then the ideal of $\bar k[x_1,\ldots, x_n]$ that is generated by $I$ is the intersection of finitely many maximal ideals of $\bar k[x_1,\ldots,x_n]$.

3. Keep the notations as in the previous question, but assume further that $k$ is algebraically closed. Show that the prime ideals containing $I$ are in natural bijection with the irreducible subvarieties of $V$, and that the minimal prime ideals containing $I$ are in bijection with the irreducible components of $V$.

4. Show that the radical of an ideal $I$ of $k[x_1,\ldots, x_n]$ is equal to the intersection of the prime ideals containing $I$.

5. Let $k$ be an algebraically closed field. Let $V$ be an affine variety in ${\mathbb A}^n$ attached to an ideal $I$ of $k[x_1,\ldots, x_n]$, and let let $\bar V$ denote its projective closure in ${\mathbb P}_n$, obtained by homogenising the collection of equations defining $V$ to obtain a homogenous ideal of $k[x_0,\ldots,x_n]$. Show that $V \mapsto \bar V$ defines a bijection between the non-empty affine varieties in ${\mathbb A}^n$ and the non-empty projective varieties in ${\mathbb P}_n$ none of whose irreducible components lie on the ``hyperplane at infinity" $$ {\mathbb P}_{n-1}^{(\infty)} := \{ (x_0:x_1:\cdots : x_n) \mbox{ such that } x_0=0 \}.$$

6. With notations as in question 5, show that if $V=V_1\cup\cdots \cup V_s$ is the decomposition of $V$ as a union of its irreducible components, then $\bar V = \bar V_1 \cup \cdots \cup \bar V_s$ is the decomposition of $\bar V$ as a union of its irreducible components.

7. Kunz, Exercise 2, page 29.

8. Kunz, Exercise 4, page 29.

9. Kunz, Exercise 4, page 37.

10. Kunz, Exercise 6, page 38.