Homework 3

Due date: July 30th, 10pm Waterloo time.

Required Exercises

For this entire homework, let $R := \mathbb{C}[x_1, \dots, x_n]$ be the polynomial ring in $n$ variables over $\mathbb{C}$, and let $I, J \subseteq R$ be ideals.

Problem 1 - Primary Ideals (30 points)

Prove the following:

1. If $\sqrt{I}$ is maximal then $I$ is primary.
2. For any $b \in R$, we have $(I \cap J) : (b) = (I : (b)) \cap (J : (b))$.
3. Let $\mathcal{P} \subset R$ be a prime ideal. If $I, J$ are $\mathcal{P}$-primary ideals, then $I \cap J$ is $\mathcal{P}$-primary.

Problem 2 - Primary Decomposition (35 points)

Let $u \subset \{x_1, \dots, x_n\}$ be an independent set with respect to $I$ and let $v := \{x_1, \dots, x_n\} \setminus u$. Prove that $I \cdot \mathbb{C}(u)[v]$ is a primary ideal if $I$ is primary.

Optional: you can also prove that $I \cdot \mathbb{C}(u)[v]$ is prime if $I$ is prime.

Problem 3 - Primary Decomposition (35 points)

Let $I = \bigcap_{i=1}^s \mathcal{Q}_i$ be an irredundant primary decomposition of $I$ and let $u \subset \{x_1, \dots, x_n\}$ be an independent set with respect to $I$. Set $v := \{x_1, \dots, x_n \} \setminus u$. Assume that $\mathcal{Q}_i \cap \mathbb{C}[u] = 0$ for $i \in [r]$ and $\mathcal{Q}_i \cap \mathbb{C}[u] \neq 0$ for $i \in [r+1, s]$. Prove that $I \cdot \mathbb{C}(u)[v] = \bigcap_{i=1}^r \mathcal{Q}_i \cdot \mathbb{C}(u)[v]$ and that this is an irredundant primary decomposition of $I \cdot \mathbb{C}(u)[v]$.