Homework 2

This homework covers the material from lectures 6 to 12.

Due date: July 4th, 10pm Waterloo time.

LaTeX template, in case you want to write it in LaTeX.

Required Exercises

Problem 1 - Completing proof of Hilbert’s theorem (25 points)

In lecture 8, we used a lemma about univariate polynomials taking integral values to prove a theorem of Hilbert showing that the Hilbert function of any finitely generated, graded module over a polynomial ring with $n$ variables eventually agrees with a polynomial of degree $< n$.

In this problem, you will prove the auxiliary lemma that we used in lecture 8. For convenience, we restate the lemma here:

Lemma: Let $H : \mathbb{N} \to \mathbb{Z}$ be a function such that its “first difference” $H’(s) := H(s) - H(s-1)$ agrees with a polynomial $Q’(s) \in \mathbb{Q}[s]$ of degree $\leq n-1$ for all $s \geq s_0$ for some $s_0 \in \mathbb{N}$. Then there exists a polynomial $Q(s) \in \mathbb{Q}[s]$ of degree $\leq n$ such that $H(s) = Q(s)$ for all $s \geq s_0$.

To prove this lemma, you can use the following fact:

Fact 1: Let $R \subset \mathbb{Q}[x]$ be the subring of rational polynomials that take integral values for sufficiently large integers. Then, $R$ is the set of rational polynomials which are integer linear combinations of the polynomials $F_k(x) := \binom{x}{k}$ for $k \in \mathbb{N}$, where $\deg F_k = k$.

Problem 2 - Exact Sequences and Dimension of Vector Spaces (25 points)

In this exercise we will explore the relationship of the dimensions of vector spaces that appear in an exact sequence.

Let $V_1, \dots, V_r$ be finite dimensional vector spaces over a field $\mathbb{K}$, such that the following sequence of $\mathbb{K}$-linear maps is exact:

$$ 0 \to V_1 \xrightarrow{\varphi_1} V_2 \xrightarrow{\varphi_2} \cdots \xrightarrow{\varphi_{r-1}} V_r \to 0. $$

Prove that the following identity holds:

$$ \sum_{i=1}^{r} (-1)^{i-1} \dim V_i = 0. $$

Problem 3 - Computing Free Resolutions (20 points)

In lecture, we saw Schreyer’s method to compute the free resolution of a module over a polynomial ring. In the first exercise, you will apply this method to compute the free resolution of a module over a polynomial ring. In the second exercise, we will see an easy example on how Hilbert’s theorem really needs the assumption that the module is over a polynomial ring.

1. Compute the minimal graded free resolution of the module $M = \mathbb{K}[x,y]/(x^2, xy)$ over the polynomial ring $\mathbb{K}[x,y]$.

2. Let $R := \mathbb{K}[x]/(x^2)$ (note that $R$ is not a polynomial ring). Compute the graded free resolution of the $R$-module $M = R/(x)$.

Problem 4 - Lemmas from Dube’s paper (30 points)

In this question, you will prove some of the lemmas from Dube’s paper that we left as exercises.

1. Prove the following properties of cone decompositions:

   1.1. $\emptyset$ is a $0$-standard cone decomposition of $\emptyset$.

   1.2. $\{(h,u)\}$ is a $\deg h$-standard cone decomposition of $C(h,u)$.

   1.3. $\{(1, \{x_1, \ldots, x_n\})\}$ is a $0$-standard cone decomposition of $S := \mathbb{K}[x_1, \ldots, x_n]$.

   1.4. If $T = A_1 \oplus A_2$ and $P_1$ and $P_2$ are $k$-standard cone decompositions of $A_1$ and $A_2$, respectively, then $P_1 \cup P_2$ is a $k$-standard cone decomposition of $T$.

2. Prove Lemma 4.10 from Dube’s paper.

3. Prove Lemma 6.2 from Dube’s paper.