\(\gdef\LT{\operatorname{LT}}\) \(\gdef\LM{\operatorname{LM}}\)

General Information

This page will be “under construction” throughout this quarter (especially any section that does not yet appear in the Course Schedule). Please feel free to point out any typos or mistakes!

Each section below has a list of “Core Ideas” and a list of “Exercises.”

The “Core Ideas” are a list of important definitions, theorems, etc, that you should know:

I’ve suggested possible readings for learning about these ideas. You can use the list of “Core Ideas” as a checklist as you do the reading. Feel free to supplement with other resources you find.
Some of the content can be found in one of our primary references. Some of it can be found in both, in which case you are welcome to read whichever you like better.
When a term doesn’t have a completely standard definition (eg, if the definition varies between our two primary references), I’ve listed the definition I’d like to use for the purposes of our class and/or made notes about any possible discrepancies I could think of. Same with notation.

The “Exercises” are for practice:

You are encouraged to do as many of the exercises as you can.
Give formal proofs whenever possible! For example, if an exercise asks you to “give a example” of something, make sure you include a proof that your example is in fact an example of that thing.
Sometimes, exercises define important concepts and develop further theory that will be used in the future. This means that, even if you don’t do all of the exercises, you should at least read through all of them.
Sometimes, exercises ask you to state a definition of a term in italics. You may have to look this up. Use any resource: one of the books, the internet, whatever.
Sometimes, you might find the solution to an exercise as a proof of a proposition somewhere in one of the two primary references. This is meant to indicate to you that you should understand the core ideas well enough that you can come up with that proof on your own.
The first few exercises in each section are usually the easiest. After the first few exercises, there’s no particular order to the exercises and you don’t need to do them linearly. Sometimes, doing later problems might give you ideas or tools that are helpful for earlier problems. Sometimes, it may even help to come back to problems you didn’t figure out before after you’ve seen later material.

Many of the exercises are stolen from primary references, sometimes with a minimal amount of adaptation.

Ring Theory Basics

Core Ideas

Definition of a ring.
- Note: We will always assume that rings are commutative and have a multiplicative identity.
Definition of an ideal.
Definition of the ideal \(\langle S \rangle\) generated by a subset \(S \subseteq R\).
Definition of a finitely generated ideal and of a principal ideal.
Definition of a quotient ring, and the fact that it is in fact a ring.
Definition of a field.
Examples of fields: \(\mathbb{Q}\), \(\mathbb{R}\), \(\mathbb{C}\), and \(\mathbb{F}_p\) for a prime \(p\).
- Note: The field \(\mathbb{F}_p\) might also be denoted \(\mathbb{Z}/p\mathbb{Z}\), \(\mathbb{Z}_p\), or \(\operatorname{GF}(p)\), depending on what you’re reading. We will use the notation \(\mathbb{F}_p\).

Possible reading: Whatever textbook you used for Math 100B or 103B or the equivalent. If you like, you might also skim through [CLO15, Appendix A.1].

Exercises

Give an example to demonstrate that the union of two ideals need not be an ideal.
Let \(I\) be an ideal in a ring \(R\). Prove that \(I = R\) if and only if \(1 \in I\).
For ideals \(I\) and \(J\) in a ring \(R\), define \(I + J = \{a + b \mid a \in I, b \in J\} \subseteq R\).
1. Prove that \(I + J\) is an ideal of \(R\).
2. Prove that \(I + J\) is the smallest ideal of \(R\) containing both \(I\) and \(J\).
3. Prove that, if \(I = \langle a_1, \dotsc, a_m \rangle\) and \(J = \langle b_1, \dotsc, b_m \rangle\), then \(I + J = \langle a_1, \dotsc, a_m, b_1, \dotsc, b_m \rangle\).
For ideals \(I\) and \(J\) in a ring \(R\), show that \(I \cap J\) is also an ideal.
For ideals \(I\) and \(J\) in a ring \(R\), define \[IJ = \left\{ \sum_i a_i b_i \mid a_1, \dotsc, a_n \in I, b_1, \dotsc, b_n \in J \right\} \subseteq R.\]
1. Prove that \(IJ\) is an ideal of \(R\).
2. Prove that, if \(I = \langle a_1, \dotsc, a_m \rangle\) and \(J = \langle b_1, \dotsc, b_n \rangle\), then \(IJ\) is generated by \(a_ib_j\) for \(i = 1, \dotsc, m\) and \(j = 1, \dotsc, n\).
3. Prove that \(IJ \subseteq I \cap J\).
4. Is it true always that \(IJ = I \cap J\)? If so, prove it. If not, give a counterexample.
Let \(I\) be an ideal in a ring \(R\). Prove that ideals of \(R/I\) are in bijection with ideals of \(R\) that contain \(I\). Hint: Consider the map \(\pi : R \to R/I\) given by \(\pi(a) = a + I\).
1. State the definition of a maximal ideal.
2. Let \(I\) be an ideal in a ring \(R\). Prove that \(I\) is maximal if and only if \(R/I\) is a field.
1. State the definitions of a prime ideal and an integral domain.
2. Let \(I\) be an ideal in a ring \(R\). Prove that \(I\) is prime if and only if \(R/I\) is an integral domain.
1. State the definitions of a radical ideal and a reduced ring.
2. Let \(I\) be an ideal in a ring \(R\). Prove that \(I\) is radical if and only if \(R/I\) is reduced.
For an ideal \(I\) in a ring \(R\), the radical of \(I\), denoted \(\sqrt{I}\) (or sometimes \(\operatorname{Rad}(I)\)), is defined by \[ \sqrt{I} = \{a \mid a^m \in I \text{ for some } m \geq 1\}.\] Prove that \(\sqrt{I}\) is an ideal, that it is radical, and that it is the smallest radical ideal containing \(I\).
Let \(I \subseteq J\) be ideals in a ring \(R\).
1. Prove that \(J/I\) is an ideal in \(R/I\).
2. Prove that \(J/I\) is maximal if and only if \(J\) is maximal.
3. Prove that \(J/I\) is prime if and only if \(J\) is prime.
4. Prove that \(J/I\) is radical if and only if \(J\) is radical.

Single Variable Polynomials

Core Ideas

Definition of the polynomial ring \(k[x]\) in one variable \(x\) over a field \(k\).
Division algorithm in a single variable polynomial ring.
The fact that a polynomial of degree \(n\) has at most \(n\) roots.
Definition of a divisor of a polynomial.
Definition of an irreducible polynomial.
Definition of the greatest common divisor of two polynomials.
Euclidean algorithm for finding greatest common divisors.
Definition of an algebraically closed field.
Fundamental theorem of algebra: \(\mathbb{C}\) is algebraically closed.
- Note: You don’t need to read a proof. Just know, and be able to use, the statement.

Possible reading: [CLO15, Theorem 1.1.7 and Section 1.5]. You may remember the definition of irreducibility from Math 100B or 103B or the equivalent, but you can also find it in [CLO15, Definition A.2.1] in a slightly more general context.

Exercises

In \(\mathbb{Q}[x]\), compute the following.
1. \(\gcd(x^4 + x^2 + 1, x^4 - x^2 - 2x - 1)\).
2. \(\gcd(x^4 + x^2 + 1, x^3 - 1)\).
Find a polynomial \(f \in \mathbb{Q}[x]\) such that \(\langle f \rangle = \langle x^3 + 2x^2 - x - 2, x^3 - x^2 - 4x + 4 \rangle\).
Prove that \(x^2+1\) is irreducible in \(\mathbb{R}[x]\), and that it is not irreducible in \(\mathbb{C}[x]\).
1. Prove that \(x^2 + x + 1\) is irreducible in \(\mathbb{F}_2[x]\).
2. Prove that \(x^3 - x - 1\) is irreducible in \(\mathbb{F}_3[x]\).
3. Read about the finite field \(\mathbb{F}_{p^n}\) for a prime \(p\) and an integer \(n \geq 2\). Then explain how the irreducible polynomials appearing in the previous two parts can be used to give explicit descriptions of \(\mathbb{F}_4\) and \(\mathbb{F}_{27}\).
1. Prove that there are infinitely many irreducible polynomials in \(k[x]\). Hint: Does your proof work even if \(k\) is finite? You could try mimicking Euclid’s proof of the infinitude of primes.
2. Prove that any algebraically closed field has infinitely many elements. Hint: What are the irreducibles over an algebraically closed field?
Suppose \(f \in k[x]\) and let \(f = a f_1^{m_1} \cdots f_r^{m_r}\) be the unique factorization of \(f\) where \(a \in k\) is the leading coefficient of \(f\), each \(f_i\) is monic and irreducible, and each \(m_i\) is a positive integer. Define the reduction of \(f\), denoted \(f_{\mathrm{red}}\), by \[ f_{\mathrm{red}} = af_1 \cdots f_r. \]
1. Prove that the radical of the ideal \(\langle f \rangle \subseteq k[x]\) is \(\langle f_{\mathrm{red}} \rangle\).
2. Suppose that \(k\) has characteristic 0, ie, that \(\mathbb{Q} \subseteq k\). Let \(f'\) be the formal derivative of \(f\). In other words, if \(f = ax^n + a_{n-1} x^{n-1} + \cdots + a_2x^2 + a_1x + a_0\), then \[f' = nax^{n-1} + (n-1)a_{n-1}x^{n-2} + \cdots + 2a_2x + a_1.\] Prove that \(f_{\mathrm{red}} = f/\gcd(f, f')\). Note: You may use standard properties of derivatives (eg, the product rule).
3. Is the statement of the previous part of this exercise true for \(k = \mathbb{F}_p\)? If so, prove it. If not, provide a counterexample.
Observe that the field \(\mathbb{C}\) is the smallest algebraically closed field containing \(\mathbb{R}\). This is expressed by saying that \(\mathbb{C}\) is an algebraic closure of \(\mathbb{R}\). It is a fact that every field has an algebraic closure. Look up and read through a proof of this fact. (Maybe these notes by Hanspeter Fischer?) You may also have to look up and read about Zorn’s lemma as you do this. Then summarize the strategy of the proof in your own words.

Multivariable Polynomials

Core Ideas

Definition of a monomial and its total degree.
Definition of a polynomial in the variables \(x_1, \dotsc, x_n\) over a field \(k\) and its total degree.
The fact that the set \(k[x_1, \dotsc, x_n]\) of all polynomials in the variables \(x_1, \dotsc, x_n\) over a field \(k\) forms a ring.
Hilbert basis theorem: Every ideal in \(k[x_1, \dotsc, x_n]\) is finitely generated.
Definition of a homogeneous polynomial.

Possible reading: [Ful08, Sections 1.1 and 1.4]. You can also look at [CLO15, Section 1.1, Definitions 1.4.1–2, and Definition 7.1.6]. The Hilbert basis theorem also appears as [CLO15, Theorem 2.5.4], but that proof requires some concepts we have not yet discussed.

Exercises

Prove that the ideal \(\langle x, y \rangle\) in \(k[x, y]\) is not principal.
Let \(S\) be the (infinite) set of all monomials in \(k[x_1, \dotsc, x_n]\) of nonzero total degree. Find a finite generating set for the ideal \(I\) generated by \(S\).
1. Let \(f(x, y) = x^2 y + y^2 x\). Show that \(f(x, y) = 0\) for all \(x, y \in \mathbb{F}_2\).
2. Find a nonzero polynomial in \(\mathbb{F}_2[x,y,z]\) which vanishes for all \(x,y,z \in \mathbb{F}_2\).
3. Find a nonzero polynomial in \(\mathbb{F}_2[x_1, \dotsc, x_n]\) which vanishes for all \(x_1, \dotsc, x_n \in \mathbb{F}_2\).
For any \(a_1, \dotsc, a_n \in k\), let \(I = \langle x_1 - a_1, \dotsc, x_n - a_n \rangle \subseteq k[x_1, \dotsc, x_n]\). Prove that \(I\) is a maximal ideal. Hint: What is the quotient ring?
1. Prove that the number of monomials in \(k[x_0, x_1, \dotsc, x_n]\) of total degree \(d\) is \(\binom{n+d}{d}\).
2. Prove that the number of monomials in \(k[x_1, \dotsc, x_n]\) of total degree at most \(d\) is \(\binom{n+d}{d}\).
Let \(R\) be a ring and let \(I \subseteq R[x]\) be an ideal in a single variable polynomial ring over \(R\). Let \(J\) be the subset of \(R\) consisting of leading coefficients of elements of \(I\). Also, for an integer \(m \geq 0\), let \(J_m\) be the subset of \(R\) consisting of leading coefficients of elements of \(I\) whose degree is at most \(m\). Observe that \[J_0 \subseteq J_1 \subseteq J_2 \subseteq \cdots \subseteq J \subseteq R.\] This exercise is about working through some of the details in the proof of the Hilbert basis theorem as it occurs in [Ful08, section 1.4].
1. Verify that \(J\) is an ideal of \(R\).
2. Suppose \(I\) is generated by \(f_1, \dotsc, f_r \in R[x]\). Is it true that \(J\) is generated by the leading coefficients of \(f_1, \dotsc, f_r\)? If so, prove it. If not, provide a counterexample.
3. Suppose that the leading coefficients of \(f_1, \dotsc, f_r \in I\) generate \(J\). Fix \(g \in I\) such that \(\deg(g) \geq \deg(f_i)\) for all \(i\). Prove that there exist \(q_1, \dotsc, q_r \in R[x]\) such that \(\deg(g - \sum q_i f_i) < \deg(g)\).
4. Verify that each \(J_m\) is an ideal of \(R\).
5. Suppose \(f_1, \dotsc, f_r \in I\) are polynomials of degree at most \(m\) whose leading coefficients generate \(J_m\). Suppose \(g \in I\) and \(\deg(g) = m\). Prove that there exist \(q_1, \dotsc, q_r \in R[x]\) such that \(\deg(g - \sum q_i f_i) < \deg(g)\).
Consider the ideal \(I = \langle x^2y^2z - 1, x^3yz^2 + 1 \rangle\) inside \(k[x, y, z]\).
1. Regarding polynomials in \(k[x, y, z]\) as polynomials in \(z\) with coefficients in \(k[x, y]\), what is the ideal \(J\) of leading coefficients of elements of \(I\)? Find a finite generating set for \(J\), and for each of the generators, find a polynomial in \(I\) whose leading coefficient is that generator.
2. For each integer \(m \geq 0\), let \(J_m\) be the ideal in \(k[x, y]\) of leading coefficients of elements of \(I\) whose degree (when regarded as a polynomial in \(z\) with coefficients in \(k[x,y]\)) at most \(m\). Find a finite generating set for \(J_m\), and for each of the generators, find a polynomial in \(I\) of degree at most \(m\) whose leading coefficient is that generator. Note: The answer depends on the integer \(m \geq 0\), but it should “stabilize.”
3. Verify that all of the polynomials in \(I\) that you’ve identified in the previous two parts do in fact generate \(I\).
Suppose \(I = \langle f_1, \dotsc, f_r \rangle \subseteq k[x_1, \dotsc, x_n]\) and \(f \in k[x_1, \dotsc, x_n]\). Prove that \(f \in \sqrt{I}\) if and only if \(\langle f_1, \dotsc, f_r, 1 - yf \rangle = k[x_1, \dotsc, x_r, y]\). Hint: Observe that \(\langle f_1, \dotsc, f_r, 1 - yf \rangle = k[x_1, \dotsc, x_r, y]\) if and only if \(1 \in \langle f_1, \dotsc, f_r, 1 - yf \rangle\). In an expression where \(1\) is written in terms of these generators, what happens when you make the substitution \(y = 1/f\)?

Affine and Projective Spaces

Core Ideas

Definition of affine \(n\)-space over a field \(k\).
- Note: We follow the notation of [Ful08, Section 1.2] in that we write \(\mathbb{A}^n(k)\) instead of \(k^n\), and when the field \(k\) can be arbitrary or it can be inferred from context, we simply write \(\mathbb{A}^n\).
Definition of the projective plane over \(\mathbb{R}\).
Definition of projective \(n\)-space \(\mathbb{P}^n(k)\) over a field \(k\).
- Note: We simply write \(\mathbb{P}^n\) when \(k\) can be arbitrary or it can be inferred from context. The homogeneous coordinates of a point \(P \in \mathbb{P}^n\) can be written \((a_0 : \cdots : a_n)\) or \([a_0 : \cdots : a_n]\).
For \(i = 0, \dotsc, n\), the fact that the subset \(U_i = \{ (a_0 : \cdots : a_n) \in \mathbb{P}^n \mid a_i \neq 0\}\) is in bijection with \(\mathbb{A}^n\) and that its complement is in bijection with \(\mathbb{P}^{n-1}\).
The fact that \(\mathbb{P}^n = U_0 \cup \cdots \cup U_n\).

Possible reading: For affine space, either [CLO15, Definition 1.1.4] or [Ful08, Section 1.2, first paragraph] is fine. For projective space, I recommend [CLO15, Sections 8.1–2 up through Corollary 8.2.3]. You’d get all the necessary ideas from [Ful08, Section 4.2], but the exposition there is much more terse.

Exercises

For each of the following pairs of lines in \(\mathbb{A}^2(\mathbb{R})\), find homogeneous coordinates for their point of intersection in \(\mathbb{P}^2(\mathbb{R})\). You can assume that \((x, y) \in \mathbb{A}^2(\mathbb{R})\) is identified with \((x : y : 1) \in \mathbb{P}^2(\mathbb{R})\).
1. \(y = x\) and \(y = -x + 1\).
2. \(y = 2x\) and \(y = 2x + 1\).
Which points of \(\mathbb{P}^2(\mathbb{R})\), if any, are not contained in \(U_0 \cup U_1\)?
Do [CLO15, Exercise 8.1.2].
Let \(S^1 = \{ (x, y) \in \R^2 \mid x^2 + y^2 = 1\}\) be the unit circle. Give a geometric construction of a surjective function \(S^1 \to \mathbb{P}^1(\mathbb{R})\). What is the preimage of a point of \(\mathbb{P}^1(\mathbb{R})\) under your function?
Suppose \(k\) is a finite field with \(q\) elements. How many points are in \(\mathbb{P}^n(k)\)? Find and prove a formula in terms of \(q\) and \(n\).
\(\mathbb{P}^1(\mathbb{C})\) is sometimes called the Riemann sphere. The point of this exercise is to explain why. Let \[S^2 = \{ (x, y, z) \mid x^2 + y^2 + z^2 = 1\} \subseteq \mathbb{R}^3\] be the unit sphere. The point \(N = (0, 0, 1)\) is the north pole of \(S^2\).
1. For any point \(P = (x, y, z) \in S^2 \setminus \{N\}\), find a formula (in terms of \(x, y, z\)) for the point where the line connecting \(P\) and \(N\) intersects the plane \(z = 0\).
2. Explain how the construction of the previous part gives a bijection \(S^2 \setminus \{N\} \to \mathbb{A}^1(\mathbb{C})\).
3. Explain how to extend \(S^2 \setminus \{N\} \to \mathbb{A}^1(\mathbb{C})\) to a bijection \(S^2 \to \mathbb{P}^1(\mathbb{C})\).

Affine Varieties

Core Ideas

Definition: For a subset \(S \subseteq k[x_1, \dotsc, x_n]\), define \[V(S) = \{ (a_1, \dotsc, a_n) \mid f(a_1, \dotsc, a_n) = 0 \text{ for all } f \in S \} \subseteq \mathbb{A}^n.\] An algebraic subset of \(\mathbb{A}^n\) is one that is of the form \(V(S)\) for some \(S\). An affine variety is an algebraic subset of \(\mathbb{A}^n\) for some \(n\).
- Note: Our definition of “affine variety” matches [CLO15, Definition 1.2.1], which is more general than the definition in [Ful08, Chapter 2].
Definition: For a subset \(X \subseteq \mathbb{A}^n\), define \[ I(X) = \{ f \in k[x_1, \dotsc, x_n] \mid f(a_1, \dotsc, a_n) = 0 \text{ for all } (a_1, \dotsc, a_n) \in X \}. \]

Possible reading: [CLO15, Sections 1.2 and 1.4] and [Ful08, Sections 1.2–3].

Exercises

1. Explain why the empty set and all of \(\mathbb{A}^n\) are both algebraic subsets of \(\mathbb{A}^n\).
2. Prove that any singleton subset of \(\mathbb{A}^n\) is algebraic. In other words, prove that for any \((a_1, \dotsc, a_n) \in \mathbb{A}^n\), the subset \(S = \{ (a_1, \dotsc, a_n)\}\) of \(\mathbb{A}^n\) containing just one element is algebraic.
What are the algebraic subsets of \(\mathbb{A}^1\)?
Determine whether or not each of the following subsets of \(\mathbb{A}^2(\mathbb{R})\) is algebraic.
1. The \(y\)-axis.
2. The graph of the function \(y = x^2\).
3. The graph of the function \(y = \cos(x)\).
4. The set of points of the form \((\cos \theta, \sin \theta)\) for some \(\theta\).
5. The set of points whose polar coordinates \((r, \theta)\) satisfy the equation \(r = \sin \theta\). Hint: Describe this set in cartesian coordinates.
6. The set of points of the form \((x, x)\) where \(x \neq 1\).
1. Prove that, if \(S \subseteq T\) are subsets of \(k[x_1, \dotsc, x_n]\), then \(V(S) \supseteq V(T)\).
2. Prove that, if \(X \subseteq Y\) are subsets of \(\mathbb{A}^n\), then \(I(X) \supseteq I(Y)\).
Let \(I = \langle S \rangle\) be the ideal generated by a subset \(S \subseteq k[x_1, \dotsc, x_n]\) and let \(\sqrt{I}\) be the radical of \(I\). Prove that \(V(S) = V(I) = V(\sqrt{I})\).
Let \(X\) be a subset of \(\mathbb{A}^n\). Prove that \(I(X)\) is an ideal in \(k[x_1, \dotsc, x_n]\), and that it is radical.
1. Prove that, if \(I\) is an ideal in \(k[x_1, \dotsc, x_n]\), then \(I \subseteq \sqrt{I} \subseteq I(V(I))\).
2. Let \(I = \langle x^2 \rangle \subseteq k[x]\). Show that \(I \subsetneq \sqrt{I} = I(V(I))\).
3. Let \(I = \langle 1+x^2 \rangle \subseteq \mathbb{R}[x]\). Show that \(I = \sqrt{I} \subsetneq I(V(I))\).
4. Let \(k\) be finite and \(I = \{0\} \subseteq k[x]\). Show that \(I = \sqrt{I} \subsetneq I(V(I))\).
1. Prove that, if \(I\) and \(J\) are ideals in \(k[x_1, \dotsc, x_n]\), then \(V(I + J) = V(I) \cap V(J)\).
2. Prove that finite intersections of algebraic subsets of \(\mathbb{A}^n\) are algebraic.
3. Are infinite intersections of algebraic subsets of \(\mathbb{A}^n\) again algebraic? If so, prove it. If not, give a counterexample.
1. Prove that, if \(I\) and \(J\) are ideals in \(k[x_1, \dotsc, x_n]\), then \(V(IJ) = V(I) \cup V(J)\).
2. Prove that finite unions of algebraic subsets of \(\mathbb{A}^n\) are algebraic.
3. Are infinite unions of algebraic subsets of \(\mathbb{A}^n\) again algebraic? If so, prove it. If not, give a counterexample.
Suppose \(V\) and \(W\) are algebraic subsets of \(\mathbb{A}^n\). Is the set-theoretic difference \(V \setminus W\) algebraic? If so, prove it. If not, give a counterexample.
Suppose \(V \subseteq \mathbb{A}^m\) and \(W \subseteq \mathbb{A}^n\) are affine varieties. Prove that the cartesian product \(V \times W\) is an affine variety in \(\mathbb{A}^{m+n}\).
Prove that, if \(k\) is finite, then every subset of \(\mathbb{A}^n(k)\) is algebraic.
Suppose \(V\) and \(W\) are algebraic subsets of \(\mathbb{A}^n\). Prove that \(V \subseteq W\) if and only if \(I(V) \supseteq I(W)\).
1. State the definition of an antitone Galois connection.
2. Prove that the \(V(-)\) and \(I(-)\) operators define an antitone Galois connection between subsets of \(k[x_1, \dotsc, x_n]\) and subsets of \(\mathbb{A}^n\).

Polynomial Maps

Core Ideas

Definition: If \(V\) is an algebraic subset of \(\mathbb{A}^n\), the coordinate ring of \(V\) is \(k[V] = k[x_1, \dotsc, x_n]/I(V)\).
- Note: Other symbols that may be used for the coordinate ring include \(\Gamma(V)\), \(\Gamma(V, \mathscr{O})\), \(\Gamma(V, \mathscr{O}_V)\), \(\mathscr{O}(V)\), or \(\mathscr{O}_V(V)\). All of these notations allude to sheaf theory, which is important in modern formulations of algebraic geometry but which you do not need to know about for this class.
Definition: If \(V\) is an algebraic subset of \(\mathbb{A}^n\), a polynomial function on \(V\) is a function \(\phi : V \to k\) such that there exists a polynomial \(f \in k[x_1, \dotsc, x_n]\) for which \(\phi(a_1, \dotsc, a_n) = f(a_1, \cdots, a_n)\) for all \((a_1, \dotsc, a_n) \in V\).
The fact that the set of polynomial functions on an affine variety \(V\) is a ring, and that is naturally isomorphic to \(k[V]\).
The fact that, if \(k\) is infinite, then two polynomials \(f, g \in k[x_1, \dotsc, x_n]\) define the same polynomial function on \(\mathbb{A}^n\) if and only if \(f = g\).
Definition of a polynomial map \(\phi : V \to W\) between two affine varieties \(V\) and \(W\).
The fact that the set of polynomial maps \(\phi : V \to W\) is in bijection with the set of \(k\)-algebra homomorphisms \(\phi^\sharp : k[W] \to k[V]\).
- Note: It’s important to be able to go back and forth through this construction. In other words, you should be able to write down what the polynomial map \(V \to W\) corresponding to a given homomorphism \(k[W] \to k[V]\), and write down what the homomorphism \(k[W] \to k[V]\) corresponding to a given polynomial map \(V \to W\) is.
Definition of an isomorphism of affine varieties.
The fact that a polynomial map \(\phi : V \to W\) is an isomorphism if and only if the corresponding homomorphism \(\phi^\sharp : k[W] \to k[V]\) is an isomorphism.

Possible reading: [CLO15, Theorem 1.1.5 and Corollary 1.1.6], [CLO15, Section 5.1 through Definition 1.1.3, Section 5.2, and Definition 5.4.1], and [Ful08, Sections 2.1–2].

Exercises

Prove that, if \(k\) is infinite, then \(k[\mathbb{A}^n] = k[x_1, \dotsc, x_n]\).
Fix an integer \(i = 1, \dotsc, n\) and let \(\pi_i : \mathbb{A}^n \to \mathbb{A}^1\) be the function given by \(\pi_i(a_1, \dotsc, a_n) = a_i\).
1. Prove that \(\pi_i\) is a polynomial map.
2. Suppose \(k\) is infinite and consider the \(k\)-algebra homomorphism \[\pi_i^\sharp : k[y] = k[\mathbb{A}^1] \to k[\mathbb{A}^n] = k[x_1, \dotsc, x_n]\] that corresponds to \(\pi_i\). What is \(\pi_i^\sharp(y)\)?
Let \(k\) be an infinite field. Consider the map \(f : \mathbb{A}^1 \to \mathbb{A}^1\) given by \(t \mapsto t^2\). Prove that \(f\) is a polynomial map, and describe the corresponding homomorphism \(f^\sharp : k[x] = k[\mathbb{A}^1] \to k[\mathbb{A}^1] = k[y]\) by saying what \(f^\sharp(x)\) is.
For each of the following functions \(\phi : \mathbb{A}^1(\mathbb{R}) \to \mathbb{A}^2(\mathbb{R})\), determine whether or not (i) \(\phi\) is a polynomial map, and (ii) the image of \(\phi\) is an algebraic subset of \(\mathbb{A}^2(\mathbb{R})\).
1. \(\phi(t) = (t, t^2)\).
2. \(\phi(t) = (t^2, t^2)\).
3. \(\phi(t) = (\cos t, \sin t)\).
For each of the functions in the previous exercise that is in fact a polynomial map, describe the corresponding \(\mathbb{R}\)-algebra homomorphism \[ \phi^\sharp : \mathbb{R}[x, y] = \mathbb{R}[\mathbb{A}^2] \to \mathbb{R}[\mathbb{A}^1] = \mathbb{R}[t] \] by stating what the values of \(\phi^\sharp(x)\) and \(\phi^\sharp(y)\) are.
Let \(V = V(y - x^2) \subseteq \mathbb{A}^2\) and let \(\phi : \mathbb{A}^1 \to V\) be the polynomial map \(\phi(t) = (t, t^2)\). Prove that \(\phi\) is an isomorphism of affine varieties.
Let \(V = V(y^2 - x^3) \subseteq \mathbb{A}^2\) and let \(\phi : \mathbb{A}^1 \to V\) be the polynomial map \(\phi(t) = (t^2, t^3)\).
1. Prove that \(\phi\) is not an isomorphism of affine varieties.
2. Prove that \(\phi\) is a bijective function.
Let \(V\) be an affine variety. An algebraic subset of \(V\) is a subset of \(V\) that is simultaneously an algebraic subset of \(\mathbb{A}^n\), where \(V\) is an algebraic subset of \(\mathbb{A}^n\).
1. Explain how an ideal \(J \subseteq k[V]\) corresponds to an algebraic subset of \(V\).
2. Suppose \(U\) is also an affine variety and \(\phi : U \to V\) is a polynomial map. If \(W \subseteq V\) is an algebraic subset of \(V\), prove that \(\phi^{-1}(W)\) is an algebraic subset of \(U\).
Suppose \(V \subseteq \mathbb{A}^n\) is an algebraic subset. Then \(V\) is said to be irreducible if it cannot be written as a union \(V = V_1 \cup V_2\) where \(V_1, V_2\) are both proper algebraic subsets of \(V\).
1. Prove that \(V(xz, yz)\) is reducible (ie, not irreducible). What does \(V(xz, yz)\) look like when \(k = \mathbb{R}\)?
2. Let \(k = \mathbb{R}\). Prove that \(V(x^2 + y^2 - z^2)\) is irreducible. What does \(V(x^2 + y^2 - z^2)\) look like?
3. Prove that \(V\) is irreducible if and only if \(I(V) \subseteq k[x_1, \dotsc, x_n]\) is a prime ideal.
4. Prove that every affine variety can be written as a finite union of irreducible algebraic subsets.
Suppose \(U, V, W\) are varieties and \(\phi : U \to V\) and \(\psi : V \to W\) are both polynomial maps. Prove that \(\psi \circ \phi : U \to W\) is also a polynomial map.

Monomial Orders

Core Ideas

Definition of a monomial order.
Definition of the multidegree, the leading coefficient, the leading monomial, and the leading term of a polynomial with respect to a monomial order.
Examples of monomial orders: lexicographic, graded lexicographic, graded reverse lexicographic.

Possible reading: [CLO15, Section 2.2]. I also strongly suggest looking at [CLO15, Corollary 2.4.6] and getting comfortable with using that statement (even if you don’t yet look through the proof, though one direction of the proof will already be accessible). Feel free to use that statement in the exercises.

Exercises

Rewrite each of the following polynomials in \(\mathbb{Q}[x, y, z]\) so that the terms are in decreasing order. Use (i) lexicographic order, (ii) graded lexicographic order, and (iii) graded reverse lexicographic order.
1. \(2x + 3y - z + x^2 + 2z^2 + x^3\).
2. \(2x^2y^8 + 3x^5yz^4 + xyz^3 - xy^4\).
Is it true for any monomial order that there are only a finite number of monomials between any fixed pair of monomials?
Prove that \(\cdots > x^2 > x > 1\) is the unique monomial order on \(k[x]\).
Suppose \(>\) is a monomial order. Define \(>^{\mathrm{gr}}\) by declaring that \(x^a \mathbin{>^{\mathrm{gr}}} x^b\) if and only if either \(|a| > |b|\), or \(|a| = |b|\) and \(x^a > x^b\).
1. Prove that \(>^{\mathrm{gr}}\) is a monomial order.
2. Prove that \((>_{\mathrm{lex}})^{\mathrm{gr}}\) is equal to \(>_{\mathrm{grlex}}\).
3. Is there a monomial order \(>\) such that \(>^{\mathrm{gr}}\) is equal to \(>_{\mathrm{grevlex}}\)? If so, say what \(>\) is and prove that \(>^{\mathrm{gr}}\) is equal to \(>_{\mathrm{grevlex}}\). If not, prove that no such monomial order exists. Hint: This is a little bit of a “trick” question.
4. Suppose that \(>\) only satisfies conditions (i) and (ii) in the definition of a monomial order. Prove that \(>^{\mathrm{gr}}\) is still a monomial order.
Suppose \(>\) is a monomial order and \(\sigma \in S_n\) is a permutation of \(\{1, \dotsc, n\}\). Define \(>^\sigma\) by declaring that \(x_1^{a_1} \cdots x_n^{a_n} \mathbin{>^\sigma} x_1^{b_1} \cdots x_n^{b_n}\) if and only if \(x_1^{a_{\sigma(1)}} \cdots x_n^{a_{\sigma(n)}} > x_1^{b_{\sigma(1)}} \cdots x_n^{b_{\sigma(n)}}\).
1. Prove that \(>^\sigma\) is a monomial order.
2. Colexicographic order (sometimes called inverse lexicographical order) is the monomial order \(>_{\mathrm{colex}}\) where \(x^{a_1} \cdots x^{a_n} \mathbin{>_{\mathrm{colex}}} x^{b_1} \cdots x^{b_n}\) if and only if the rightmost nonzero entry of \((a_1 - b_1, \dotsc, a_n - b_n)\) is strictly positive. Prove that \(>_{\mathrm{colex}}\) is equal to \((>_{\mathrm{lex}})^\sigma\) for some \(\sigma \in S_n\).
3. State the definition of a left (resp. right) group action of a group on a set. Then prove that the \(>^\sigma\) construction defines an action of \(S_n\) on the set of monomial orders. Is it a left action or a right action?
4. State the definition of faithful and free group actions. Is this group action faithful? Is it free?
5. State the definition of a transitive group action. Is this group action transitive?
Do [CLO15, Exercise 2.4.9].

Multivariable Division

Core Ideas

Algorithm for dividing a polynomial \(f \in k[x_1, \dotsc, x_n]\) by a list of polynomials \(F = (f_1, \dotsc, f_s)\) with respect to a given monomial order to obtain a list of quotients \((q_1, \dotsc, q_s)\) and a remainder \(r\) such that \(f = q_1 f_1 + \cdots + q_s f_s + r\).

Possible reading: [CLO15, Section 2.3]. Note that [CLO15, Definition 2.6.3] introduces the notation \(\bar{f}^F\) for the remainder when \(f\) is divided by \(F = (f_1, \dotsc, f_s)\), and that this can be useful notation.

Exercises

Divide the given polynomial \(f\) by the given list \(F\) using (i) lexicographic order, (ii) graded lexicographic order, and (iii) graded reverse lexicographic order.
1. \(f = x^7 y^2 + x^3 y^2 - y + 1\) and \(F = (xy^2 - x, x - y^3)\).
2. \(f = x^7 y^2 + x^3 y^2 - y + 1\) and \(F = (x - y^3, xy^2 - x)\).
3. \(f = xy^2 z^2 + xy - yz\) and \(F = (x - y^2, y - z^3, z^2 - 1)\).
Suppose \[ \begin{aligned} f_1 &= 2xy^2 - x, \\ f_2 &= 3x^2 y - y - 1, \end{aligned} \] and let \(F = (f_1, f_2)\). Find an element \(f \in \langle f_1, f_2 \rangle\) such that \(\bar{f}^F \neq 0\). Make sure to specify what monomial order you are using in order for \(\bar{f}^F\).
Let \(k\) be an infinite field and let \(I = \langle y - x^2, z - x^3 \rangle \subseteq k[x, y, z]\).
1. Prove that \(I = I(V(I))\). Hint: Suppose \(f \in I(V(I))\). On the one hand, consider \(f(t, t^2, t^3)\) for any \(t\). On the other hand, consider dividing \(f\) by the generators of \(I\) with respect to colexicographic order.
2. Prove that \(I\) is a prime ideal. Note: Thus, \(V(I)\) is an irreducible affine variety. It is called the (affine) twisted cubic.
Do [CLO15, Exercise 2.3.11].
For any list \(F = (f_1, \dotsc, f_s)\) of polynomials in \(k[x_1, \dotsc, x_n]\), prove that the remainder function \(f \mapsto \bar{f}^F\) is a \(k\)-linear function on \(k[x_1, \dotsc, x_n]\).
Let \(k\) be an infinite field. Fix integers \(m, n \geq 2\), let \(\phi : \mathbb{A}^1 \to \mathbb{A}^3\) be the polynomial map given by \(\phi(t) = (t, t^m, t^n)\), and let \(V\) be its image.
1. Prove that \(V\) is an algebraic subset of \(\mathbb{A}^3\).
2. What is \(I(V)\)? Find a finite set of generators.
3. Prove that \(V\) is an irreducible affine variety.
Suppose \(F = (f_1, \dotsc, f_s)\) is a list of polynomials.
1. Is it necessarily true that \(\bar{f}_1^F = 0\)?
2. Is it necessarily true that \(\bar{f}_i^F = 0\) for all \(i = 1, \dotsc, s\)?

Monomial Ideals

Core Ideas

Definition of a monomial ideal.
Basic facts about how to determine if a monomial or a polynomial is contained in a given monomial ideal.
Dickson’s lemma: Every monomial ideal is generated by a finite set of monomials.
Definition of the minimal basis of a monomial ideal, and the fact that it exists and is unique.
The fact that the affine variety corresponding to a monomial ideal is a finite union of coordinate subspaces.
Definition of dimension for the affine variety corresponding to a monomial ideal.

Possible reading: [CLO15, Section 2.4] and [CLO15, Section 9.1].

Exercises

Let \(I = \langle x^6, x^2y^3, xy^7 \rangle\). Plot the set of exponent vectors \((a, b)\) of monomials \(x^a y^b\) appearing in elements of \(I\).
Do [CLO15, Exercise 2.4.4].
Find the dimension of \(V(I)\) for each of the following monomial ideals \(I\).
1. \(I = \langle xy, yz, xz \rangle \subseteq k[x, y, z]\).
2. \(I = \langle wx^2z, w^3y, wxyz, x^5y^5 \rangle \subseteq k[w,x,y,z]\).
Prove that, if \(I \subseteq k[x_1, \dotsc, x_n]\) is a monomial ideal, then so is its radical \(\sqrt{I}\).
Use the Hilbert basis theorem to provide an alternative proof of Dickson’s lemma.
- Note: The proof of the Hilbert basis theorem in [CLO15, Section 2.5] makes use of Dickson’s lemma. However, you’ve already read through a proof of the Hilbert basis theorem in [Ful08, Section 1.4] that didn’t use Dickson’s lemma, so there is no logical circularity here!
Let \(I\) be a monomial ideal in \(k[x_1, \dotsc, x_n]\). Prove that the following statements are equivalent:
1. \(V(I) = \{(0, \dotsc, 0)\}\).
2. \(\dim V(I) = 0\).
3. For each \(i = 1, \dotsc, n\), there exists an integer \(\ell_i \geq 1\) such that \(x_i^{\ell_i} \in I\).
In Ravi Vakil’s game of Chomp, a cookie is placed at every integer point \((m, n)\) in the first quadrant of the plane (ie, with \(m, n \geq 0\)). Two players take turns choosing a point \((m, n)\) that still has a cookie on it, and eating that cookie and all of the cookies above or to the right of that cookie (ie, every cookie \((m', n')\) with \(m' \geq m\) and \(n' \geq n\)). The \((0,0)\) cookie is poisoned, and the player who eats it loses. Use the Hilbert basis theorem and/or Dickson’s lemma to prove that any game of Chomp ends after finitely many moves.

Gröbner Bases

Core Ideas

Definition of the ideal of leading terms \(\langle \LT(I) \rangle\) of an ideal \(I \subseteq k[x_1, \dotsc, x_n]\), and the fact that it is a monomial ideal.
Definition of a Gröbner basis of an ideal \(I \subseteq k[x_1, \dotsc, x_n]\), and the fact it generates \(I\). Also, the fact that Gröbner bases exist.
The fact that the remainder upon division by a Gröbner basis of an ideal is independent of the choice of Gröbner basis.
Definition of a reduced Gröbner basis of an ideal, and the fact that it exists and is unique.
Definition of the S-polynomial of two nonzero polynomials.
Buchberger’s criterion: Suppose \(I \subseteq k[x_1, \dotsc, x_n]\) is the ideal generated by \(G = \{f_1, \dotsc, f_r\}\). Then \(G\) is a Gröbner basis for \(I\) if and only if there is no remainder when \(S(f_i, f_j)\) is divided by \(G\) for all \(i \neq j\).

Possible reading: [CLO15, Sections 2.5–6, Definition 2.7.4, and Theorem 2.7.5]. My suggestion is that you black-box Buchberger’s criterion at first pass (ie, familiarize yourself with using the statement before returning to the proof).

Exercises

Let \(f_1 = xy^2 - xz + y, f_2 = xy - z^2\), and \(f_3 = x - yz^4\) in \(k[x, y, z]\). Using graded lexicographic order, find an example of a polynomial \(f \in \langle f_1, f_2, f_3 \rangle\) such that \(\LT(f) \notin \langle \LT(f_1), \LT(f_2), \LT(f_3) \rangle\).
Fix a monomial order and an ideal \(I \subseteq k[x_1, \dotsc, x_n]\). Suppose \(G\) and \(G'\) are both Gröbner bases for \(I\). Prove that \(\bar{f}^G = \bar{f}^{G'}\) for all \(f \in k[x_1, \dotsc, x_n]\).
Fix a monomial order and an ideal \(I \subseteq k[x_1, \dotsc, x_n]\). Prove that, for any \(f \in k[x_1, \dotsc, x_n]\), there exists a unique pair of polynomials \(g, r \in k[x_1, \dotsc, x_n]\) such that \(f = g + r\), \(g \in I\), and no term of \(r\) is divisible by any element of \(\LT(I)\).
The set \(G = \{y - x^2, z - x^3\}\) is a Gröbner basis for the ideal \(I = \langle G \rangle\) for some monomial order. Which monomial order? Note: \(V(I)\) is the (affine) twisted cubic.
Let \(I = \langle wz - xy, wy - x^2, xz - y^2 \rangle \subseteq k[w, x, y, z]\). Note: The affine variety \(V(I)\) is the “cone of the projective twisted cubic.”
1. Let \(\phi : k[w, x, y, z] \to k[s, t]\) be the homomorphism given by \(w \mapsto s^3, x \mapsto s^2 t, y \mapsto st^2, z \mapsto t^3\). Prove that \(I = \ker \phi\).
2. Prove that \(I\) is prime.
Let \(I \subseteq k[x_1, \dotsc, x_n]\) be a monomial ideal. Show that the minimal basis of \(I\) is the reduced Gröbner basis for \(I\) with respect to any monomial order.
Let \(I \subseteq k[x_1, \dotsc, x_n]\) be an ideal and suppose that \(I\) has a Gröbner basis (with respect to some monomial ordering) \(G = \{g_1, \dotsc, g_r\}\) such that \(\LT(g_i)\) is square-free (ie, the power on every variable is at most 1). Prove that \(I\) is radical. Possible hint: First prove that, if \(f \in \sqrt{I}\), then \(\LT(f)\) is divisible by \(\LT(g_i)\) for some \(i\). Then use that to prove that \(G\) is a Gröbner basis for \(\sqrt{I}\).
Suppose \(I \subseteq k[x_1, \dotsc, x_n]\) is an ideal.
1. Prove that, if \(\langle \LT(I) \rangle\) is radical, then so is \(I\).
2. Is the converse true? If so, prove it, If not, provide a counterexample.
Fix an ideal \(I \subseteq k[x_1, \dotsc, x_n]\).
1. Fix a monomial order. Prove that the monomials that are not in \(\langle \LT(I) \rangle\) form a basis for the quotient ring \(k[x_1, \dotsc, x_n]/I\) as a vector space over \(k\).
2. Prove that, if \(k[x_1, \dotsc, x_n]/I\) is finite dimensional as a vector space over \(k\), then \(V(I)\) is a finite set. Hint: Fix \(i = 1, \dotsc, n\) and prove that there are only finitely many possibilities for the \(i\)th coordinate of a point in \(V(I)\) by considering \(1, x_i, x_i^2, x_i^3, \dotsc\) in the quotient ring.
Suppose \(I\) is an ideal in the single variable polynomial ring \(k[x]\). Prove that the reduced Gröbner basis for \(I\) with respect to the unique monomial order on \(k[x]\) is the singleton set consisting of the unique monic generator of \(I\).
Fix a monomial order. Suppose \(I\) is a principal ideal in \(k[x_1, \dotsc, x_n]\) and that \(G\) is a finite subset of \(k[x_1, \dotsc, x_n]\).
1. Is it true that \(\langle \LT(I) \rangle\) is also a principal ideal? If so, prove it. If not, give a counterexample.
2. Formulate and prove a statement beginning with “\(G\) is a Gröbner basis for \(I\) if and only if…”
3. Formulate and prove a statement beginning with “\(G\) is the reduced Gröbner basis for \(I\) if and only if …”
For a \(r \times n\) matrix \[A = \begin{bmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{r,1} & a_{r,2} & \cdots & a_{r,n} \end{bmatrix}\] with entries in \(k\), let \(S_A \subseteq k[x_1, \dotsc, x_n]\) be the finite set of \(r\) linear polynomials \[ \begin{aligned} f_1 &= a_{1,1} x_1 + \cdots + a_{1,n} x_n \\ &\;\;\vdots \\ f_r &= a_{r,1} x_1 + \cdots + a_{r,n} x_n. \\ \end{aligned} \] Note that \(f_i\) is the \(i\)th entry in the vector that results from multiplying \(A\) on the left by the column vector of variables \((x_1, \dotsc, x_n)\). Let \(I_A\) be the ideal generated by \(S_A\).
1. Prove that, if \(A\) is row equivalent to \(B\), then \(I_A = I_B\).
2. Prove that, if \(A\) is in row echelon form, then \(S_A\) is a Gröbner basis for \(I_A\) with respect to lexicographic order.
3. Prove that, if \(A\) is in reduced row echelon form, then \(S_A\) is the reduced Gröbner basis for \(I_A\) with respect to lexicographic order.

Buchberger’s Algorithm

Core Ideas

Buchberger’s algorithm for computing the reduced Gröbner basis for an ideal.

Possible reading: [CLO15, Section 2.7].

Exercises

Let \(I = \langle x^2 - y, y + x^2 - 4 \rangle\) in \(\mathbb{R}[x, y]\).
1. Find the reduced Gröbner basis for \(I\) with respect to lexicographic order.
2. Enumerate the points of \(V(I)\).
Let \(I = \langle x^2 + y^2 + z^2 - 1, x^2 + z^2 - y, x - z \rangle\) in \(\mathbb{C}[x, y, z]\).
1. Find the reduced Gröbner basis for \(I\) with respect to lexicographic order.
2. Enumerate the points of \(V(I)\).
Enumerate the points of \(V(x^2 + y^2 + z^2 - 1, x^2 + y^2 + z^2 - 2x, 2x - 3y - z)\).
Let \(I = \langle y - x^2 , xz - y^2 \rangle\) in \(\mathbb{R}[x, y, z]\).
1. Find the reduced Gröbner basis for \(I\) with respect to colexicographic order.
2. Sketch a picture of \(V(I)\).
Let \(I \subseteq k[x_1, \dotsc, x_n]\) be an ideal. Let \(G_{\mathrm{grlex}}\) and \(G_{\mathrm{grevlex}}\) be the reduced Gröbner bases of \(I\) with respect to graded lexicographic order and graded reverse lexicographic order, respectively. Must \(G_{\mathrm{grlex}}\) and \(G_{\mathrm{grevlex}}\) have the same number of elements?
Suppose \(a, b, c\) are real numbers satisfying the following equations. \[ \begin{aligned} a + b + c &= 3 \\ a^2 + b^2 + c^3 &= 5 \\ a^3 + b^3 + c^3 &= 7 \end{aligned} \]
1. What is \(a^4 + b^4 + c^4\)?
2. What is \(a^5 + b^5 + c^5\)?
3. Can you find a formula for \(a^k + b^k + c^k\) for any positive integer \(k\)?
Explain in your own words why it makes sense to say that Buchberger’s algorithm for finding reduced Gröbner bases simultaneously generalizes both (i) the euclidean algorithm for finding greatest common divisors of single variable polynomials, and (ii) the row reduction algorithm for solving homogeneous linear systems of equations. Hint: You might find it helpful to read through the exercises above, even if you haven’t finished doing all of them yet.

Affine Nullstellensatz

Core Ideas

Nullstellensatz: If \(k\) is algebraically closed, then \(I(V(I)) = \sqrt{I}\) for any ideal \(I \subseteq k[x_1, \dotsc, x_n]\).
The fact that, if \(k\) is algebraically closed, then the operations \(V(-)\) and \(I(-)\) are inverse bijections between the set of radical ideals in \(k[x_1, \dotsc, x_n]\) and the set of algebraic subsets of \(\mathbb{A}^n(k)\).

Possible reading: Either [CLO15, Sections 3.1–2 through Theorem 3.2.7] or [Ful08, Sections 1.7–10], you can choose! My suggestion is that you black-box the nullstellensatz at first pass (ie, familiarize yourself with using the statement before returning to the proof).

Exercises

Note: For all of the following exercises, assume that \(k\) is algebraically closed.

The “strong” nullstellensatz says that \(I(V(I)) = \sqrt{I}\) for any ideal \(I \subseteq k[x_1, \dotsc, x_n]\). The “weak” nullstellensatz says that \(V(I) = \emptyset\) implies \(I = k[x_1, \dotsc, x_n]\). The reading shows that, with some work, the “weak” nullstellensatz implies the “strong” one. Prove the other direction. In other words, use the statement of the “strong” nullstellensatz to prove the “weak” one.
Suppose \(I, J \subseteq k[x_1, \dotsc, x_n]\) are ideals. Prove that \(I + J = k[x_1, \dotsc, x_n]\) if and only if \(V(I) \cap V(J) = \emptyset\).
Let \(V = V(x^2 + 1, x^2 - 2xy + y^2)\). Determine whether or not each of the following polynomials is contained in the ideal \(I(V) \subseteq k[x, y]\).
1. \(y^2 + 1\)
2. \(x - y\)
3. \(y^2 - 1\)
Suppose \(f \in k[x_1, \dotsc, x_n]\) is a irreducible polynomial.
1. Prove that \(I(V(f)) = \langle f \rangle\).
2. Prove that \(V(f)\) is irreducible affine variety.
3. Construct an explicit example of an irreducible \(f \in \mathbb{R}[x_1, \dotsc, x_n]\) for some \(n \geq 1\) such that \(I(V(f)) \neq \langle f \rangle\). Can you construct an example where \(V(f)\) is nonempty?
1. Prove that irreducible algebraic subsets of \(\mathbb{A}^n\) are in bijection with prime ideals in \(k[x_1, \dotsc, x_n]\).
2. Construct an explicit example of a prime ideal \(I \subeteq \mathbb{R}[x_1, \dotsc, x_n]\) for some \(n \geq 1\) such that \(V(I)\) is nonempty and reducible.
1. Prove that every maximal ideal in \(k[x_1, \dotsc, x_n]\) is of the form \(\langle x_1 - a_1, \dotsc, x_n - a_n \rangle\) for some \(a_1, \dotsc, a_n \in k\).
2. Give an explicit example of a maximal ideal in \(\mathbb{R}[x_1, \dotsc, x_n]\) for some \(n \geq 1\) that is not of the form \(\langle x_1 - a_1, \dotsc, x_n - a_n \rangle\) for some \(a_1, \dotsc, a_n \in \mathbb{R}\).
Prove that algebraic subsets of any affine variety \(V\) are in bijection with radical ideals in \(k[V]\).
Let \(I \subseteq k[x_1, \dotsc, x_n]\) is an ideal. Prove that \(V(I)\) is a finite set if and only if the quotient ring \(k[x_1, \dotsc, x_n]/I\) is finite dimensional as a vector space over \(k\). Hint: If \(V(I)\) is finite, fix a monomial order and an index \(i = 1, \dotsc, n\) and then prove that some sufficiently large power of \(x_i\) is contained in \(\langle \LT(I) \rangle\) by considering the polynomial \[f_i = \prod_{(a_1, \dotsc, a_n) \in V(I)} (x_i - a_i).\] What does that tell you about the set of monomials that are not contained in \(\langle \LT(I) \rangle\)?

Affine Hilbert Function

Core Ideas

Definition of the Hilbert function \(\mathrm{HF}_I\) of an ideal \(I \subseteq k[x_1, \dotsc, x_n]\).
The fact that \(\mathrm{HF}_I(s)\) is equal to the number of monomials of degree at most \(s\) that are not in \(I\) whenever \(I\) is a monomial ideal.
The fact that the complement of the set of monomials in a monomial ideal can be written as a finite disjoint union of translates of coordinate subspaces.
- Note: The book proves this statement without the word “disjoint,” but the stronger statement above is true.
The fact that the set of monomials of degree at most \(n\) in \(T = \alpha + [e_{i_1}, \dotsc, e_{i_m}] \subseteq \mathbb{N}^n\) for some \(\alpha \in \mathbb{N}^n\) is \(\binom{m+s-|\alpha|}{s-|\alpha|}\).
The fact that any ideal \(I\) has the same Hilbert function as its ideal of leading terms \(\langle \mathrm{LT}(I) \rangle\) with respect to any graded monomial order.
The fact that the Hilbert function of an ideal \(I \subseteq k[x_1, \dotsc, x_n]\) is eventually a polynomial, called the Hilbert polynomial \(\mathrm{HP}_I\) of \(I\), whose leading coefficient is of the form \(g/d!\) for some integer \(g \geq 1\), where \(d\) is degree of the Hilbert polynomial. Then \(d\) is called the dimension of \(I\) and \(g\) is called the degree of \(I\).
Definition of the index of regularity of an ideal.
The fact that the dimension of \(I\) is equal to that of \(\sqrt{I}\).
Definition: The Hilbert series \(\mathrm{HS}_I\) of an ideal \(I \subseteq k[x_1, \dotsc, x_n]\) is the formal series defined by \[\mathrm{HS}_I(z) = \sum_{s = 0}^\infty \mathrm{HF}_I(s)z^s.\]

Possible reading: [CLO15, Sections 9.2 through Proposition 9.2.7] and [CLO15, Section 9.3 through Theorem 9.3.8].

Exercises

Let \(I = \langle xy \rangle \subseteq k[x, y]\). For any integer \(s \geq 0\), give explicit bases for the finite dimensional \(k\)-vector spaces \(I_{\leq s} = I \cap k[x, y]_{\leq s}\) and \(k[x, y]_{\leq s} / I_{\leq s}\).
Suppose \(I \subseteq k[x_1, \dotsc, x_n]\) is an ideal whose Hilbert polynomial \(\mathrm{HP}_I\) has degree \(d \geq 0\). Prove that the leading coefficient of \(\mathrm{HP}_I(s)\) is \(g/d!\) for some positive integer \(g\). This integer is called the degree of \(I\). Hint: Explain why it suffices to consider the case when \(I\) is a monomial ideal, and then consider [CLO15, Proposition 9.2.7].
- Note: The word “degree” is overloaded: the degree of \(I\) is distinct from the degree of \(\mathrm{HP}_I\)! Make sure you’re using the correct definition of degree in the following exercises.
For each of the following ideals \(I \subseteq k[x, y]\), (i) compute \(\mathrm{HF}_I(s)\) for all integers \(s \geq 0\), (ii) compute \(\mathrm{HP}_I\), (iii) compute the index of regularity of \(I\), and (iv) compute the degree of \(I\). Note: See exercise 2 for the relevant definition of degree.
1. \(I = \langle xy - 1 \rangle\) (ie, the ideal of the hyperbola).
2. \(I = \langle x^2 + y^2 - 1 \rangle\) (ie, the ideal of the circle).
3. \(I = \langle y - x^4 \rangle\) (ie, the ideal of the graph of a quartic).
1. Let \(f \in k[x, y]\) be a polynomial of total degree \(d \geq 0\). Prove that the degree of the ideal \(\langle f \rangle \subseteq k[x, y]\) is equal to \(d\). Note: See exercise 2 for the relevant definition of degree.
2. Can you generalize the result of the previous part for \(f \in k[x_1, \dotsc, x_n]\)?
Suppose \(I, J \subseteq k[x_1, \dotsc, x_n]\) are ideals such that \(V(I) = V(J)\). Is it true that \(\mathrm{HF}_I = \mathrm{HF}_J\)? Is it true that \(\mathrm{HP}_I = \mathrm{HP}_J\)?
Let \(I = \langle y - x^2, z - x^3 \rangle \subseteq k[x, y, z]\) be the ideal of the affine twisted cubic.
1. Find \(\mathrm{HF}_I(s)\) for all integers \(s \geq 0\). Hint: The given set of generators for \(I\) is a Gröbner basis with respect to a lexicographic order, but that is not a graded order! You might find it convenient to start by computing a Gröbner basis with respect to graded reverse lexicographic order.
2. Find the Hilbert polynomial, dimension, degree, and index of regularity of \(I\).
This problem is about the zero ideal \(\{0\} \subseteq k[x_1, \dotsc, x_n]\).
1. Prove that the Hilbert function is given by \(\displaystyle \mathrm{HF}_{\{0\}}(s) = \binom{n+s}{s}\).
2. Explain in your own words why the index of regularity of \(\{0\}\) is 0.
3. Prove that the Hilbert series has the property that \(\displaystyle \mathrm{HS}_{\{0\}}(z) = \frac{1}{(1-z)^{n+1}}\) whenever \(|z| < 1\).
This problem is about the ideal \(I = \langle x^\alpha \rangle \subseteq k[x_1, \dotsc, x_n]\) generated by a single monomial \(x^\alpha\) of total degree \(d\).
1. Prove that the Hilbert function is given by \[\mathrm{HF}_I(s) = \binom{n+s}{n} - \binom{n+s-d}{s-d},\] where the second binomial coefficient is defined to be 0 when \(s < d\).
2. What is the index of regularity of \(I\)?
3. Prove that \(\displaystyle \mathrm{HS}_I(z) = \frac{1-z^d}{(1-z)^{n+1}}\) whenever \(|z| < 1\).
Let \(I = \langle x_1 - a_1, \dotsc, x_n - a_n \rangle\). Prove that \(\mathrm{HF}_I(s) = 1\) for all integers \(s \geq 0\).
Let \(V = \{a_1, a_2, a_3\} \subseteq \mathbb{A}^n(k)\) be the affine variety consisting of three distinct points in affine space and let \(I = I(V)\).
1. Prove that \(\mathrm{HP}_I(s) = 3\).
2. What is the index of regularity of \(I\)? Hint: The answer depends on whether or not the three points \(a_1, a_2, a_3\) are colinear in \(\mathbb{A}^n(k)\).
Fix a graded monomial order \(>\) and let \(I\) be an ideal in \(k[x_1, \dotsc, x_n]\). Recall that any \(f \in k[x_1, \dotsc, x_n]\) can be written uniquely in the form \(f = g + r\) where \(g \in I\) and no term of \(r\) is in \(\langle \mathrm{LT}(I) \rangle\). Is it true that \(f\) having total degree at most \(s\) implies that \(r\) has total degree at most \(s\)? What about the converse?
For ideals \(I, J \subseteq k[x_1, \dotsc, x_n]\), prove that \(\mathrm{HF}_{I \cap J} + \mathrm{HF}_{I + J} = \mathrm{HF}_I + \mathrm{HF}_J\). Possible hint: It may be helpful to use [CLO15, Proposition 9.3.1].

Projective Varieties

Core Ideas

Note: Assume throughout that \(k\) is infinite.

Definition of “\(f(P) = 0\)” for \(f \in k[x_0, \dotsc, x_n]\) and \(P = (a_0 : \cdots : a_n) \in \mathbb{P}^n\). Namely, we say that \(f(P) = 0\) if and only if \(f(a_0, \dotsc, a_n) = 0\) for every coordinate representation \((a_0 : \cdots : a_n)\) of \(P\) [Ful08, Section 4.2, page 45].
Definition of a homogeneous ideal in \(k[x_0, \dotsc, x_n]\), and the fact that homogeneous ideals are precisely the ideals that are generated by sets of homogeneous polynomials.
Definition: For a subset \(S \subseteq k[x_0, \dotsc, x_n]\), let \[V(S) = \{ P \in \mathbb{P}^n \mid f(P) = 0 \text{ for all } f \in S \}. \] An algebraic subset of \(\mathbb{P}^n\) is one that is of the form \(V(S)\) for some \(S\). A projective variety is an algebraic subset of \(\mathbb{P}^n\) for some \(n\).
- Note 1: Typically, one only considers sets \(S\) of homogeneous polynomials, but the definition of \(f(P) = 0\) given in [Ful08, Section 4.2, page 45] lets us make sense of this for any subset. If you’re curious, exercises below ask you to explore what this means in this “greater generality.”
- Note 2: If ambiguity can arise between the projective and affine versions of this construction, we write \(V_p(S)\) and \(V_a(S)\), respectively, to disambiguate.
- Note 3: When \(S\) is either a set of homogeneous polynomials or a homogeneous ideal, the affine variety \(V_a(S)\) is called the cone of the projective variety \(V_p(S)\).
Definition: For a subset \(X \subseteq \mathbb{P}^n\), let \[I(X) = \{ f \in k[x_0, \dotsc, x_n] \mid f(P) = 0 \text{ for all } P \in X \}. \]
- Note: If ambiguity can arise between the projective and affine versions of this construction, we write \(I_p(X)\) and \(I_a(X)\), respectively, to disambiguate.

Possible reading: [Ful08, Section 4.2, first page] and [CLO15, Proposition 8.2.4, Definition 8.2.5, and Section 8.3 through Theorem 8.3.5].

Exercises

Note: For all of the following exercises, assume that \(k\) is infinite, that \(V = V_p\), and that \(I = I_p\).

1. Explain why the empty set and all of \(\mathbb{P}^n\) are both algebraic subsets of \(\mathbb{P}^n\).
2. Prove that any singleton subset of \(\mathbb{P}^n\) is algebraic. In other words, prove that for any \((a_0 : \cdots : a_n) \in \mathbb{P}^n\), the subset \(S = \{ (a_0 : \cdots : a_n)\}\) of \(\mathbb{P}^n\) containing just one element is algebraic.
What are the algebraic subsets of \(\mathbb{P}^1(k)\)?
Using homogeneous coordinates \((x : y : z)\) for \(\mathbb{P}^2(\mathbb{C})\), list all of the points of each of the following intersections.
1. \(V(y^2z - x(x-2z)(x+z)) \cap V(y^2 + x^2 - 3xz)\).
2. \(V(y^2z - x(x-2z)(x+z)) \cap V(y^2 + x^2 - 2xz)\).
1. Show that the formula \[ \phi(a : b) = (a^2 : ab : b^2) \] gives a well-defined function \(\phi : \mathbb{P}^1 \to \mathbb{P}^2\). Note: What you’re being asked to show is that the output point in \(\mathbb{P}^2\) does not depend on the choice of homogeneous coordinates for the input point in \(\mathbb{P}^1\).
2. Prove that \(\phi\) is an injective function and that its image is equal to \(V(xz - y^2) = \{ (x : y : z) \mid xz = y^2 \} \subseteq \mathbb{P}^2\).
3. If \(k = \mathbb{R}\), draw a picture of the image of \(\phi\).
4. Explain why the formula \(\psi(x : y : z) = (y : z)\) does not define an inverse function \(\psi : V(xz - y^2) \to \mathbb{P}^1\) for \(\phi\).
1. Show that the formula \[ \sigma((x_1 : x_2), (y_1 : y_2)) = (x_1y_1 : x_1y_2 : x_2y_1 : y_1y_2) \] gives a well-defined function \(\sigma : \mathbb{P}^1 \times \mathbb{P}^1 \to \mathbb{P}^3\).
2. Prove that \(\sigma\) is injective.
3. Prove that the image of \(\sigma\) is equal to \(V(ad - bc) = \{ (a : b : c : d) \mid ad = bc \} \subseteq \mathbb{P}^3\). Note: The image of \(\sigma\) is a Segre variety. It is also a determinantal variety, for reasons that may be apparent from the appearance of the determinant formula “\(ad - bc\)” above!
1. Show that \[ \phi(a : b) = (a^3 : a^2b : ab^2 : b^3) \] is a well-defined function \(\phi : \mathbb{P}^1 \to \mathbb{P}^3\).
2. Prove that \(\phi\) is injective.
3. Let \(C\) be the image of \(\phi\). Using homogeneous coordinates \((w : x : y : z)\) for \(\mathbb{P}^3\), prove that \(C\) is equal to \(V(S)\) for \(S = \{wy - x^2, wz - xy, xz - y^2\}\). Note: \(C\) is the (projective) twisted cubic.
4. Prove that, for any two-element subset \(S' \subsetneq S\), the algebraic subset \(V(S') \subseteq \mathbb{P}^3\) is the union of \(C\) with a line. Note: In other words, you are showing that all three polynomials in \(S\) are in fact needed to cut out \(C\).
1. Suppose \(f \in k[x_0, \dotsc, x_n]\) and \(f = f_0 + \cdots + f_d\) where each \(f_i\) is homogeneous of degree \(i\). Prove that \(V(f) = V(f_0, \dotsc, f_d)\).
2. Prove that, for any subset \(S \subseteq k[x_0, \dotsc, x_n]\), there exists a subset \(S'\) of homogeneous polynomials such that \(V(S) = V(S')\).
If an ideal \(I \subseteq k[x_0, \dotsc, x_n]\) is homogeneous, must \(\sqrt{I}\) also be homogeneous?
Let \(I = \langle S \rangle\) be the ideal generated by a set \(S \subseteq k[x_0, \dotsc, x_n]\) of homogeneous polynomials, and let \(\sqrt{I}\) be the radical of \(I\). Prove that \(V(S) = V(I) = V(\sqrt{I})\).
Let \(X\) be a subset of \(\mathbb{P}^n\). Prove that \(I(X)\) is a homogeneous ideal in \(k[x_0, \dotsc, x_n]\), and that it is radical.
Suppose \(I\) is a homogeneous ideal. Prove that the reduced Gröbner basis of \(I\) with respect to any monomial order consists of homogeneous polynomials. Hint: Carefully analyze what happens when one works through Buchberger’s algorithm starting from a homogeneous generating list.
Let \(I\) be a homogeneous ideal in \(k[x_0, \dotsc, x_n]\). Let \(V = V_p(I) \subseteq \mathbb{P}^n\) and \(C = V_a(I) \subseteq \mathbb{A}^{n+1}\).
1. Prove that, if \((a_0 : \cdots : a_n) \in V\), then \((a_0, \dotsc, a_n) \in C\).
2. Prove that, if \((a_0, \dotsc, a_n) \in C\) and \((a_0, \dotsc, a_n) \neq (0, \dotsc, 0)\), then \((a_0 : \dotsc : a_n) \in V\).
3. Prove that if \(V\) is nonempty, then \((0, \dotsc, 0) \in C\).
4. Show by example that, if \(V\) is empty, then \((0, \dotsc, 0)\) may or may not be in \(C\).

Homogenization and Dehomogenization

Core Ideas

Definition of dehomogenization of a homogeneous polynomial \(f \in k[x_0, \dotsc, x_n]\) with respect to \(x_0\). More generally, the dehomogenization with respect to \(x_j\) for any \(j = 0, 1, \dotsc, n\).
The fact that, if \(f_1, \dotsc, f_r \in k[x_0, \dotsc, x_n]\) are homogeneous and if we identify \(U_0\) with \(\mathbb{A}^n\), then \(V_p(f_1, \dotsc, f_r) \cap U_0 = V_a(g_1, \dotsc, g_r)\) where \(g_i \in k[x_1, \dotsc, x_n]\) is the dehomogenization of \(f_i\) with respect to \(x_0\). More generally, the same fact for \(U_j\) and the dehomogenization with respect to \(x_j\) for any \(j = 0, \dotsc, n\).
Definition of homogenization \(f^h \in k[x_0, \dotsc, x_n]\) of a polynomial \(f \in k[x_1, \dotsc, x_n]\).
Definition of homogenization \(I^h \subseteq k[x_0, \dotsc, x_n]\) of an ideal \(I \subseteq k[x_1, \dotsc, x_n]\).
The fact that the homogenization of a Gröbner basis for an ideal \(I \subseteq k[x_1, \dotsc, x_n]\) with respect to a graded monomial order yields a set of generators for \(I^h\).
Definition of the projective closure of an algebraic subset \(V \subseteq \mathbb{A}^n\) as \(V_p(I_a(V)^h)\), and the fact that it is the smallest algebraic subset of \(\mathbb{P}^n\) containing \(V\) when \(\mathbb{A}^n\) is identified with \(U_0 \subseteq \mathbb{P}^n\).
The fact that, if \(k\) is algebraically closed, then the projective closure of \(V_a(I)\) is \(V_p(I^h)\).
The fact that there is nothing special about the index 0 in any of the above bullet points, i.e., that everything above can be done with any index \(j = 0, 1, \dotsc, n\).

Possible reading: [CLO15, Section 8.2 starting from Proposition 8.2.6, and Section 8.4] or [Ful08, Sections 2.6 and 4.3].

Exercises

Let \(I = \langle y - x^2, z - x^3 \rangle \subseteq k[x, y, z]\). In all of the parts below, homogenization refers to homogenizing with respect to \(w\) to obtain polynomials in \(k[w, x, y, z]\).
1. Prove that \(\langle wz - x^2, w^2z - x^3 \rangle \subsetneq I^h\) by exhibiting an explicit polynomial \(f \in I\) such that \(f^h \notin \langle wz - x^2, w^2z - x^3 \rangle\).
2. Prove that \(G = \{y - x^2, z - x^3\}\) is not a Gröbner basis for \(I\) with respect to any graded monomial order. Possible hint: Use Buchberger’s criterion.
3. Compute the reduced Gröbner basis \(G_{\mathrm{grevlex}}\) for \(I\) with respect to graded reverse lexicographic order. You should find that \(G_{\mathrm{grevlex}} = \{f_1, f_2, f_3\}\) for three distinct polynomials \(f_1, f_2, f_3 \in k[x, y, z]\).
4. Compute the reduced Gröbner basis \(G_{\mathrm{grlex}}\) for \(I\) with respect to graded lexicographic order. You should find that \(G_{\mathrm{grlex}} = \{f_1, f_2, f_3, f_4\}\), where \(f_1, f_2, f_3\) are the same polynomials you found in the previous part.
5. With \(f_1, f_2, f_3, f_4\) as above, find polynomials \(q_1, q_2, q_3 \in k[x, y, z]\) such that \(f_4 = q_1 f_1 + q_2 f_2 + q_3f_3\).
6. For the polynomial \(f\) you named in part (a), find polynomials \(q_1, q_2, q_3 \in k[w, x, y, z]\) such that \(f^h = q_1 f_1^h + q_2 f_2^h + q_3 f_3^h\).
Let \(V = V_p(wy - x^2, wz - xy, xz - y^2) \subseteq \mathbb{P}^3\). Let \(U = \{(w : x : y : z) \mid w \neq 0\}\) and identify \(U\) with \(\mathbb{A}^3\) with coordinates \((x, y, z)\).
1. Prove that \(U \cap V = V_a(y - x^2, z - x^3)\).
2. Prove that \(V\) is the projective closure of \(V_a(y - x^2, z - x^3)\).
1. Construct an explicit example of an ideal \(I \subseteq \R[x_1, \dotsc, x_n]\) such that \(V_a(I) = \emptyset\) but \(V_p(I^h) \neq \emptyset\).
2. Explain why no such ideal exists if we replace \(\mathbb{R}\) with \(\mathbb{C}\).
1. Is it true that \((fg)^h = f^h g^h\) in \(k[x_0, \dotsc, x_n]\) for any pair of polynomials \(f, g \in k[x_1, \dotsc, x_n]\)?
2. Suppose \(I \subseteq k[x_0, \dotsc, x_n]\) is a homogeneous ideal. Prove that \(I\) is prime ideal if and only \(fg \in I\) for homogeneous \(f, g \in k[x_0, \dotsc, x_n]\) implies that \(f \in I\) or \(g \in I\).
3. Suppose \(I \subseteq k[x_1, \dotsc, x_n]\) is an ideal. Prove that \(I\) is prime if and only if \(I^h \subseteq k[x_0, \dotsc, x_n]\) is prime.
Let \(k\) be algebraically closed. Prove that the formula \[ \phi(a : b) = (a^n : a^{n-1}b : a^{n-2}b^2 : \cdots : ab^{n-1} : b^n) \] gives a well-defined injective function \(\phi : \mathbb{P}^1 \to \mathbb{P}^n\) whose image is \(V(S)\), where \(S \subseteq k[x_0, \dotsc, x_n]\) is the set of all of the \(2 \times 2\) subdeterminants of the \(2 \times n\) matrix \[ \begin{pmatrix} x_0 & x_1 & x_2 & \cdots & x_{n-1} \\ x_1 & x_2 & x_3 & \cdots & x_n \end{pmatrix} \] obtained by choosing any pair of columns of this matrix.
1. Show that the formula \[\nu(x : y : z) = (x^2 : xy : y^2 : xz : yz : z^2)\] gives a well-defined function \(\nu : \mathbb{P}^2 \to \mathbb{P}^5\).
2. Is \(\nu\) injective?
3. Let \(k\) be algebraically closed. Prove that the the image of \(\nu\) is an algebraic subset of \(\mathbb{P}^5\). Note: This may be somewhat challenging. You first have to find a set \(S\) of homogeneous polynomials such that the image of \(\nu\) is \(V_p(S)\), and then prove that the image is in fact \(V_p(S)\). The image of \(\nu\) is a Veronese surface.
For this problem, fix using graded reverse lexicographic order on both \(k[x_0, \dotsc, x_{n-1}]\) and \(k[x_0, \dotsc, x_{n-1}, x_n]\). Also, homogenization of polynomials in \(k[x_0, \dotsc, x_{n-1}]\) referes to homogenization with respect to \(x_n\). Let \(I \subseteq k[x_0, \dotsc, x_{n-1}]\) be an ideal, so that \(I^h \subseteq k[x_0, \dotsc, x_{n-1}, x_n]\).
1. Prove that, if \(G\) is a Gröbner basis for \(I\), then \(G^h\) is a Gröbner basis for \(I^h\).
2. If \(G\) is the reduced Gröbner basis for \(I\), must \(G^h\) be the reduced Gröbner basis for \(I^h\)?

Projective Nullstellensatz

Core Ideas

Weak projective nullstellensatz: Suppose \(k\) is algebraically closed. If \(I\) is a homogeneous ideal, then \(V(I) = \emptyset\) if and only if \(\sqrt{I} \supseteq \langle x_0, \dotsc, x_n \rangle\).
Strong projective nullstellensatz: Suppose \(k\) is algebraically closed. Then \(I(V(I)) = \sqrt{I}\) for any homogeneous ideal \(I \subseteq k[x_0, \dotsc, x_n]\) such that \(V(I) \subseteq \mathbb{P}^n\) is nonempty.
The fact that, if \(k\) is algebraically closed, then \(V(-)\) and \(I(-)\) are inclusion-reversing bijections between algebraic subsets of \(\mathbb{P}^n\) and radical homogeneous ideals contained in \(\langle x_0, \dotsc, x_n \rangle \subseteq k[x_0, \dotsc, x_n]\).

Possible reading: [CLO15, Section 8.3 starting from Theorem 8.3.9] and/or [Ful08, Section 4.2].

Exercises

Note: For all of the following exercises, assume that \(k\) is algebraically closed, and that \(V = V_p\) and \(I = I_p\). There are not many exercises here. If none of the above appeal to you for your reading assignment, you are welcome to choose an exercise from one of the previous sections that you haven’t already done yet!

Give an example of a homogeneous ideal \(I \subseteq k[x, y]\) such that \(I(V(I)) \neq \sqrt{I}\).
Prove that the only two radical homogeneous ideals \(I \subseteq k[x_0, \dotsc, x_n]\) such that \(V(I) = \emptyset\) are \(I = \langle x_0, \dotsc, x_n \rangle\) and \(I = k[x_0, \dotsc, x_n]\).
A projective variety \(V\) is irreducible if it cannot be written as a union \(V = V_1 \cup V_2\) where \(V_1, V_2\) are proper algebraic subsets of \(V\).
1. Prove that, if \(I\) is a prime homogeneous ideal, then \(V(I)\) is irreducible.
2. Prove that, if \(I\) is a radical homogeneous ideal and \(V(I)\) is irreducible, then \(I\) is prime.
3. Give an example of a non-radical (hence non-prime) homogeneous ideal such that \(V(I)\) is irreducible. Note that your example should work over an algebraically closed field!
4. Explain why prime homogeneous ideals in \(k[x_0, \dotsc, x_n]\) are in bijection with irreducible algebraic subsets of \(\mathbb{P}^n(k)\).
Prove the statement of the weak projective nullstellensatz given above (namely, that if \(I\) is a homogeneous ideal, then \(V(I) = \emptyset\) if and only if \(\sqrt{I} \supseteq \langle x_0, \dotsc, x_n \rangle\)). Note: Don’t use the statement of [CLO15, Theorem 8.3.8]. There’s a more direct argument for this statement than the one given in the proof of that theorem. You’re welcome to inspect the proof given there for ideas, but it’s not strictly necessary, and in fact, you may actually get more ideas by studying the proof of [CLO15, Theorem 8.3.9].

Projective Hilbert Function

Core Ideas

Definition of the Hilbert function of a homogeneous ideal.
- Note: If ambiguity can arise between the projective and affine versions of this construction, we write \(\mathrm{HF}^p_I\) and \(\mathrm{HF}^a_I\), respectively, to disambiguate.
The fact that any homogeneous ideal \(I\) has the same Hilbert function as its ideal of leading terms \(\langle \mathrm{LT}(I) \rangle\) for any monomial order.
The fact that the Hilbert function of a homogeneous ideal \(I \subseteq k[x_1, \dotsc, x_n]\) is eventually a polynomial, called the Hilbert polynomial \(\mathrm{HP}_I\).
- Note: If ambiguity can arise between the projective and affine versions of this construction, we write \(\mathrm{HP}^p_I\) and \(\mathrm{HP}^a_I\), respectively, to disambiguate.
Definition of dimension, index of regularity, and degree of a homogeneous ideal.
The fact that, if \(I \subseteq k[x_0, \dotsc, x_n]\) is a homogeneous ideal, then \[ \mathrm{HF}^p_I(s) = \mathrm{HF}^a_I(s) - \mathrm{HF}^a(s-1) \] for all integers \(s \geq 1\).
The fact that if \(I \subseteq k[x_1, \dotsc, x_n]\) is an ideal and \(I^h \subseteq k[x_0, x_1, \dotsc, x_n]\) is its homogenization with respect to \(x_0\), then \(\mathrm{HF}^p_{I^h}(s) = \mathrm{HF}^a_I(s)\).

Possible reading: [CLO15, Section 9.3 starting from page 492].

Exercises

Note: For all of the following, use the projective Hilbert function (not the affine one). There are not many exercises here. If none of the above appeal to you for your reading assignment, you are welcome to choose an exercise from one of the previous sections that you haven’t already done yet!

Compute the (projective!) Hilbert polynomial, dimension, and degree of the following homogeneous ideals.
1. \(\langle x^2 - y^2, x^3 - x^2y + y^3 \rangle\)
2. \(\langle y^2 - xz, x^2y - z^2w, x^3-yzw \rangle\)
3. \(\langle wy - x^2, xz - y^2, wz - xy \rangle\)
4. \(\langle ad - bc \rangle\)
Suppose \(I \subseteq J\) are homogeneous ideals in \(k[x_0, \dotsc, x_n]\). Prove that the dimension of \(I\) is at least the dimension of \(J\).
1. Suppose \(k\) is algebraically closed and \(f \in k[x_0, \dotsc, x_n]\) is nonconstant, homogeneous, and irreducible. What is the dimension of the homogeneous ideal \(\langle f \rangle\)?
2. Is your answer from part (a) still true without assuming that \(k\) is algebraically closed?
3. Is your answer from part (a) still true without assuming that \(f\) is irreducible?
Let \(I\) be a homogeneous ideal in \(k[x_0, \dotsc, x_n]\) and let \(V = V_p(I)\).
1. Prove that \(V\) is a finite set if and only if the dimension of \(I\) is 0.
2. Suppose \(V\) is finite. Prove that the cardinality of \(V\) is at most the degree of \(I\).
3. Give examples to show that the cardinality of \(V\) is sometimes equal to the degree of \(I\), and sometimes it’s strictly less.