# Math 6170 (Spring 2019)

Algebraic geometry

## Office hours

See my calendar.

## Textbooks

I will probably assign reading from all of the following books. For the most part, I plan to follow Gathmann.

Algebraic Geometry by Robin Hartshorne

The Red Book of Varieties and Schemes by David Mumford

Algebraic Geometry II (a penultimate draft) by David Mumford and Tadao Oda

Algebraic Geometry by Andreas Gathmann

The Rising Sea: Foundations of Algebraic Geometry by Ravi Vakil

Commutative Algebra by Michael Atiyah and Ian MacDonald

## Grading

Any students who submits 4 or more problem sets over the course of the semester will receive an A. Any student who submits between 2 and 4 problem sets will receive an A-. Students who submit fewer problem sets but are otherwise active participants in the class will receive passing grades.

## Homework

- For Monday, 22 April: Read Gathmann, §§8.3 and 8.4.
- For Friday, 19 April: In class, we proved that the definition of \( H^2(X, F) \) is independent of the choice of flasque resolution and constructed a long exact sequence up to \( H^2 \). Complete the argument to demonstrate that \( H^p(X,F) \) is independent of the choice of flasque resolution for all \( p \geq 2 \) and construct the long exact sequence in all degrees.
- For Monday, 15 April: Find all \( \mathbf Z \)-torsors on \( S^1 \). Hint: prove that \( \mathbf Z \)-torsors are equivalent to covering spaces with deck transformation group \( \mathbf Z \).
- For Friday, 12 April: Read Gathmann, §8.1–2.
- For Wednesday, 10 April: Read Gathmann, §8.1.
- For Monday, 8 April: Read Gathmann, §7.2.
- For Friday, 5 April: Read Gathmann, §6.1 and Atiyah–MacDonald, Chapter 11.
- For Wednesday, 3 April: Read Gathmann, §6.1 and Atiyah–MacDonald, Chapter 11.
- For Monday, 1 April: Do Arbarello, Cornalba, Griffiths, and Harris, p. 32, Exercises A-1 and A-2.
- For Wednesday, 20 March: Read Gathmann, §4.4. and Mumford–Oda, §V.3
- For Monday, 18 March: Read Gathmann, §4.4. and Mumford–Oda, §V.3
- For Wednesday, 13 March: Read Gathmann, §4.4.
- For Monday, 1 March: Read Gathmann, §4.3.
- For Friday, 29 February: Read Atiyah–MacDonald, pp. 62–64. Read Gathmann, §4.3.
- For Wednesday, 27 February: Read Gathmann, §4.2–4.3.
- For Friday, 22 February: Read Gathmann, §5.4 and Mumford–Oda, §I.4.
- For Wednesday, 20 February: Read Gathmann, §4.1 and Hartshorne, §II.4, Lemma 4.5.
- For Monday, 18 February: Read Gathmann, §3.4.
- For Wednesday, 13 February: Read Hartshorne, §II.2, pp. 76–77. Read Vakil, §4.5. We will do Exercise 4.5.E in class.
Do Hartshorne, §I.2, #2.1.

Do Gathmann, §3.5, #3.5.1–3

- For Wednesday, 6 February: Read Vakil, §4.5.
- For Monday, 4 February: Read Gathmann, §3.1.
- Show that \( X = D(x) \cup D(f(x)) \).
- Define \( D(x) \to U_0 \) sending \( s \mapsto \frac{y}{x} \) and \( D(y) \to U_1 \) sending \( t \mapsto \frac{y}{f(x)} \). Show that these glue to a map \( X \to \mathbf P^1 \).
- Let \( k = \mathbf R \) and make your own choice for \( f(x) \). Draw a picture of the graph of \( X(\mathbf R) \to \mathbf P^1(\mathbf R) \).
- For Wednesday, 30 January: Read Gathmann, §2.5.
- For Wednesday, 30 January: Review Gathmann, §2.4, particularly Examples 2.4.4 and 2.4.5 and Lemma 2.4.7, and Vakil, §4.4.
Do Vakil, §4.3, #4.3.A (hint: use the exercise below), 4.3.D, 4.3.F, 4.3.G.

If \( X \) is a scheme, let \( B = \mathcal O_X(X) \). Construct a canonical morphism \( X \to \operatorname{Spec} B \) and show that any morphism from \( X \) to an affine scheme \( \operatorname{Spec} A \) factors uniquely as \( X \to \operatorname{Spec} B \to \operatorname{Spec} A \).

- For Monday, 28 January: Suppose that \( \varphi : F \to G \) is a morphism of sheaves on \( X \). Prove that \( \varphi_p : F_p \to G_p \) is injective for all \( p \in X \) if and only if \( \varphi_U : F(U) \to G(U) \) is injective for all open subsets \( U \subset X \). Prove the same statement with "injective" replaced by "bijective". What happens if you replace "injective" with "surjective"?
Do the exercises in Friday's lecture notes.

- For Friday, 25 January: Read Gathmann, §2.3–2.4 and Mumford–Oda, §I.3.
- For Wednesday, 23 January: Read Gathmann, §2.2–2.3 and Mumford, §I.4.
- For Friday, 18 January: Read Gathmann, §2.1–2.2 and Mumford–Oda, §I.2. Do Gathmann, §1.4, #1.4.4, 1.4.6. Prove that \( \operatorname{Spec} A \) is irreducible if and only if \( A \) has a unique minimal prime ideal.
- For Wednesday, 16 January: Read Gathmann, §1.3. Do §1.4, #1.4.1—1.4.3, 1.4.5. Read Gathmann, §2.1 and/or Mumford, §I.3.
- For Monday, 14 January: Make sure you can access the textbooks. Review the commutative algebra in Gathmann, §§1.1–1.2 and/or Mumford, §§I.1–I.2.

Prove that any morphism of \( G \)-torsors is an isomorphism.

Suppose that \( \varphi : F \to G \) is a homomorphism of sheaves of groups on \( X \). Let \( P \) be an \( F \)-torsor. Show that \( \varphi_\ast P \) can be constructed by either of the following formulas: \( \varphi_\ast P = \underline{\operatorname{Hom}}_{F^{\mathrm op}}(P^{\rm op}, G) = (G \mathop\times P) / F \). In the first formula, \( P^{\rm op} \) is the sheaf of right \( F \)-sets \( P \) with \( x . f = f^{-1} . x \) and \( \underline{\operatorname{Hom}}_{F^{\mathrm op}} \) indicates the sheaf of homomorphisms of right \( F \)-sets. In the second formula, \( F \) acts by \( f . (g, x) = (gf^{-1},f.x) \). Part of this problem is to specify what the \( G \)-actions are on these things and what the \( F \)-equivariant map \( P \to \varphi_\ast P \) is.

Prove that a finite product of quasicoherent sheaves is quasicoherent but that \( \prod_{i = 1}^\infty \mathcal O_X \) is not quasicoherent for most affine schemes \( X \).

Suppose that \( F \) is quasicoherent on an affine scheme \( X \). Show that the restriction of \( F \) to an open subset \( U \) is quasicoherent on \( U \).

Let \( X \) be a scheme over an affine scheme \( \operatorname{Spec} A \). Show that there is a universal \( A \)-derivation \( \mathcal O_X \to \Omega_{X/A} \) and that if \( U = \operatorname{Spec} B \) is an affine open subset of \( X \) then \( \Omega_{X/A}(U) = \Omega_{B/A} \). Prove that \( \Omega_{X/A} \) is quasicoherent.

Do Gathmann, §6.7, #6.7.1—3.

Prove that a surjective endomorphism of a noetherian ring is an isomorphism.

Suppose that \( A \to B \to C \) are ring homomorphisms, where \( A \to C \) is smooth and \( B \to C \) is surjective with ideal \( I \). Show that the sequence of \( C \)-modules \( 0 \to I/I^2 \to C \otimes_B \Omega_{B/A} \to \Omega_{C/A} \to 0 \) is split exact. (We proved this in class in the case where \( B \) is a polynomial ring over \( A \). Hint: use the square zero extension \( B/I^2 \to C \) as we did in class on Wednesday.)

Suppose that \( A \to B \) and \( A \to C \) are ring homomorphisms and \( D = B \otimes_A C \). Prove that \( \Omega_{D/C} \simeq D \otimes_B \Omega_{B/A} \simeq C \otimes_A \Omega_{B/A} \). I suggest using universal properties.

Show using the infinitesimal criterion for smoothness that \( C = k[x,y] / (xy) \) is not smooth over \( k \). Hint: use \( D = k[t] / (t^2) \) and \( D' = k[t] / (t^3) \).

Suppose that \( A \to B \to C \) is a sequence of ring homomorphisms. Prove that there is an exact sequence \( C \otimes_B \Omega_{B/A} \to \Omega_{C/A} \to \Omega_{C/B} \to 0 \).

Do Gathmann, §4.6, #4.6.4, 4.6.12. Let \( A \) be a ring and \( I \subset A \) an ideal, and let \( X = \operatorname{Spec} A \). Let \( \pi : Y \to X \) be the blowup of \( X \) at \( I \). Prove that \( \pi^{-1} Z = \operatorname{Proj} \sum_{n=0}^\infty I^n / I^{n+1} \).

Do Gathmann, §4.6, #4.6.7, 4.6.8, 4.6.12.

Do Gathmann, §4.6, #4.6.5.

Prove the following statement, used in lecture on Friday: Suppose that \( \varphi : A \to B \) is an integral injection of integral domains, with \( A \) integrally closed, that \( p \) is a prime ideal of \( A \), and that \( f \in \varphi(p) B \). Show that \( f \) satisfies a monic polynomial with coefficients in \( p \). (Hints: Write \( f = \sum b_i x_i \) with \( x_i \in p \) and \( b_i \in B \). Then let \( B' \subset B \) be the subring of \( B \) generated by the \( b_i \). Show that \( B' \) is a finite \( A \)-module and that \( f B' \subset p B' \). Write the matrix of \( f \) in terms of a finite set of generators of \( B' \) and show that \( f \) satisfies the characteristic polynomial of this matrix.)

Let \( k \) be a field and let \( A = \operatorname{Spec} k[x,y] / (y^2 - x^3 - x^2) \). Prove that the integral closure \( B \) of \( A \) in its field of fractions is isomorphic to \( k[t] \). Compute the fibers of the map \( \operatorname{Spec} B \to \operatorname{Spec} A \). Do the same for \( A = \operatorname{Spec} k[x,y] / (y^2 - x^3) \).

Prove that if \( f : X \to Y \) is affine and \( Y \) is affine then \( X \) is affine. (Hint: Let \( X' = \operatorname{Spec} \Gamma(X, \mathcal O_X) \). Prove that \( X \to X' \) is an isomorphism by finding an open cover \( Y = \bigcup U_i \) such that \( X \mathop\times_Y U_i \to X' \mathop\times_Y U_i \) is an isomorphism for all \( i \).

Suppose that \( f : X \to Y \) is an affine morphism of schemes. Prove that \( f' : X \mathop\times_Y Y' \to Y' \) is affine for all maps \( Y' \to Y \).

Suppose that \( U, V \) are open subschemes of \( X \). Prove that \( U \cap V = (U \times V) \mathop\times_{X \times X} \Delta_X \).

Conclude from the other exercises that the intersection of two affine subschemes of a separated scheme is affine.

Let \( f_1, \ldots, f_N \) be a complete list of the homogeneous polynomials of degree \( d \) in \( n + 1 \) variables, \( x_0, \ldots, x_n \). Show that the formula \( \mathbf x \mapsto (f_1(\mathbf x), \ldots, f_N(\mathbf x)) \) defines a morphism of projective spaces \( \mathbf P^n \to \mathbf P^N \). Show that this comes from a homomorphism of graded rings. Compute \( N \).

Suppose that \( k \) is a field and that \( A \) is a finite \( k \)-algebra. Prove that \( \operatorname{Spec} A \) is discrete. Now suppose that \( B \) is an integral \( k \)-algebra. Prove that \( \operatorname{Spec} B \) is totally disconnected. Finally, suppose that \( f : X \to Y \) is an integral morphism of schemes. Prove that the fibers of \( f \) are totally disconnected.

Let \( \alpha, \beta : Z \to X \) be morphisms in some category. Show that the equalizer of \( \alpha \) and \( \beta \) is the fiber product of \( (\alpha, \beta) : Z \to X \times X \) and the diagonal map \( \Delta_X = (\mathrm{id}_X,\mathrm{id}_X) : X \to X \times X \). Conclude that a scheme is separated if and only if the diagonal map \( \Delta_X : X \to X \times X \) is a closed embedding. More generally, show that \( f : X \to Y \) if \( \Delta_f = (\mathrm{id}_X, \mathrm{id}_X) : X \to X \mathop\times_Y X \) is a closed embedding.

Do Gathmann, §5.6, #5.6.2, 5.6.5, 5.6.14.

Suppose that \( f : \mathbb P^n_k - \{ q \} \to \mathbb P^{n-1}_k \) is projection away from a closed point \( q \) and that \( Z \subset \mathbb P^n_k \) is a closed subset not containing \( q \). Complete the proof (sketched in class) that specializations lift along \( f \big|_Z : Z \to f(Z) \). Show also that \( f(Z) \) is quasicompact and conclude that \( f(Z) \) is closed. Conclude that \( \dim Z = \dim f(Z) \).

Do Hartshorne, Ch. II, §4, #4.2.

Prove that the affine line with a doubled origin is not separated and show directly that it does not satisfy the valuative criterion for properness.

A morphism of schemes \( f : X \to Y \) is called *affine* if there is an open cover of \( Y \) by affine schemes \( V \) such that \( f^{-1} V \) is affine. Show that an affine morphism of schemes satisfies the valuative criterion for separatedness and show that an affine morphism is separated.

Prove that a separated morphism of schemes satisfies the valuative criterion for separatedness.

Complete the proof that projective space satisfies the valuative criterion for properness by showing that if \( R \) is a valuation ring with field of fractions \( K \) and \( \alpha, \beta : \operatorname{Spec} R \to \mathbf P^n_A \) are two maps whose restriction to \( \operatorname{Spec} K \) are the same then the images of \( \alpha \) and \( \beta \) are both contained in the same chart \( D^+(x_i) \) for some \( i \).

Do Gathmann, §3.5, #3.5.4–5.

Let \( X \) be a separated scheme and let \( U = \operatorname{Spec} A \) and \( V = \operatorname{Spec} B \) be affine open subschemes of \( X \). Prove that \( U \cap V \) is affine. (Hints: let \( Z = operatorname{Spec} (A \otimes B) \) and construct two maps \( \varphi_1, \varphi_2 : Z \to X \) induced from the inclusions \( U \subset X \) and \( V \subset X \). Show that the construction from class gives an ideal \( I \subset A \otimes B \) such that \( V(I) = |\operatorname{eq}(\varphi_1,\varphi_2)| \). Show that \( \operatorname{Spec}(A \otimes B / I) \) is isomorphic to \( U \cap V \).

Construct a scheme with two affine open subsets \( U_1 \) and \( U_2 \) such that \( U_1 \cap U_2 \) is not affine.

Use each of the automorphisms \( \phi \) of \( \operatorname{Spec} k[x,x^{-1}] \) you constructed in the homework for Wednesday to construct a scheme \( X_\phi \) by gluing two copies of \( \mathbf A^1 \) along \( \mathbf A^1 - \{ 0 \} = \operatorname{Spec} k[x,x^{-1}] \). Up to isomorphism, how many different schemes can you construct this way? (Hint: you will want to think about whether \( \phi \) lifts to an automorphism of \( \mathbf A^1 \).)

Compute \( \Gamma(\mathbf P_A^1, \mathcal O_{\mathbf P_A^1}) \) where \( A \) is a commutative ring. You can assume \( A \) is a field or even an algebraically closed field if that makes it easier.

Let \( k \) be a field and let \( f \in k[x] \) be a quadratic polynomial such that \( f(0) \neq 0 \). Let \( X = \operatorname{Spec} k[x,y] / (y^2 - x f(x)) \). Let \( U_0 = \operatorname{Spec} k[s] \) and let \( U_1 = \operatorname{Spec} k[t] \), glued together along \( \operatorname{Spec} k[s,s^{-1}] \simeq \operatorname{Spec} k[t,t^{-1}] \), with \( s = t^{-1} \).

Compute all maps from the scheme \( \operatorname{Spec} k[x,x^{-1}] \), where \( k \) is a field, to itself. Can you solve the same problem if \( k \) is an arbitrary integral domain? An arbitrary commutative ring? (I had to do the latter in a paper, recently.)

Do Gathmann, §2.6, #2.6.3, 2.6.8–12.

If \( A \) is a commutative ring and \( M \) is an \( A \)-module and \( U \subset X = \operatorname{Spec} A \) is open, let \( \tilde M(U) \) be the set of locally compatible tuples \( (x_p)_{p \in U} \in \prod_{p \in U} M_p \). Show that \( \tilde M \) is a sheaf on \( X \). (See Mumford–Oda, §I.2 for details.) Do Gathmann, §2.6, #2.6.8, 2.6.9.

Let \( S \) be a set with more than one element. Show that the constant presheaf \( F(U) = S \) is not a sheaf on any topological space. Let \( F' \) be the subpresheaf of \( F \) such that \( F'(\varnothing) \) is a 1-element set and \( F'(U) = F(U) \) for all nonempty \( U \). Show that \( F' \) is a sheaf if and only if \( X \) is irreducible.

## Lecture Notes

You can look ahead at my notes for the next few classes. Warning: I usually make changes to the notes a few hours before the lecture.

- Friday, 19 April: quasicoherent sheaves on affine schemes and Čech cohomology
- Wednesday, 17 April: higher sheaf cohomology
- Monday, 15 April: flasque resolutions
- Friday, 12 April: torsors and sheaf cohomology
- Wednesday, 10 April: abelian categories of sheaves
- Monday, 8 April: quasicoherent sheaves
- Friday, 5 April: dimension of smooth schemes
- Wednesday, 3 April: tangent space and Hilbert polynomial
- Monday, 1 April: infinitesimal criterion for smoothness (thanks to Justin Wilson for providing notes)
- Friday, 22 March: differential criterion for smoothness (thanks to Leo Herr for providing notes)
- Monday, 18 March: Kähler differentials (thanks to Jonathan Quartin for providing notes)
- Monday, 11 March: blowups
- Monday, 4 March: dimension of affine schemes and blowups
- Friday, 1 March: the going down theorem
- Wednesday, 27 February: separated and proper morphisms, valuative criteria
- Monday, 25 February: fiber products, dimension of projective space over a field
- Friday, 22 February: dimension of projective space (over an algebraically closed field), fiber products
- Wednesday, 20 February: proper morphisms are closed, dimension of projective space
- Monday, 18 February: properness of projective space
- Friday, 15 February: (some) maps between projective spaces
- Wednesday, 13 February: end of the construction of Proj
- Monday, 11 February: topology and structure sheaf of projective space
- Friday, 8 February: projective space (the Proj construction)
- Wednesday, 6 February: the multiplicative group and its actions
- Monday, 4 February: morphisms to affine schemes and separated schemes
- Friday, 1 February: gluing morphisms of schemes
- Wedneday, 30 January: schemes are locally ringed spaces, the projective line
- Monday, 28 Janaury: ringed spaces, locally ringed spaces, schemes
- Friday, 25 January: morphisms, stalks, and pushforward of sheaves
- Wednesday, 23 January: sheaves and the structure sheaf of an affine scheme
- Friday, 18 January: functions on affine schemes
- Wednesday, 16 January: irreducibility and dimension
- Monday, 14 January: review of commutative algebra