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 Wednesday, 24 April: Let \( L \) be the blowup of \( \mathbf A^{n+1} \) at the origin. Describe canonical maps \( L \to \mathbf P^n \) and \( L \to \mathbf A^{n+1} \) and combine these to give a closed embedding \( L \to \mathbf P^n \times \mathbf A^{n+1} \) (Hint: this is almost the definition of the blowup!). Let \( f : \operatorname{Spec} k \to \mathbf P^n \) be a \( k \)-point of \( \mathbf P^n \) with homogeneous coordinates \( (\xi_0, \ldots, \xi_n) \). Show that the map \( f^{-1}(L) \to f^{-1}(\mathbf P^n \times \mathbf A^{n +1}) = \mathbf A^{n+1}_k \) is the inclusion of the line \( \{ (\lambda \xi_0, \ldots, \lambda \xi_n ) \: \big| \: \lambda \in k \} \) in \( k^{n+1} \). Check that the formula \( (\xi_0, \ldots, \xi_n, \sigma) \mapsto (\xi_0, \ldots, \xi_n, x_i(\sigma)^{-1} \sigma) \) gives an isomorphism \( L \big|_{D^+(x_i)} \simeq D^+(x_i) \times \mathbf A^1 \) for all \( i \).

Recall that \( S(d)^\sim \) is the sheaf on \( \mathbf P^n \) whose sections over \( D^+(f) \) are the homogeneous elements of degree \( d \) in \( \mathbf Z[x_0, \ldots, x_n, f^{-1}] \). Show that if \( g\in S(d)^\sim(D^+(f)) \) then \( g \) induces a map \( \varphi_g : L \big|_{D^+(f)} \to D^+(f) \times \mathbf A^1 \). Verify that \( \varphi_g \) is a degree \( d \) function, in the sense that \( \varphi_g \big|_{D^+(f) \cap D^+(x_i)} : L \big|_{D^+(f) \cap D^+(x_i)} \simeq \mathbf A^1 \times (D^+(f) \cap D^+(x_i)) \to \mathbf A^1 \times (D^+(f) \cap D^+(x_i)) \) is a homogeneous linear function of degree \( d \) for all \( i \).

Check that the map \( g \mapsto \varphi_g \) induces an isomorphism between \( S(d)^\sim \) and the sheaf \( \mathcal O(d) \) of homogeneous linear functions of degree \( d \) on \( L \).

Compute the Čech cohomology of \( \mathcal O(d) \) on \( \mathbf P^n \) using the cover \( \{ D^+(x_i) \} \). Suggestion: if it's not obvious what to do in general, do it for small values of \( n \) to start to get the idea. We will complete this calculation in class on Wednesday.

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 \( \mathbf Z \).

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.

For Friday, 12 April: Read Gathmann, §8.1–2.
For Wednesday, 10 April: Read Gathmann, §8.1.

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.

For Monday, 8 April: Read Gathmann, §7.2.

Do Gathmann, §6.7, #6.7.1—3.

For Friday, 5 April: Read Gathmann, §6.1 and Atiyah–MacDonald, Chapter 11.

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.

For Wednesday, 3 April: Read Gathmann, §6.1 and Atiyah–MacDonald, Chapter 11.

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 \( D' = k[t] / (t^3) \).

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

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

For Monday, 18 March: Read Gathmann, §4.4. and Mumford–Oda, §V.3

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 \( \pi^{-1} Z = \operatorname{Proj} \sum_{n=0}^\infty I^n / I^{n+1} \).

For Wednesday, 13 March: Read Gathmann, §4.4.

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

For Monday, 1 March: Read Gathmann, §4.3.

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 \( A = \operatorname{Spec} k[x,y] / (y^2 - x^3) \).

For Friday, 29 February: Read Atiyah–MacDonald, pp. 62–64. Read Gathmann, §4.3.

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.

For Wednesday, 27 February: Read Gathmann, §4.2–4.3.

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.

For Friday, 22 February: Read Gathmann, §5.4 and Mumford–Oda, §I.4.

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 \( \dim Z = \dim f(Z) \).

For Wednesday, 20 February: Read Gathmann, §4.1 and Hartshorne, §II.4, Lemma 4.5.

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 \( i \).

For Monday, 18 February: Read Gathmann, §3.4.

Do Gathmann, §3.5, #3.5.4–5.

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.

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 \( U \cap V \).

For Monday, 4 February: Read Gathmann, §3.1.

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 \( s = t^{-1} \).

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.

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.

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.

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.

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.

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, 26 April: cohomology of invertible sheaves on projective space, extending sections by twisting up
Wednesday, 24 April: cohomology of invertible sheaves on projective space
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

Math 6170 (Spring 2019) Algebraic geometry