Schemes

Professor A.J. de Jong, Columbia university, Department of Mathematics.

This semester I am teaching the course on schemes. If you are interested, please email me, tell me a little about yourself (academically), and I will add you to my email list.

The first lecture will be Jan 18 unfortunately on zoom. If I haven't email you about this yet, then email me, see above.

The lectures will be Tuesday and Thursday 10:10am-11:25am Location: 507 Mathematics Building

TA: Noah Olander

Office hours: 1 - 2:30 PM on Wednesdays.

Exam. Thu, May 12 2022 9:00am-12:00pm in MATH 507

Very rough outline of the course. In the first few lectures I will talk about the language of schemes. The idea is to discuss this minimally, enough to introduce some of the concepts needed to discuss the topics below. Then I will turn the course into a sequence of mini-topics. I hope to say something about

Moduli spaces: we can talk about moduli of varieties and about moduli of vector bundles. Just enough material so you have an idea what this even means.
Coherent modules and their cohomology. This is a very important technical tool, but also really fun in and of itself. I hope to be able to discuss some of the more fun aspects of the story.
Rational points and heights. Rudimentary introduction. I personally wonder about the question: what is really required to have an adequate theory of heights?
Tiny bit of intersection theory. We can discuss B\'ezout. We can talk about interscection theory on surfaces. We can discuss why it is hard to get an intersection theory on chow groups of a smooth projective variety.
Group schemes. What is a group scheme? Why do we not understand group schemes over the dual numbers? What is a representation of a group scheme. Why is an elliptic curve a group scheme? What is an abelian variety? How do we prove Mordell-Weil using heights?

Let me know if there are other basic topics you would be interested in hearing about.

Lectures.

First lecture = zoom lecture on Jan 18. If you haven't gotten a zoom link emailed to you by 10 AM on Jan 18, please send me a reminder.
Second lecture = zoom lecture on Jan 20. If you haven't gotten a zoom link emailed to you by 10 AM on Jan 18, please send me a reminder.

Material covered

What is a scheme? It is a locally ringed space such that every point has an open neighbourhood isomorphic (as a locally ringed space) to an affine scheme.
What is an affine scheme? It is the output of the "Spec" construction applied to a ring A. Its underlying topological space is the spectrum of A as discussed in the Fall. Its structure sheaf is constructed from A by the rule that the ring of sections over the principal open D(a) is the principal localization A_a of A at a.
What is a morphism of schemes? It is a morphism of locally ringed spaces.
Given a ring map A → B we obtain a corresponding morphism of affine schemes in the opposite direction. Thus a contravariant functor from the category of rings to the category of affine schemes.
The functor from rings to affine schemes is an anti-equivalence.
Pⁿ(k) for k a field as a topological space
Projective space Pⁿ_R over a ring R as a scheme
Closed subsets V₊(F) and complementary opens D₊(F) of projective space
Value of the structure sheaf of projective space on D₊(F)
Stratification of Pⁿ by Aⁿ, A^{n - 1}, ..., A⁰
Non-Euclidean geometry: any two lines in P² meet!
Category of objects over a fixed object
Category of schemes over a ring
We defined affine morphisms.
To check affineness of a morphism is suffices to consider an affine open covering of the target.
We defined finite morphisms, integral morphisms, and closed immersions as affine morphisms which have some properties for the ring map.
To check a morphism is, finite, integral, or a closed immersion, it suffices to consider an affine open covering of the target.
Spec(A/I) → Spec(A) is a closed immersion
We constructed the Fermat hypersurface over a ring R for any ring and any dimension and any degree as a closed subscheme of projective space.
We discussed closed subschemes of projective space and the relationship with homogeneous ideals in the graded polynomial ring over any ring R.
Projective schemes over a ring R are schemes over R which admit a closed immersion into Pⁿ_R for some n.
We defined morphisms locally of finite type
We discussed in some detail the proof of the lemma that to check a morphism of schemes is locally of finite type it suffices to check the existence of only some affine opens where the condition holds. See Lemma Tag 01T2 for a variant of the statement I wrote down in my lecture.
We talked about fibre products in any category
We talked about fibre products in the category of schemes
Fibre products of affines are affine and given by tensor products of rings
Fibre products of schemes are compatible with taking open subschemes
The fibre product of affine n-space and affine m-space is affine (n + m)-space
We talked about the base change functor in any category which has fibre products
We talked about base change for schemes and how for example starting with a scheme over the integers you get schemes over the complex numbers, the rational numbers, and fields of characteristic p, in particular finite fields.
We talked about scheme theoretic fibres
The topological space of a scheme theoretic fibre is homeomorphic to the fibre on the map of topological spaces.
A morphism X → S of schemes is separated if the corresponding diagonal morphism is a closed immersion
A morphism of affine schemes is separated
If X → S is separated and S is an affine scheme then the intersection of two affine opens in X is an affine open of X
Projective space over R is separated over R and we proved this for the case of P^1
We defined universally closed morphisms
We stated the theorem that Pⁿ_R is universally closed over R.
We proved the theorem that Pⁿ_R is universally closed over R.
We talked about families; we talked about what is a family of Riemann surfaces in complex analytic geometry. Moduli of such. We identified proper submersions of complex manifolds as good examples of families.
We defined a proper morphism of schemes.
We gave some examples of proper morphisms of schemes.
We defined what is a "proper flat family"
We defined a "proper smooth family" as a proper flat family whose fibres are smooth; we didn't yet define what it means to have smooth fibres
We talked about the "universal" family of conics in P^2.
We defined the following properties of morphisms of schemes: quasi-compact, locally of finite presentation, locally quasi-finite, unramified, smooth, etale, and finite type.
The definition of a smooth morphism is that it is a morphism of schemes which is locally of finite presentation, flat, and has smooth fibres (see next point).
We still haven't formally defined what it means for a scheme X over a field k to be smooth over k. (Whatever the condition actually is it will certainly imply that X is locally of finite type over k.)
We discussed the fact that all the properties P of morphisms we have seen so far are preserved by composition
We discussed the fact that all the properties P of morphisms we have seen so far are preserved by arbitrary base change
The property P(f) = "f is quasi-compact and has dense image" is preserved by flat base change but not arbitrary base change.
We discussed how flat proper families are stable under arbitrary base change.
We discussed how smooth proper families are stable under arbitrary base change.
A scheme X is integral if it is nonempty and for every nonempty affine open U of X the ring O_X(U) is a domain.
A scheme X is reduced if for every open U of X the ring O_X(U) has no nonzero nilpotent elements.
A scheme X is irreducible if the topological space of X is irreducible.
Lemma: X is integral IFF X is irreducible + reduced. See Tag 01ON
Definition: Let k be a field. A variety X is a scheme over k such that X is of finite type over k, X is separated over k, and X is integral.
Definiton: A variety X is called smooth, proper, projective if and only if the morphism X → Spec(k) has the corresponding property.
Warning: in the definition of a variety some people require also that X remains integral after any base change to a field extension k'/k
Task of AG: classify all varieties
Examples of varieties: Aⁿ_k and Pⁿ_k
Affine variety: Spec(k[x_1, ..., x_n]/p) for some prime ideal p of k[x_1, ..., x_n]
Projective variety: V(P) ⊂ Pⁿ_k for some homogeneous prime ideal P of k[T_0, ..., T_n] such that for some i the element T_i is not in P.
Specific example: Fermat hypersurfaces.
Moduli: How many varieties? What are families of them?
(Q1) What are some discrete, topological, cohomological invariants of varieties?
(Q2) Which of these invariants are locally constant in proper flat families?
Example 1: the dimension of fibres is locally constant in proper flat families. Proof next lecture.
Example 2: the Euler characteristic of the structure sheaf is locally constant in proper flat families. This will be discussed when we turn to cohomology of coherent sheaves.
Not easy to think of "new" discrete invariants of proper schemes over fields which are locally constant in proper flat families ("new" means not some combination of examples 1, 2).
We talked about dimension of schemes locally of finite type over a field. Here we can bring all the results about dimension from the lectures last semester to bear. See Tag 06LF
We talked about lifting generalizations along a flat morphism of schemes, see Tag 03HV
We talked about the canonical morphism Spec(O_{X, x}) → X, see Tag 01J7
We proved the the dimension of fibres is locally constant in proper flat families. The proof in the Stacks project is different (somehow more difficult -- because it is a combination of a result for flat morphisms and a result for proper ones), but if you want to take a look, see Tag 0d4J
We talked about moduli of curves
We talked about coarse moduli schemes
We talked about fine moduli schemes
We talked about M_g
We talked about Hilbert schemes
We talked about moduli of triples of points in A^1
If X is a ringed space, then Mod(X) is the category of O_X-modules. This is an abelian category. Exactness of a complex of modules can be checked at stalks.
Theorem: Mod(X) has enough injectives
We define H^i(X, F) as the ith right derived functor of functor sending F to H^0(X, F) = Γ(X, F) = F(X).
Theorem: the singular cohomology of a reasonable topological space X (for example a CW complex or a manifold) with coefficients in an abelian group A is equal to the cohomology of the constant sheaf with value A on X.
Mayer-Vietoris
We studied extension by zero also known as j-lower-shriek or in notation j_!
We proved that the restriction of an injective module I to an open is an injective module on that open.
We proved that an injective module I is flasque, which means that all restriction mappings are surjective.
We proved locality of cohomology, see Tag 01E3
LEMMA: on an affine scheme X the structure sheaf O_X has vanishing higher cohomologies.
We computed the cohomology of the structure sheaf on Pⁿ_R
We defined invertible modules on a locally ringed space X
We defined finite locally free modules on a locally ringed space X
We defined quasi-coherent modules on a locally ringed space X
We constructed the quasi-coherent module F_M associated a to an A-module M on Spec(A). A key property is that F_M(D(f)) = M_f and in particular F_M(Spec(A)) = M. We discussed how Hom_{O_X}(F_M, G) = Hom_A(M, G(X)). We discussed how Mod_A is equivalent to the category of quasi-coherent modules. We discussed how H^p(X, F_M) = 0 for p > 0.
On a scheme X extensions of quasi-coherent modules are quasi-coherent.
We talked about tensor products of O_X-modules on a ringed space X. The tensor product is commutative and associative. So it defines a symmetric monoidal structure on the category of all O_X-modules with unit O_X.
The tensor product of quasi-coherent modules is quasi-coherent.
The tensor product of finite locally free modules is finite locally free (of rank equal to the product of the ranks).
We defined Pic(X) as the set of isomorphism classes of invertible O_X-modules with group structure given by tensor products.
We "computed" Pic(Spec(A)) in the sense that it is equal to the group of isomorphism classes of invertible A-modules, in other words we have Pic(Spec(A)) = Pic(A) with Pic(A) defined as in Tag 0AFW
Pic(Spec(R)) = 0 if R is a UFD, see Tag 0BCH
We started computing Pic(P¹_k) but we didn't finish.
We talked about glueing sheaves, see Tag 00AK
We reviewed invertible modules on (locally) ringed spaces, see Tag 01CR (note that Lemma 0B8M says that invertible modules as in Definition 01CS are the same thing as locally free of rank 1 if the ringed space is a locally ringed space).
Using glueing of modules we proved that the Picard group Pic(P¹_k) of the scheme P¹_k is Z.
We explicitly constructed the Serre twists O(d) of the structure sheaf on projective n space. More generally, we explained how to a graded S-module M one associates a quasi-coherent module on projective space, if S = R[T_0, ..., T_n]. For more discussion on this you can read for example Hartshorne, Algebraic Geometry, Chapter II, Proposition 5.11 and continue reading there.
We proved that O(d) is an invertible module on projective space.
We proved that the invertible modules O(d) generate the Picard group of P¹_k.
It is true that O(d) ⊗ O(e) = O(d + e).
We computed the global sections of O(d) on Pⁿ_R for any d, n, R.
We computed all the cohomology of O(d) on P¹_R for any d and R.
We formulated the result on the cohomology of Pⁿ_R for any n, d, and R.
We talked about Cech cohomology on presheaves of abelian groups, about the comparison with cohomology for abelian sheaves, and about the Cech-to-Cohomology spectral sequence, see Tag 01ED, Tag 01EH, and Tag 01EO.
We formulated the agreement of Cech cohomology with cohomology for a quasi-coherent module on a separated scheme with respect to an affine open covering.
We explained how the agreement in the previous point leads to the calculation of the cohomology of the Serre twists of the structure sheaf on P^n.
We computed the genus of a planar curve using the cohomology of the Serre twists of the structure sheaf of P^2.
We proved that H^p(X, F) = H^p(Y, f_*F) if f : X → Y is an affine morphism of schemes and F is a quasi-coherent module.
We talked about morphisms to projective space in terms of (L, s_0, ..., s_n), see for example Tag 01ND
We gave some examples of this correspondence for A^1 mapping to P^1 and P^1 mapping to P^2
We discussed a theoretical example where we start with a smooth curve X over an algebraically closed field k and a finite dimensional k-subvectorspace V of the function field k(X). This always leads to a morphism to a projective space if dim_k(V) > 1
We proved that if V is large enough then we get an immersion from X into projective space and if X is also proper then we get a closed immersion.
Corollary: a proper curve is projective in the setting above
We discussed invertible modules L on smooth projective curves X over algebraically closed fields. We proved these are always of the form O_X(D) for some divisor D on X. We defined the degree of L as chi(X, L) - chi(X, O_X). We proved that the degree of O_X(D) is the degree of D as a divisor. We started proving some inequalities on the dimensions of the cohomology groups of L. Finally, we proved that if h^0(L) = deg(L) + 1 then the curve X has to be P^1.
For discussion on heights, see heights.pdf
We talked about Bezout in projective space, as explained in Hartshorne, Chapter I, Section 7.

Problem sets. If you email your problem sets, please email them to Noah. Handing in could be to me personally or under my door in the building. Some of the exercises will be impossible, so it should not be your goal to do each and every one of them. Moreover, these exercises are not always doable purely with the material discussed in the course -- sometimes you'll have to look up things online or in books and use what you find.

Due 01-27-2022
1. This exercise is trying to get you to think about the construction of the structure sheaf of an affine scheme in a perhaps more familiar setting. Let X = R be the real numbers viewed as a topological space with the usual topology (from analysis). A subset U of X is called an open interval if there exist a < b real numbers such that U = {x in X : a < x < b}. The open intervals form a basis for the topology on X. Next:
  1. For an open interval U let A(U) be the set of polynomial maps U → R, and
  2. for open intervals U, V with V ⊂ U let A(U) → A(V) be restriction of functions.
  Show that there is a unique sheaf O_X of rings on X which comes equipped with isomorphisms A(U) → O_X(U) of rings compatible with the restriction mappings for A and O_X. What is the value of O_X over an arbtrary open of X? (The difficult part is to show uniqueness somehow.)
2. Let k be an algebraically closed field. Carefully prove that the two definitions of closed subsets of Pⁿ(k) as defined in lecture 2 agree. (Reminder: one of the definitions uses the covering by affine spaces and the other definition uses vanishing sets of homogeneous polynomials.)
3. Let X be a locally ringed space. Let A = O_X(X) be the global sections of the structure sheaf. Let x ∈ X be a point. The construction of the stalk of O_X gives us a canonical map A = O_X(X) → O_{X, x}. Thus the inverse image of the maximal ideal m_x ⊂ O_{X, x} in A gives us a prime ideal p_x ⊂ A. In this way we get a map X → Spec(A), x ↦ p_x. Prove that this map is continuous.
4. Let f : X → Y be a morphism of locally ringed spaces. Then f induces a ring map B = O_Y(Y) → O_X(X) = A. Show that the construction of the previous exercise is compatible with this, i.e., that we have a commutative square with corners X, Y, Spec(A), Spec(B).
5. NOT AN EXERCISE: the construction x ↦ p_x in exercise 3 actually is the continuous map underlying a morphism X → Spec(A) of locally ringed spaces. A locally ringed space X is an affine scheme exactly when this morphism is an isomorphism. Exercise 4 then says that sending X to A = O_X(X) is a quasi-inverse to the "Spec" functor from rings to affine schemes.
6. Work out Question 078U in some case you like (for example feel free to choose your graded ring to be the graded polynomial ring if you like and to choose some particular homogeneous f of positive degree). To understand what the questions asks for, please read a bit more in Section 0280.
Due 02-03-2022
1. Let X be a scheme which has a covering X = U_1 ∪ U_2 such that U_1, U_2, and U_1 ∩ U_2 are affine opens of X. Let f ∈ Γ(X, O_X) be a global section of the structure sheaf. Let W ⊂ X be the set of points where f does not vanish, i.e., f is not in the prime ideal p_x from Exercise A.3, equivalently, the image of f in O_{X, x} is not in the maximal ideal m_x. Then W is open in X (by exercise A.3). Carefully argue that O_X(W) is isomorphic to the principal localization Γ(X, O_X)_f of Γ(X, O_X) at f. Remark: this result is true whenever X is quasi-compact and quasi-separated, but I really only want you to prove it in the case given.
2. Let P be a property of rings, i.e., given a ring P(A) is either true or false. Assume that
  1. for a ring A and prime ideal p of A there is an f in A, f not in p such that P(A_f) is true.
  2. for a ring A if P(A) then P(A_f) for all f in A
  3. for a ring A and f, g in A which generate the unit ideal if P(A_f) and P(A_g), then P(A).
  Prove that P(A) is always true.
3. Do exercise 02FK. Note that X finite type over Z means X locally of finite type over Z and that X is a finite union of affine opens
4. Do exercise 069T.
5. Let R → A be a finite ring map (so A is finite as an R-module). Show that Spec(A) is a projective scheme over R as defined in the lectures.
Due 02-10-2022
1. Let R be a ring. Let A = R[x_1, ..., x_n]. Let f ∈ A be an element. Assume the coefficients of f generate the unit ideal in R. Show that A/fA is flat over R. (Suggestion: either find a more general statement in the Stacks project and quote that or find a direct proof under some special assumption, for example if f is a monic polynomial in one of the variables.)
2. Let R = Z[a,b,c,d,e, f] and ξ = a + bx + cy + dx^2 + exy + fy^2 in A = R[x, y]. Show that A/ξA is not flat over R.
3. Let k be a field. Let C be a conic over k; I mean C is the closed subschemes of P²_k defined by single nonzero homogeneous polynomial Q in k[T_0, T_1, T_2] of degree 2. Let Q_0, Q_1, Q_2 be the partial derivatives of Q with respect to T_0, T_1, T_2. Let us say C is smooth if V_+(Q, Q_0, Q_1, Q_2) is empty.
  1. Give an example of a reducible C.
  2. Give an example of an irreducible C which is not smooth.
  3. Give an example of a C which is not smooth, yet regular. (This is hard; can only happen in char 2; suggest skipping this. A scheme is regular if all the local rings are regular local rings.)
  4. Assume C is smooth and has a k-rational point (which means that Q has a nontrivial k-rational solution). Prove that C is isomorphic to P¹_k.
  5. If k is the field R of real numbers, give an example of a smooth conic C which does not have a R-rational point.
  6. For every odd prime number p give an example of a smooth conic C over the p-adic number field Q_p which does not have a Q_p-rational point.
  7. Conclude that there exists a conic C over a field k which is not isomorphic to P¹_k but such that there exists a finite extension k'/k such that the base change of C to Spec(k') is isomorphic to P¹_k'
  8. Show that there are infinitely many pairwise non-isomorphic conics over the rational numbers Q. (This is difficult without more theory perhaps although there may be a simple trick I don't remember now.)
4. Let k be an algebraically closed field (your choice). Show there exists some smooth projective curve (curve = variety of dimension 1) over k which is not isomorphic to P¹_k. (There are several ways to do this; but I want you to think about how you would do this with the least possible amount of theory you can think of. My idea would be to make a C with an automorphism group G which does not fit into the automorphism group of P^1_k --- this automorphism group is PGL_2(k). I bet there are other, better ways.)
Due 02-17-2022
1. Let k be a field. Let G be the automorphism group of A¹_k as a scheme over k. Show that G is the "ax + b group".
2. Give an example of a ring R and an automorphism of the scheme A¹_R over R which is not linear, i.e., is not in the "ax + b group".
3. Let R be a Noetherian ring. Suppose that Z is a closed subscheme of A¹_R = Spec(R[x]) which is finite and flat over R. Prove that Z is defined by a monic polynomial in x over R. (Please restrict your ring if it helps.)
4. Let X be a topological space with finitely many, say n, points. Prove that H^i(X, F) = 0 for i ≥ n and any abelian sheaf F on X. (The inequality isn't optimal, right?)
5. Give an example of a non-quasi-coherent module on the affine line A¹_k over a field k.
Due 02-24-2022
1. Let A → B be a ring map. Let f : Y → X be the corresponding morphism on spectra.
  1. For a B-module N, denote F_N the quasi-coherent O_Y-module associated to N, denote res(N) = N viewed as an A-module, and finally denote F_{res(N)} the quasi-coherent O_X-module associated to res(N). Carefully show that there is an isomorphism f_*(F_N) = F_{res(N)}.
  2. Using adjoint functors (i.e., certain formulas we discussed in the lectures), show that the result in part (a) formally implies that f^*F_M = F_{M ⊗_A B} for an A-module M.
2. Choose your favorite field k. Find a finite type k-algebra A whose Picard group Pic(A) you can compute, yet is not trivial.
3. Compute Pic(X) if X is the affine line over a field with 0 doubled. (Please feel free to substitute X with any non-affine finite type scheme X over k with nontrivial Picard group Pic(X) that you can determine, except for P¹_k which we will do in the lectures.)
4. Let X be a curve over a field k. (Recall: curve = variety of dimension 1.) Let x be a closed point of X. Let I ⊂ O_X be the ideal sheaf of x, i.e., the O_X-module of O_X whose sections over U open are exactly those f in O_X(U) such that f is zero at x provided x is in U; if x is not in U then no condition. Show that I is an invertible O_X-module if and only if O_{X, x} is a discrete valuation ring.
5. Let L_n, n = 1, 2, 3, ... be invertible modules on some locally ringed space X. Let E = ⨁ L_n be the direct sum of all of them. Why might it not be true that E is "locally free of countable rank"? Can you give an example? (This is probably quite hard.)
Due 03-03-2022
1. Choose 4 or 5 exercises from Sections Tag 0D8P (except for 0D8R, 0D8S, or 0D8T), Tag 0DAI, or Tag 0DB3. Feel free to substitute exercises from Hartshorne on cohomology.
Due 03-10-2022. No problems.
Due 03-31-2022
1. Compute the degree of the image of the d-uple embedding from P^n into P^N where N = {n + d choose n} - 1.
2. Compute the degree of the Segre embedding P^n x P^m into P^{nm + n + m}.
3. Find two curves (recall that curves are irreducible and reduced) in P^2 of degrees 10 and 11 which intersect in exactly one point (set theoretically).
4. Suppose given 3 quadrics Q_i in P^3 whose intersection is scheme theoretically a line L and finitely pairwise distinct points P_1, ..., P_r. What is r? (Do an example if you don't want to prove it in general.)
Reading.
1. Robin Hartshorne, Algebraic Geometry
2. Ravi Vakil, Foundations of algebraic geometry
3. Stacks project