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.
- P^{n}(k) for k a field as a topological space
- Projective space P^{n}_{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 P^{n} by A^{n},
A^{n - 1}, ..., A^{0}
- Non-Euclidean geometry: any two lines in P^{2} 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 P^{n}_{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 P^{n}_{R}
is universally closed over R.
- We proved the theorem that P^{n}_{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:
A^{n}_{k} and
P^{n}_{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) ⊂ P^{n}_{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
P^{n}_{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(P^{1}_{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(P^{1}_{k}) of the scheme
P^{1}_{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 P^{1}_{k}.
- It is true that O(d) ⊗ O(e) = O(d + e).
- We computed the global sections of O(d) on
P^{n}_{R} for any d, n, R.
- We computed all the cohomology of O(d) on
P^{1}_{R} for any d and R.
- We formulated the result on the cohomology of
P^{n}_{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
- 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:
- For an open interval U let A(U) be the set of polynomial maps
U → R, and
- 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.)
- Let k be an algebraically closed field. Carefully prove that the two
definitions of closed subsets of P^{n}(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.)
- 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.
- 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).
- 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.
- 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
- 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.
- Let P be a property of rings, i.e., given a ring P(A) is either
true or false. Assume that
- 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.
- for a ring A if P(A) then P(A_f) for all f in A
- 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.
- 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
- Do exercise 069T.
- 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
- 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.)
- 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.
- Let k be a field. Let C be a conic over k; I mean C
is the closed subschemes of P^{2}_{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.
- Give an example of a reducible C.
- Give an example of an irreducible C which is not smooth.
- 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.)
- 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 P^{1}_{k}.
- 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.
- 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.
- Conclude that there exists a conic C over a field k
which is not isomorphic to P^{1}_{k} but such that
there exists a finite extension k'/k such that the base change
of C to Spec(k') is isomorphic to P^{1}_{k'}
- 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.)
- 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 P^{1}_{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
- Let k be a field. Let G be the automorphism group of
A^{1}_{k} as a scheme over k.
Show that G is the "ax + b group".
- Give an example of a ring R and an automorphism of
the scheme A^{1}_{R} over R which
is not linear, i.e., is not in the "ax + b group".
- Let R be a Noetherian ring. Suppose that Z is a closed subscheme
of A^{1}_{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.)
- 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?)
- Give an example of a non-quasi-coherent module on
the affine line A^{1}_{k} over a field k.
- Due 02-24-2022
- Let A → B be a ring map. Let f : Y → X
be the corresponding morphism on spectra.
- 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)}.
- 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.
- Choose your favorite field k. Find a finite type k-algebra A
whose Picard group Pic(A) you can compute, yet is not trivial.
- 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 P^{1}_{k} which we will do
in the lectures.)
- 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.
- 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
- 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
- Compute the degree of the image of the d-uple embedding
from P^n into P^N where N = {n + d choose n} - 1.
- Compute the degree of the Segre embedding
P^n x P^m into P^{nm + n + m}.
- 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).
- 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.
- Robin Hartshorne, Algebraic Geometry
- Ravi Vakil, Foundations of algebraic geometry
- Stacks project