Schemes
Professor A.J. de Jong,
Columbia university,
Department of Mathematics.
This semester I am teaching the course on schemes.
The lectures will be on Tuesday and Thursday from
11:40AM -- 12:55PM in Room 507.
The TA is Carl Lian who will be in the help room on
Monday 10:30-12 and Thursday 4:30-6.
For last years course on schemes webpage click
here.
However, this year we will do something completely different.
Please visit the first class of the course to find out more, or
you can read about it in
this blog post.
It is very
important to attend the lectures. You will not be able to
pick up the material by just reading this webpage
Review of lectures and suggested readings
- Lecture I: we discussed prerequisites and we found most
participants had some knowledge of commutative algebra
and sheaves on topological spaces. It turned out that most
of the participants are interested in learning about schemes.
We then started discussing the questions "What is a scheme?"
and "What is a morphism of schemes?". We reviewed and discussed
the following topics:
- topological spaces and continuous maps of topological spaces,
- categories of sheaves of different types on topological spaces,
- pushforward and pullback of sheaves under a continuous map of
topological spaces,
- ringed spaces and morphisms of ringed spaces,
- local rings and local homomorphisms of local rings,
- locally ringed spaces and morphisms of locally ringed spaces,
- the spectrum of a ring as a set and as a topological space,
- the spectrum of C[x] where C is the complex numbers, and
- the value of the structure sheaf on the spectrum of a ring
on the basic opens of the spectrum.
Please review this material and prepare to ask questions about
arguments or constructions you do not understand when you do this.
Some links to material in the Stacks project:
- Spectrum of a ring
00DY
- Sheaves
006A
- Sheaves of modules
01AC
(this will come later)
- Sheaves on a spectrum
handout (pdf)
(also see the fun result here:
0F1A)
- Locally ringed spaces
01HA
01HD
(skip closed immersions)
- Affine schemes
01HR,
01HX
These references should be enough to keep you busy till Thursday.
But I encourage you to read in *any* reference you want. In fact,
it could be really fun to compare different arguments
and different definitions we get from different references.
- Lecture II: we discussed the following
- functoriality of the spectrum: given a ring map how
does this give you a map of spectra (as a map of sets, continuity,
and on the structure sheaves),
- surjections of rings give closed maps on spectra,
- the localization of a ring at an element gives
an open immersion of spectra,
- fully faithfulness of the spectrum as a functor from the
(opposite of the) category of rings to the category of locally
ringed spaces,
- Hausdorff topological spaces as those topological spaces
whose diagonal is closed,
- separated schemes as those schemes whose diagonal morphism
is a closed immersion,
- affine morphisms of schemes (just the definition; there's lots
missing here),
- closed immersions as affine morphisms which affine locally
correspond to surjections of rings,
- Spec(Z) is the final object in the category of schemes,
- Spec(C) is the final object in the category of schemes
over C (which means you replace rings everywhere by C-algebras
in everything we've said sofar),
- any non-affine scheme gives rise to a nonaffine morphism
by looking at the morphism to Spec(Z),
- P^1_Z was constructed and we computed the global sections
of its structure sheaf,
- the affine line with zero doubled was constructed and
we computed the global sections of its structure sheaf,
- we proved the previous two examples are not affine.
Please try to think of questions of a different flavor for next time!
- Lecture III: we discussed the following
- Yoneda lemma
- fibre products in any category
- examples of fibre products in sets, topological spaces, vector spaces,
- pushouts are the dual notion to fibre products
- pushouts of rings are tensor products
- fibre products of affine schemes are spectra of tensor products of rings
- example of fibre product of affine lines
- fibre products as glueing fibre products of affine schemes
- P^1 times P^1 as a closed subscheme of P^3 (in terms of points)
- P^1 is separated using the above (kind of a cheat)
- the morphism from punctured affine space to projective space
Please do some examples of the things we discussed yourself.
- Lecture IV: we discussed the following
- We reviewed the sheaf condition for the structure sheaf on
an affine scheme, in particular we discussed if A is a ring
and f, g are elements of A which generate the unit ideal in A,
then the sequence with terms 0, A, A_f, A_g, A_{fg} is exact.
- Let C be a category which has all fibre products and
let P be a property of morphisms in C.
We discussed what it means when we say a morphism
in C is "universally P".
- We discussed proper maps of topological spaces and
universally closed maps of topological spaces
and we discussed how these are the same, see
005M
(our definitions were not literally the same as in the Stacks project...
because the definitions in the Stacks project follow in this instance
the conventions of Bourbaki which are slightly unnatural for us in
this case).
- We proved that the real line mapping to a point is not
a universally closed map (and hence not proper).
- Motivated by our discussion of proper maps of topological
spaces we defined a morphism of schemes to be proper if it
is finite type, separated, and universally closed.
- We proved that the affine line over a field is not
proper over the field.
- We discussed affine classical varieties V over an
algebraically closed field k, in particular we discussed
- the Zariski topology on k^n,
- V is an irreducible closed subset of k^n in Zariski topology,
- regular functions on V are maps of sets from V to k which
locally in the Zariski topology on V can be written as quotients
of polynomials,
- V corresponds to a prime ideal p of k[x_1, ..., x_n], namely
the set of polynomials vanishing on V,
- the k-algebra k[V] of regular functions on V is the same
as the quotient of k[x_1, ..., x_n] by p
- morphisms of classical affine algebraic varieties from V to V'
correspond 1-to-1 to k-algebra maps k[V'] to k[V].
- We discussed how a classical affine variety V over k correspond
to an affine, integral scheme X of finite type over Spec(k) and we
discussed how points of X correspond to closed (classical)
subvarieties of V
- In order to state the previous point we discussed how you can
talk about connected topological spaces,
irreducible topological spaces,
connected components of topological spaces,
irreducible components of topological spaces,
generic points of irreducible topological spaces,
sober topological spaces (each irreducible closed subset
has a unique generic point)
- We discussed how schemes are sober via the example of
spectrum of a ring and we applied this to see the fact about
points of the scheme X associated to the classical variety V
mentioned above.
- Lecture V: we discussed the following
- we discussed the standard proof that the ring of regular functions on
a classical affine algebraic variety V is the quotient of the polynomial ring
by the (prime) ideal of functions vanishing on V,
- given an affine scheme S = Spec(A) and an A-module M we defined
the O_S-module \widetilde{M} associated to M,
- we characterized \widetilde{M} as the unique sheaf of O_S-modules
such that for any O_S_module F we have
Hom_{sheaves of modules}(\widetilde{M}, F) = \Hom_A(M, F(S)),
- in other words the functor \widetilde{M} is the left adjoint to the
functor of global sections which goes from the category of all
O_S-modules to the category of A-modules,
- an important observation is that the functor which assigns to M
the sheaf \widetilde{M} is exact; this follows because the stalk
of \widetilde{M} at p is the localizations M_p and because localization
is exact
- given a scheme X we defined an O_X-module F to be quasi-coherent if
for every affine open U = Spec(A) of X the restriction F|_U is of the
form \widetilde{M} for some A-module M,
- we observed that with this definition it isn't clear that
given S = Spec(A) and an A-module M, the sheaf \widetilde{M}
on Spec(A) is a quasi-coherent module!
- more importantly perhaps, it isn't clear why, given an open covering
of a scheme X by affine opens U_i = Spec(A_i), to check quasi-coherence
of an O_X-module F, it suffices to check that F|_{U_i} is isomorphic
to \widetilde{M_i} for some A_i-modules M_i,
- anyway, we discussed pushforward and pullbacks of the modules
\widetilde{M} under morphisms of affine schemes,
- more precisely, given a ring map A → B we found that pushforward
along Spec(B) → Spec(A) corresponds to the restriction functor
Mod_B → Mod_A and we found that pullback along Spec(B) → Spec(A)
corresponds to the base change functor Mod_A → Mod_B which sends
M to M ⊗ B.
- we finally were able to conclude that \widetilde{M} is indeed
a quasi-coherent module (please make sure you understand how this
follows)
- Motivation for defining quasi-coherent modules: (1) these are the
algebraic geometers version of modules over rings, (2) vector bundles
are special types of quasi-coherent modules, (3) we can look at cohomology
of quasi-coherent modules and try to find invariants of algebraic
varieties in that manner.
Here are some references to material on quasi-coherent modules in the
Stacks project
- The approach in the Stacks project is to first define a notion of
quasi-coherent module on any ringed space as those sheaves of modules
which locally are the cokernel of a map of free modules. This is
done in
01BD
- Then it is shown that on an affine scheme one obtains exactly
the sheaves \widetilde{M} from the lecture, see
01I6
- Because quasi-coherent modules where defined in general
(without referencing affine schemes or whatnot) we then
see that a quasi-coherent module on a scheme is affine locally
of the form we want (this is an advantage of doing things this way).
- Lecture VI: we discussed the following
- Very briefly we discussed why given a curve C and a point p of C
there exists a regular function on C - {p} which blows up at p. The
corresponding algebra statement for affine curves would be: (*)
given a 1 dimensional Noetherian domain A and a maximal ideal p of A there
exists an element f of the fraction field of A which is not contained in the
local ring A_p but is contained in the local ring A_q for all prime
ideals q of A which are not equal to p.
- We gave the definition of a finite locally free O_X-module on a
ringed space (X, O_X).
- We defined an invertible O_X-module to be a finite locally free
O_X-module of rank 1.
- We defined the Picard group of X, denoted Pic(X), to be the
set of isomorphism classes of invertible O_X-modules with addition
given by tensor product.
- We implicitly accepted the fact that this is indeed a group, i.e.,
that every invertible O_X-module has an ``inverse''. It turns out that
the inverse of an invertible O_X-module L is given by the sheaf-hom
from L to O_X, so L^{⊗ -1} = SheafHom_{O_X}(L, O_X).
- We proved that if X is a scheme then every finite locally free
module is quasi-coherent. In particular every invertible module is
quasi-coherent.
- We concluded if X = Spec(A), then the Picard group is the set
of isomorphism classes of certain A-modules. Let's call the modules
you obtain in this manner invertible A-modules.
- We showed that if M is an invertible A-module, then for every
prime ideal p of A the localization M_p is isomorphic to A_p.
- The module M = \sum (1/p)Z, seen as a Z-submodule of Q, has the
property that M_{(p)} is isomorphic to Z_{(p)} for all prime numbers p
but M is not an invertible module (as we'll see soon).
- We concluded that we were missing a property of invertible modules.
After looking at the example above, we decided that we needed to look
for finitely generated A-modules.
- On a ringed space we defined a module to be of finite type
if it can locally be generated by finitely many sections.
- Lemma: Let A be a ring and let M be an A-module. TFAE: (i) M is a finite
type A-module, (ii) the O_{Spec(A)}-module \widetilde{M} associated to
M is of finite type.
- A module M over a domain A is said to be torsion free if
every nonzero element of A is a nonzerodivisor on M.
- Let A be a domain and M an A-module. TFAE: (i) M is torsion free,
(ii) M_p is a torsion free A-module for all prime ideals p of A.
- We conlcude that over a domain an invertible module is torsion free.
- (**)
Let A be a domain and let M be a finite type torsion free A-module.
Then M is isomorphic to a submodule of a free A-module.
- Lemma: Let A be a domain and let M be an invertible A-module.
Then M is isomorphic to an ideal of A (as an A-module).
Proof: by the previous lemma we see that there is a nonzero map
M → A. Localizing at the prime (0) of A we see that it is injective
because M_{(0)} is a 1-dimensional vector space.
- CONCLUSION: Let A be a PID. Then every invertible A-module is
isomorphic to A as an A-module and the Picard group of Spec(A) is trivial.
- Addendum: in fact the Picard group of Spec(A) is trivial if
A is a UFD. The proof uses the same ingredients and then uses that if
I is an ideal which is also an invertible module, then any minimal
prime over I has height 1 and hence is principal (in a UFD).
- The Picard groups of Spec(Z) and the affine spaces A^n_k over
fields k (or even if k is a UFD) are all trivial.
In the discussion above, I never said that for general rings A the
invertible modules are exactly the finite type A-modules all of
whose localizations at primes are free of rank 1. This is true, but
it is actually a bit tricky in the non-Noetherian case
(see first reference below)!
- Algebra: finite locally free modules, finite flat modules,
finite projective modules, and relations between these, see
00NV
especially the first lemma.
- Counter examples
052H
(finite flat, not finite locally free)
0CBZ
(ideal whose localizations are free but not invertible).
- The definition of an invertible O_X-module in the
Stacks project is slightly different (better!).
The difference vanishes for locally ringed spaces. See discussion in
01CR.
- Correspondingly, for modules over rings we have the
same thing, see
0AFW.
- The Picard group of a UFD is trivial, see
0BCH.
- Lecture VII: we discussed the following
- very briefly the example of an ideal whose localizations are free
of rank 1 but which is not invertible as a module,
- pullbacks of quasi-coherent sheaves along morphisms of schemes
are quasi-coherent,
- Lemma : pushforwards of quasi-coherent sheaves along quasi-compact and
separated morphisms are quasi-coherent,
- we defined quasi-compact morphisms of schemes as those
morphisms such that the inverse image of a quasi-compact open
is a quasi-compact open,
- any base change of a quasi-compact morphism is quasi-compact,
- we defined separated morphisms as those morphisms whose
diagonals are closed immersions,
- any base change of a separated morphism is separated,
- we proved that given a separated morphism of schemes X → Y
and affine opens U, U' of X mapping into a common affine open of Y,
then U ∩ U' is an affine open of X,
- we briefly discussed how you can characterize separated morphisms
in terms of intersections of affine opens + something on the rings,
- we stated and proved that kernels, cokernels, and images of
maps between quasi-coherent modules on schemes are quasi-coherent,
- arbitrary direct sums of quasi-coherent modules are quasi-coherent,
- using all of the above we proved the lemma on pushforwards
of quasi-coherent modules,
- we discussed the rings A_n = k[x, 1/(x - t_1)(x - t_2)...(x - t_n)]
where t_1, ..., t_n are pairwise distinct elements of a field k,
- we proved that the group of units A_n^* is isomorphic to
the unit group k^* of k times a free abelian group of rank n,
- we proved that A_n is not isomorphic to A_m if n is not equal m,
- we observed that A_n is a PID for all n,
- we stated but did not prove that if k is algebraically closed
and A is a finite type k-algebra which is a Dedekind domain
(equivalently A is normal of dimension 1) and if the fraction field
of A is a purely transcendental extension of k, then A is isomorphic
to A_n for some n and some t_1, ..., t_n pairwise distinct elements of k.
- we concluded that if you do exercise 078S below, then you will have
found an finite type k-algebra A which is a Dedekind domain
whose fraction field is not isomorphic to a
purely transcendental extension of k.
- said another way we find there exists some extension field K/k
with the following properties
- K is finitely generated as a field extension of k,
- the transcendence degree of K over k is 1,
- K is not isomorphic to k(x) as a field extension of k.
Perhaps it would be a good idea to ask what this means geometrically
about 1 dimensional affine varieties over k in a future lecture.
- Lecture VIII: The motivating question for this
lecture was: "What is the relationship between the Picard
group and the divisor class group of a curve?" To answer this
question we discussed the following
- Let K be a number field with ring of integers O_K. We said:
- The class group Cl(O_K) is the group of fractional
ideals modulo principal fractional ideals,
- A fractional ideal is a nonzero finitely generad O_K-submodule of K,
- A principal fractional ideal (f) is just all O_K-multiples of f where
f is a nonzero element of K,
- Any fractional ideal is an invertible O_K-module,
- Any invertible O_K-module is isomorphic (as a module)
to a fractional ideal (this is because we showed that any
invertible A-module is isomorphic to an ideal of A for a domain A),
- Two fractional ideals give the same class in CL(O_K) if
and only if they are isomorphic as O_K-modules,
- Combining all of the above we conclude Pic(O_K) = Cl(O_K),
- Any fractional ideal I can be uniquely written in the form
p_1^{n_1} ... p_r^{n_r} where the p_i are pairwise distinct
maximal ideals of O_K.
- For a nonzero element f of K we have
(f) = p_1^{n_1} ... p_r^{n_r} where n_i = v_i(f)
where v_i : K^* → Z is the valuation on K associated
to the discrete valuation ring O_{K, p_i}. Moreover
we have v(f) = 0 if v = v_p is the valuation associated to a
prime ideal not equal to any of the p_i.
- We conclude that
Cl(O_K) = Pic(O_K) = (free abelian group on maximal ideals of O_K)/
(group of principal divisors) where a principal divisor
associated to f in K^* is the formal sum div(f) = sum_p v_p(f)[p].
- Let k be a field. A variety is a scheme
separated and of finite type over Spec(k) which is
reduced and irreducible.
- A scheme X is irreducible if and only if the underlying
topological space of X is an irreducible topological space
- A scheme X is reduced if and only if there are
no nozero local sections of the structure sheaf of X which
are nilpotent. Clearly, it is equivalent to assume the local
rings O_{X, x} to be reduced rings (no nonzero nilpotents)
for all points x of X.
- A scheme X is integral if it is nonempty and for
every nonempty affine open U = Spec(A) the ring A is a domain.
- Lemma: a scheme X is integral iff X is reduced and irreducible
Tag 01ON
- Let X/k be a variety. The function field k(X) of
X is
- the stalk of the structure sheaf at the generic point of X, or
- the residue field of the generic point of X, or
- the fraction field of A if U = Spec(A) is any nonempty
affine open subscheme of X, or
- the fraction field of O_{X, x} for any point x of X.
You should double check that
all of these things are the same.
- The dimension of a scheme X is the Krull dimension
of the underlying topological space of X.
- The Krull dimension (sometimes called the combinatorial
dimension) of a topological space X is the supremum of the lengths
of chains if irreducible closed subsets.
- If X is a sober topological space (for example the topological
space underlying a scheme) then the Krull dimension is the
supremum of the lengths n of chains x_n ⇝ x_{n - 1} ⇝ ...
⇝ x_0 of points x_i of X.
- Let X be a topological space. Let x, y in X. We say
- x specializes to y or
- y is a specialization of x or
- x is a generalization of y or
- y generalizes to x,
notation x ⇝ y, if and only if y is in the closure
of the subset {x} of X.
- If the topological space X has an open covering by U_i
then we have dim(X) = sup dim(U_i).
- Fact. If X/k is a variety, then dim(X) is equal to the
transcendence degree of k(X) over k.
This follows from the corresponding algebra fact and the
fact that we can reduce to affine varieties by
the previous point.
- A variety X is called a curve if it has dimension 1.
- A variety X is called a surface if it has dimension 2.
- A variety X is called a threefold if it has dimension 3.
- Let X be an affine variety of dimension d over an
algebraically closed field k. Write X = Spec(A) and write
A = k[x_1, ..., x_n]/(f_1, ..., f_m). Consider the matrix
T = (df_i/dx_j) of partial derivatives. We define the
singular locus of X, denoted Sing(X) to be the closed
subset of X cut out by the (n - d) x (n - d) minors of T.
- We say x in X is a nonsingular point of X or a
smooth point of X if x is not contained in Sing(X).
- We proved that if x is a smooth point of X then
the local ring O_{X, x} is regular. (Our argument only
worked for closed points and it moreover really used that
k is algebraically closed.)
- Still over an algebraically closed ground field the converse
is also true: if O_{X, x} is a regular local ring, then x is
a smooth point of X.
- Let X be a curve over an algebraically closed field.
The following are equivalent
- x is a nonsingular point of X,
- x is a smooth point of X,
- O_{X, x} is regular,
- O_{X, x} is regular of dimension 1,
- the maximal ideal m_x ⊂ O_{X, x} can be generated by 1 element,
- O_{X, x} is a discrete valuation ring,
- O_{X, x} is a normal domain,
- O_{X, x} has finite global dimension,
- the residue field kappa(x) of the local ring
O_{X, x} has finite projective dimension over O_{X, x},
- kappa(x) has projective dimension 1 over O_{X, x},
- X → Spec(k) is smooth at x (this characterization
wasn't discussed in the lecture; we'll come back to it)
- add your favorite characterization here.
- From the discussion above, if X is a nonsingular curve
(in other words, the singular locus is empty)
over an algebraically closed field, then for any x in X
we obtain a valuation v_x = ord_x : k(X)^* → Z by using
that O_{X, x} is a discrete valuation ring and that
k(X) = Frac(O_{X, x}), see above.
- Let X be a curve. A Weil divisor on X is a (finite)
formal linear combination sum n_x [x] of closed points x of X
with integer coefficients n_x. The abelian group of Weil divisors
is denoted Div(X).
- Let X be a nonsingular curve over an algebraically closed field.
There is a group homomorphism div : k(X)^* → Div(X) mapping
f to div(f) = sum ord_x(f) [x].
- Let X be a nonsingular curve over an algebraically closed field.
The Weil divisor class group of X, denoted Cl(X), is the cokernel
of the map div above. Thus we have an exact sequence
k(X)^* →
Div(X) →
Cl(X) → 0.
OK, so there is a lot of new material here; take your time digesting this.
Just one point I wanted to make about working over non-algebraically closed
fields. Let X/k be an affine variety over a field k which is not
algebraically closed. We say a point x of X is nonsingular if
O_{X, x} is regular. We say X → Spec(k) is smooth at x
if the matrix of partial derivatives of the equations for X has the
"correct" rank at x. It is always true that "smooth at x" implies
"nonsingular at x" but if k is a non-perfect field, then the reverse
implication isn't true in general.
- Lecture IX:
- The affine line over the field F_p with p elements has infinitely
many points if you think of it as a scheme.
- Notation: if S is a scheme and we say "let X be a scheme over S" then
we mean that X is a scheme which comes endowed with a morphism X → S.
In this situation the morphism X → S
is called the structure morphism. If k is a ring (often a field)
and we say "let X be a scheme over k" then we mean that X is a scheme
which comes endowed with a morphism X → Spec(k).
- If X is a scheme over a scheme S, then X(S) denotes the
set of morphisms S → X of schemes such that the composition
with the structure morphism is the identity on S. We often call
X(S) the set of S-valued points. If X is a scheme over a ring
k, then we similarly denote X(k) the set of morphisms Spec(k) → X
whose composition with the given morphism X → Spec(k) is the
identity. We call an element of X(k) an k-valued point.
If k is a field, we will call an element of X(k) a k-valued point.
- An important variation on the theme above is that if we
have X/S and S'/S, then X(S') denotes the set of morphisms
S' → X such that the composition S' → X → S
is equal the given morphism S' → S. Again we say that
X(S') is the set of S'-valued points of X. Similarly, if
we are given a ring map k → k' (often a field extension)
we can consider the set X(k') of k'-valued points.
- Let A be a local ring of dimension > 0 with residue field k.
Then the morphism Spec(k) → Spec(A) is surjective on closed
points but not surjective.
- Let X be a scheme of finite type over a field k. For a point
x of X we set δ(x) equal to the transcendence degree of
the residue field of x over k. Then we have
- δ(x) = 0 if and only if x is a closed point,
- δ is a dimension function, that is
- if x ↝ x' then δ(x) ≥ δ(x'), and
- if x ↝ x' is an immediate specialization (the
points aren't equal and there is no point strictly between), then
δ(x) = δ(x') + 1,
- if f : X → X' is a morphism of f.t. schemes/k, then
δ(f(x)) ≤ δ(x),
- δ(x) is equal to the dimension of the closure of {x} in X
(you can easily deduce this from properties a and b),
- many things can be added here, but they often trivially
follow from a, b, c.
- Let X be a scheme of finite type over a field k. Using
the dimension function δ above it is easy to show
if W is a locally closed subset of X then (a) the closed
points of W are the closed points of X which are contained
in W, and (b) these points are dense in W.
- Let f : X → Y be a morphism of schemes of finite type
over a field. Then f(closed point) = closed point.
We proved this in the affine case by invoking the Hilbert Nullstellensatz
but it also easily follows by using 6.a and 6.c above.
- A morphism X → Y of schemes of finite type over a field
is surjective if and only if it is surjective on closed points.
The statement makes sense by the previous point; we proved it
in class using the correspondence between points of the scheme X
and irreducible closed subsets of the set of closed points of X.
My suggestion: try to prove this using the properties of δ
stated above alone; if it is not possible, please tell me so I
can add the additioanl property we need to get this to work.
- Let k be an algebraically closed field of characteristic
not 2 or 3. Let X be the spectrum of k[x, y]/(y^2 - x^3 + x).
By explicitly finding relations in Cl(X) and using the result of
one of the exercises (slightly generalized) we proved that the map
{closed points of X} → Cl(X)
is injective and moreover every element but the zero element of
Cl(X) is in the image. This shows that there is a natural abelian
group structure on {closed points of X} ∪ {0} which is
given by the usual addition law for points on elliptic curves.
In particular, there is no need to check the associative property
for this group law using formulae!
References: there is a lot of very general stuff related to this
material in the Stacks project. For example, you can look up
- dimension functions in
Tag 02I8,
- dimension theory of finite type algebras over a field in
Tag 00OO
Tag 07NB
- Jacobson topological spaces (spaces with lots of closed points) in
Tag 005T,
- Jacobson rings in
Tag 00FZ
- Lecture X: The goal of this lecture was to finish off
the discussion of the Picard group and the Weil divisor class
group of a nonsingular curve.
- The (ridiculously) general theory, see
0BE0 and
02SE,
says that
- for any locally Noetherian integral scheme
there is a notion of Weil divisors and one can
define the group Div(X) of Weil divisors,
- for a nonzero element f of the function field of X
there is an associated principal Weil divisor div(f)
defined in terms of orders of vanishing of f at the
generic points of Weil divisors,
- the collection of all principal Weil divisors
is a subgroup of Div(X) and the quotient of Div(X)
by this subgroup is denoted Cl(X) and is
called the Weil divisor class group of X,
- for any invertible O_X-module L there exist nonzero
meromorphic or rational sections s and we can define
the Weil divisor div_L(s) associated to s in terms of
the orders of vanishing of s at the generic points of
Weil divisors,
- this construction defines a group homomorphism
c_1 : Pic(X) → Cl(X) which in general is neither injective
nor surjective.
- for nonsingular curves over algebraically closed
fields the theory works in exactly the same way AND
the resulting map c_1 : Pic(X) → Cl(X) is an isomorphism,
- a nonzero rational (or meromorphic) section s of an
invertible module L on a curve X is just a nonzero element
of the stalk of L at the generic point of X,
- we define div_L(s) = sum div_{P, L}(s) [P]
where for every closed point P of X we choose
a generator e_P of L in a neighbourhood of P, we write
s = f_p e_p for some rational function f_P on X, and we set
ord_{P, L}(s) = ord_P(f_P),
- a fun exercise is to show that if div_L(s) = 0
then s is a regular nowhere vanishing section of L,
- then one shows that the divisor class of div_L(s)
is independent of the choice of s because s is well defined
up to multiplication by a nonzero rational function f and
because div_L(fs) = div(f) + div_L(s)
- the map c_1 sends L to the divisor class of div_L(s)
where s is as above,
- you can check that c_1 is additive by showing that
div_{L ⊗ L'}(s ⊗ s') = div_L(s) + div_{L'}(s'),
- if c_1(L) = 0 the we see that div_L(s) + div(f) = 0 for
some nonzero rational function f, hence div_L(fs) = 0.
By the above we find that fs is a trivializing section of L, i.e.,
L defines the 0 element of Pic(X),
- finally we showed that if I ⊂ O_X is the ideal sheaf
of a closed point P of X, then c_1(I) = - [P], and hence the
map c_1 is surjective.
- From now on, given a Weil divisor D on X we set O_X(D)
equal to the unique (up to unique isomorphism) invertible module
which has a nonzero rational section 1_D such that div_{O_X(D)}(1_D) = D.
For example, if D = sum -[P_i] for some pairwise distinct points
P_1, ..., P_n, then O_X(D) is the ideal sheaf of {P_1, ..., P_n}.
- We computed that Pic(P^1_k) = Z if k is an algebraically
closed field by explicitly finding enough relations among the
points (viewed as Weil divisors). We also observed that
div(f) is always a divisor of degree 0.
- Here the degree deg(D) of a Weil divisor D = sum n_i [P_i]
(on a curve over an algebraically closed field)
is defined as the sum of the integers n_i.
- We speculated (?!?) that if X is a nonsingular projective curve
over an algebraically closed field, the the degree of a principal
Weil divisor should be 0.
- Lecture XI: The goal of this lecture was to start
talking about cohomology of sheaves, but we found ourselves
discussing derived functors in general instead.
- If F is a sheaf of O_X-modules on a locally ringed space X
let us define the fibre of F at a point x of X to be
the k(x)-vector space F_x/m_x F_x. This is not a universally
accepted terminology. The fibre is a vector space over the
residue field k(x).
- We first discussed the following question: let X be a reduced
scheme and let F be a finite type quasi-coherent O_X-module such
that all fibres have the same finite dimension r. Show that F is
finite locally free of rank r.
To answer this first prove that around every point there is an
open neighbourhood generated by r sections (use
Nakayama and
01B8).
Then use that there cannot be any relations between these
local sections as X is a reduced scheme and we know each fibre
has dimension r.
- We also very briefly discussed the relationship
between invertible modules and line bundles,
between finite locally free modules and vector bundles as usually defined, and
between quasi-coherent modules and vector bundles as defined in EGA
and the Stacks project. In each case one gets an anti-equivalence of
categories when properly formulated.
The last correspondence is discussed in
01M1;
observe how this is a bit of a cheat because on the
geometric side we use an additional structure on the
structure sheaf (namely a grading). There is a way to
say all of this completely geometrically.
- Let F : A → B be an additive functor between abelian
categories. The formal definition of an abelian category can
be found here.
- Question: How does one define/construct the left/right
derived functors of F?
- Answer: use resolutions by nice objects.
- A left resolution of an object M of A is an exact
complex ... → P_1 → P_0 → M → 0 in A.
- A right resolution of an object M of A is an exact
complex 0 → M → I^0 → I^1 → ... in A.
- We try to define L_kF(M) = H_k(F(P_*)) as the
kth left derived functor of F.
- We try to define R^qF(M) = H^q(F(I^*)) as the
kth right derived functor of F.
- In order to make this well defined and a functor, we need some
functoriality property of resolutions. The usual solution
to this is to take left resolutions by projectives and right
resolutions by injectives.
- Please look up the definition of projective and injective
objects of an abelian category A. Please look up what it means
to have enough projectives or enough injectives in A.
- Please look up uniqueness up to homotopy of projective
resolutions and injective resolutions. You can look
here and
here for example.
- The upshot of this is that if A has enough projectives,
then we obtain well defined left derived functors L_kF
and if A has enough injectives, then we obtain well defined
right derived functors R^qF.
- If F is right exact, then L_0F = F.
- If F is left exact, then R^0F = F.
- We defined Tor^R_i(M, N) to be the ith left derived functor
of the functor F(-) = M ⊗ (-). Then for R = k[x, y] the
polynomial ring over a field k, we computed
Tor^R_i(k, k) to be a k-vector space of
dimension ..., 0, 1, 2, 1 in degrees ..., 3, 2, 1, 0.
- Ask yourself some questions, such as what happens
when you take the derived functors of an exact functor?
- Lecture XII: We continued discussing cohomology.
- First we briefly discussed the fact that the category Ab(X) of
sheaves of abelian groups on a topological space X in general doesn't
have enough projectives. Neither does the category Mod(O_X) of O_X-modules
on a ringed space (X, O_X) nor the category QCoh(O_X) of quasi-coherent
modules on a scheme X.
- On the other hand, these abelian categories all have enough
injective objects and hence we can define the qth cohomology group
H^q(X, F) as the qth right derived functor R^qGamma(X, -) applied
to the abelian sheaf F.
- An object L of an abelian category A with enough injectives
is said to be F-acyclic if R^qF(L) = 0 for q > 0.
Leray's acyclicity lemma says that one may compute the right
derived functors R^qF using right resolutions by acyclic objects.
(It is a good exercise to try to prove this by a method called
dimension shifting.)
- A sheaf of abelian groups F on a topological space X
is called flasque or flabby if the restriction maps
are all surjective. If this is true then F is acyclic for
the functor Gamma(X, -), in other words, H^q(X, F) = 0 for all q > 0.
This is proved here
(please don't read about the general construction of derived categories,
just read the argument on the linked page and translate it into the language
we have used in class).
- Upshot: we may compute cohomology of abelian sheaves
using flasque resolutions.
- The construction of enough injective abelian sheaves (or injective
sheaves of modules) and also of enough flasque abelian sheaves
uses the following simple construction: given for each point x of
a topological space an abelian group A_x
we can consider the presheaf G defined by the rule
U ↦ ∏_{x in U} A_x. Then G is a
flasque abelian sheaf! If A_x is an injective abelian group
(i.e., an injective object of the category of abelian groups)
for all x, then G is an injective abelian sheaf (injective object
in the category of abelian sheaves). Using this it is straightforward
to embed any abelian sheaf in a flasque or in an injective abelian sheaf
and I explained this a bit more in the lecture.
- We prove the following lemma: given a continuous map f : X → Y
of topological spaces, the pushforward f_*I of an injective abelian
sheaf is an injective abelian sheaf. We used a clever lemma on adjoint
functors to prove this, see
015Z
- If 0 → F_1 → F_2 → F_3 → 0 is a short
exact sequence of Ab(X), then we get a long exact cohomology sequence
of cohomology involving boundary maps H^q(X, F_3) → H^{q + 1}(X, F_1)
defined using the next point
- Given a short exact sequence 0 → M_1 → M_2 → M_3 → 0
of an abelian category A having enough injectives, there exists a short
exact sequence 0 → I_1^* → I_2^* → I_3^* → 0
of injective resolutions M_i → I_i^*. Then each of the short exact
sequences 0 → I_1^q → I_2^q → I_3^q → 0 is a split
short exact sequence. It follows that on applying an additive functor
F : A → B we obtain a short exact sequence
0 → F(I_1^*) → F(I_2^*) → F(I_3^*) → 0
of complexes of B. Then taking cohomology we obtain the long
exact sequence of right derived functors
0 → R^0F(M_1) → R^0F(M_2) → R^0F(M_3) →
R^1F(M_1) → R^1F(M_2) → R^1F(M_3) → ...
- A trickier thing to prove is that the boundary
maps R^qF(M_3) → R^{q + 1}F(M_1) are well defined.
- Going back to our cohomology of topological spaces, we want
to show that cohomology has something to do with the topology of X.
The key step to doing this is the following point.
- Given an injective abelian sheaf I on X and an open U of X, then
I|_U is an injective abelian sheaf on U. To prove this use the same
clever lemma about abelian categories as above and use that restriction
to U has an exact left adjoint, namely, extension by zero. See
01E0.
- Thus given an injective resolution
0 → F → I^* of an abelian sheaf F on X
the restriction 0 → F|_U → I^*|_U is one too.
Hence there are obvious maps H^q(X, F) → H^q(U, F).
We denote these maps ξ ↦ ξ|_U.
Please make very sure you understand what this means.
- Locality of cohomology: Given ξ in H^q(X, F) where q > 0
there is an open covering X = ⋃ U_i such that
ξ|_{U_i} is zero for all i. Same link as above.
- Cech cohomology: given an open covering 𝒰 : X = ⋃ U_i
and a total ordering on the indices, we can define the Cech
complex C^*(𝒰, F) whose degree p term is the product of
the sections of F over the p = 1 fold intersections of distinct opens.
We define H^q(𝒰, F) = H^q(C^*(𝒰, F)). (There should be
a ``check'' on the H but I don't know how to do this in html.)
Some refs:
01ED,
09UD,
01FG (this
is the section discussing the canonical isomorphism between
ordered and unordered Cech complexes -- it is terrible)
- It is easy to show that H^0(X, F) = H^0(𝒰, F)
for any open covering 𝒰.
This corresponds exactly to the sheaf condition for F.
- It turns out that there always is an injective map
H^1(𝒰, F) → H^1(X, F) and that the image
consists exactly of the elements ξ which die on the
members of the covering. Hence locality of cohomology
tells us that all elements of H^1(X, F) come from the
first Cech cohomology group for some open covering of X.
This is our first evidence that cohomology has something
to do with the combinatorics of interesections of opens of X.
- Lecture XIII: The question asked this lecture
was: Why is the cohomology of a coherent sheaf on a proper
variety finite dimensional?
It turns out that this is a very difficult question. In fact, you
can say that a lot of the theory of (quasi-)coherent modules and their
cohomology builds towards answering this question. The discussion
that follows is a "top down", in other words, we discuss the ingredients
needed to prove the theorem in reverse order.
- An important theorem is Chow's lemma. This lemma says that given
a proper variety X there is a projective variety Y and a surjective
morphism f : Y → X of varieties.
- For the proof of the theorem it is desirable to have variants of
Chow's lemma. A good variant is the following: let X be a separated
and finite type scheme over a Noetherian ring A. Then there exists
a morphism f : Y → X such that
- f is a proper morphism of schemes,
- f is an isomorphism over a dense open U of X and
- Y is quasi-projective over A which we will define to mean that
there exists a locally closed immersion from Y into projective space
over A.
In this situation, the morphism f is actually projective (in all
possible meanings of the word projective) and moreover, if X is
proper over A, then Y will be projective over A, in other words
the locally closed immersion into projective space will be a closed immersion.
See 02O2
- Another important ingredient is to use devissage of coherent modules.
Let X be a Noetherian scheme. Consider the abelian category Coh(O_X) of
coherent modules on X. Let P be a property of coherent modules.
Assume the property P satisfies the following properties:
- given a short exact sequence of coherent modules
0 → F_1 → F_2 → F_3 → 0
and P(F_i) holds for 2-out-of-3 then it holds for the third,
- for every closed irreducible and reduced subscheme Z of X
we have P(O_Z)
then P holds for every object of Coh(O_X). This is just an example satement;
the idea is to use induction on the dimension of the support of objects
of Coh(O_X) and to try to filter any coherent module such that the graded
pieces are "easier". This is discussed in
01YC and
in particular see
01YI for
a slight generalization of the statement above (which is useful
in the discussion below).
- Another very important ingredient is the Leray spectral sequence
and its friends. One version is that given a morphism f : Y → X
of topological spaces and a sheaf of abelian groups F on Y, then we
have a spectral sequence with E_2 terms E_2^{p, q} = H^p(X, R^qf_*F)
converging to H^{p + q}(Y, F). Some remarks
- The functors R^qf_* are the right derived functors of the
pushforward functor f_* constructed exactly as in the previous lecture.
- Oten the only thing that matters is the following consequence: there
is a finite (separated and exhaustive) filtration on H^n(Y, F) whose
graded pieces are subquotients of the E_2^{p, q} with p + q = n.
- So for example, if we know that E^{p, q} is finite (as a module
or something), then we'll know the same thing for H^n(X, F).
- Suppose that we know for some n that H^n(Y, F) is finite and
E_2^{p, q} is finite for pairs (p, q) with p < n and q > 0.
Then E_2^{n, 0} is finite. (This is related to the construction of the
subquotients in the statement above.)
- Similarly given composable morphisms f and g one has a spectral
sequence E_2^{p, q} = R^pg_*R^qf_*F converging to R^{p + q}(g \circ f)_*F.
- How do we combine the above three ingredients to prove the finiteness
for coherent modules on a proper variety X given a similar finiteness
statement for projective morphisms? The idea is to choose, for every
irreducible and reduced closed subvariety Z of X a morphism
f : Y → Z as in the refined version of Chow's lemma.
Then f_*O_Y is a coherent module on Z (this is already a nontrivial fact,
see below) which is isomorphic to O_Z on an open of Z.
Using the Leray spectral sequence we reduce to showing that
R^qf_*O_Y is coherent for all q and that H^n(Y, O_Y) is finite for all n.
- The discussion above reduces us to proving the following
two statements
- given a (locally) projective morphism f : X → Y of
Noetherian schemes, then R^qf_*F is coherent on Y for F coherent on X, and
- given a closed subscheme X of P^n_A where A is Noetherian and
given a coherent O_X-module F, show that H^n(X, F) is a finite A-module
for all n.
- In 6a the sheaf R^qf_*F is the sheaf associated to the presheaf
which sends V to H^q(f^{-1}V, F). Thus if we have 6b and if we define
locally projective to mean locally embeddable in a projective space, then
it is sort of clear how to deduce 6a from 6b.
- A very important and very often used fact about cohomology of
quasi-coherent modules is that H^n(X, F) = 0 for n > 0
when X is an affine scheme and F is a quasi-coherent module.
I want to stress here that this is a nontrivial fact and stated
in this generality it is not in Hartshorne IIRC.
- Suppose f : X → Y is an affine morphism and F quasi-coherent
on X. Then since R^qf_*F is the sheaf associated to the presheaf
V ↦ H^q(f^{-1}V, F) and since we have a basis of affine opens V
for which f^{-1}V is affine, then we see that R^qf_*F = 0 for q > 0.
- To prove 6b we consider the closed embedding i : X → P^n_A
and we use the Leray spectral sequence and the vanishing of higher
direct images along the affine morphism i (recall this is how
we defined closed immersions in class: as affine morphisms with an
additional property) to see that H^n(X, F) = H^n(P_A^n, i_*F).
- The previous point now reduces us to the following problem:
given a coherent module F on P^n_A with A a Noetherian ring, show
that H^q(P^n_A, F) is a finite A-module for all q.
We need three further results before we can complete the proof.
- If O(d) denotes the d-th twist of the structure sheaf on
P^n_A, then we can explicitly compute the cohomology of this and it
is a finite A-module in each degree.
- If a scheme X has an open covering by n + 1 affine opens such that
all (multiple) intersections of these opens are also affine (for example
this is true if X is separated), then H^q(X, F) = 0 for q > n.
This can be deduced by using Mayer-Vietoris n times starting with the
vanishing of cohomology on affines we stated above. It is a good
exercise to work this out yourself.
- Given any coherent module F on P^n_A there exist integers
r, d_1, ..., d_r and a surjection
Π : O(d_1) ⊕ ... ⊕ O(d_r) → F.
- The end of the proof is to use descending induction on
the cohomological degree. Namely, we know the cohomology of F
is zero in degrees > n. Thus the finiteness holds in degree n + 1
and higher. Denote G is the kernel of the surjection Π. This is
a coherent module as the category of coherent modules is an
abelian subcategory of the category of all modules. Then the long
exact sequence of cohomology shows H^q(F) is sandwiched between
H^q(O(d_1) ⊕ ... ⊕ O(d_r)) which is finite by direct
computation and H^{q + 1}(G) which is finite by induction.
This finishes the proof.
Observe that this step doesn't work if A is not Noetherian, but lots
of the other things I said above also don't work for nonNoetherian
schemes and modules. Anyway, this is a lot of material to grok in
just one lecture. It seems to me we should (perhaps?)
turn to some more computional questions next time to see what
is really going on.
- Lecture XIV: During this meeting we tried to have more of
a discussion by limiting the length of the answers. This also means
we covered more different topics and I may have forgotten a few in the
list below; if you remember one of these, please send me a quick email. Thanks!
- Question: What is the relationship between Pic(X) and H^1(X, O_X^*)?
Answer: they are the same. More precisely, given any open covering
𝒰 : X = ⋃ U_i of a locally ringed space X there is a
1-to-1 correspondence between isomorphism classes of rank 1 locally free
O_X-modules L which are trivial on each U_i and elements of
H^1(𝒰, O_X^*). Namely, you map L to the Cech cohomology class
of the cocycle defined by the gleuing functions. Please make sure
you understand this fairly well because it is used very often.
- Question: What is the analogue of the Whitney embedding theorem
in algebraic geometry? Answer: there is none. More precisely, here
are some different answers
- give a base field k there isn't a known list of (proper) varieties
such that every (proper) variety has a closed embedding into one of the list
- it turns out that there is a countable collection
of proper varieties such that every proper variety embeds into one
of them, but this is a mere existence statement and nobody has any
clue what the varieties on the list look like
- in some sense the point of looking at (quasi-)projective varieties
is that you can embed them into an easy/simple variety, namely
projective space
- Question: what are some tools that you use to compute cohomology
of sheaves in algebraic geometry? Answer: often you are completely stuck.
Question: what about excision? Answer: you can do cohomology with supports.
- If Z is a closed subset of a topological space X with open complement
U, then we always have a long exact sequence of compactly supported
cohology H^i_c(U) → H^i_c(X) → H^i_c(Z) → H^{i + 1}_c(U)
- The above is similar to the sequence
H_i(U) → H_i(X) → H_i(X, U) → H_{i - 1}(U)
in homology where one sometimes has excision
- To do exision with cohomology as we've been working with
in this course, you use cohomology with supports. The functors
H^i_Z(X, -) are defined as the right derived functors of H^0_Z(X, -).
Here H^0_Z(X, F) ⊂ H^0(X, F) is the subgroup of sections whose
support is contained in the closed subset Z. This is a left exact
functor. There is a long exact sequence
H^i_Z(X, F) → H^i(X, F) → H^i(U, F) → H^{i + 1}_Z(X, F).
Finally, if X' ⊂ X is an open subspace containing Z then
we have H^i_Z(X', F) = H^i_Z(X, F) which is a very simple way to
do excision.
- Question: what is the relationship between Cohomology and Cech Cohomology?
Answer: there are several ways to look at this.
- The most satisfying answer to me is that there is a Cech-to-Cohomology
spectral sequence. More precisely, given an open covering
𝒰 : X = ⋃ U_i of a topological space X and an abelian
sheaf F on X there is a spectral sequence
E_2^{p, q} = H^p(𝒰, H^q(-, F)) converging to H^{p + q}(X, F).
- If F is a flasque sheaf, then all higher Cech cohomology groups
are zero (for any open covering of any open).
- Suppose we have a basis B for the topology of X closed
under finite intersections and consisting of quasi-compact opens.
If F is an abelian sheaf such that its higher Cech cohomology groups
are zero for finite open covers of elements of B by elements of B,
then H^q(U, F) = 0 for all U in B and all q > 0.
See for example
01EO and
09SV. You
can prove b for finite open coverings without using the
spectral sequence using a fun induction argument; this is a
good excercise or you can read a proof of (something close to)
this in Vakil's notes. Then you can prove c using b
and a dimension shifting argument; ask me if you are in doubt.
- Let X be an affine scheme, let 𝒰 : X = ⋃ U_i
be a finite affine open covering, and let F be a quasi-coherent
module. Then we proved that the higher Cech cohomologies of F
with respect to 𝒰 are trivial.
- Combinging the previous results we concluded that
the higher cohomologies of a quasi-coherent module on an
affine scheme are zero.
- Given a curve C of degree d in P^2_k over a field k
with homogeneous equation F of degree d we considered
the short exact sequence
0 → O(-d) → O → O_C → 0
of coherent modules on P^2. Then we decided this
would allow you to compute H^0 and H^1 of O_C by
the long exact cohomology sequence and our knowledge
of the cohomology groups of the twists of the structure
sheaf on P^n_k.
- Lecture XV: During this meeting we tried to discuss an
example, rather than more general theory.
- The question: how does the number of nodes on a curve
influence the genus?
- The short answer: when you add a node, then you add 1 to the genus.
- Before interpreting the answer given above, please consider
that our definitions are as follows.
- Let X be a proper $1$-dimensional scheme over a field k such
that H^0(X, O_X) = k. Then the genus of X is g = \dim_k H^1(X, O_X).
- This invariant is sometimes called the arithmetic genus
of X.
- If X is as in 4 and X is smooth over k, then everybody agrees
that g is the genus of X.
- My advisor Frans Oort and many other people have a different
definition of a variety over a field k. These people say: besides
being integral and separated, of finite type over k, it has to be
the case that the base change X \times_{Spec(k)} \Spec(\overline{k})
to the algebraic closure \overline{k} is a variety too.
Let us call such a thing a FO-variety.
- We discussed how X = P^1_{Q(i)} is a variety over Q in
the sense defined above (earlier in the course), but not a
FO-variety over Q. Of course, if we view X as a variety
over Q(i) then it is a FO-variety and a fortiori a variety.
- We proved that P^1_{Q(i)} is a projective scheme over Q
(if you don't remember how, then this is a great exercise).
- Returning to the genus, suppose that X is a FO-curve over k.
Then it turns out that there is a unique smooth and projective
FO-curve X' over k which is birational to X. It turns out that
X' is just the normalization of X in case X is projective
(or equivalenty proper).
- If X and X' are as in 10 then we define
the geometric genus of X to be the genus of X'
(as defined in 4 above).
- In the lecture we computed an example of a pair C = X and X'
as above as follows:
- let k be any field of characteristic not 2 or 3,
- let X' = P^1_k.
- consider the map ψ X' → P^2_k given on homogeneous
coordinates by [s, t] ↦ [s^3 + t^3 : s^2t : st^2]
- because X' is proper over k the image of ψ is closed
- we found the equation for the image being
X_0X_1X_2 = X_1^3 + X_2^2.
- denote C the curve in P^2 defined by the equation
X_0X_1X_2 = X_1^3 + X_2^2
- we considered the restriction of C to the open
where X_0 is not zero
- using affine coordinates x_1 = X_1/X_0 and x_2 = X_2/X_0
we obtained x_1x_2 = x_1^3 + x_2^3 as the equation for this
affine piece
- the inverse image of this open of C in X' = P^1_k
is the open given by s^3 + t^3 not zero
- this affine open is the spectrum of the ring
(k[s, t, 1/(s^3 + t^3)])_0
- this ring has generators A = s^3/(s^3 + t^3),
B = s^2t/(s^3 + t^3),
C = st^2/(s^3 + t^3),
D = t^3/(s^3 + t^3),
- equations among these are A + D = 1,
AD = BC, B^2 = AC, C^2 = BD,
- we computed the singularities of the curve x_1x_2 = x_1^3 + x_2^3
and we found it has a unique singular point corresponding to x_1 = x_2 = 0
- the map ψ on the affine pieces above is given by the map
x_1 ↦ B and x_2 ↦ C
- the inverse image of the singular point is therefore
the locus B = C = 0 in the affine curve above
- this produces the points (A, B, C, D) = (1, 0, 0, 0) and
(A, B, C, D) = (0, 0, 0, 1); these are the points 0 and ∞
on X' = P^1_k. This suggests that one gets the curve C by glueing
0 and ∞ to each other on X' = P^1_k (this is true in a very
precise sense which you can ask me about later)
- the genus of X' is 0
- the genus of C is 1
- finally we computed the genus of a plane curve C in P^2_k
- first we defined a plane curve to be a closed subscheme
of P^2_k which is also a curve (in the sense above)
- then we proved that given a plane curve there exists an
integer d and a homogeneous polynomial F of degree d such that
C is the zero set of F
- we mentioned the fact that in this situation it is actually
the case that C is Proj(k[X_0, X_1, X_2]/(F)) as a scheme
- then we observed that there is a short exact sequence
of graded modules 0 → k[X_0, X_1, X_2](-d) →
k[X_0, X_1, X_2] → k[X_0, X_1, X_2]/(F) → 0
- given a graded module M the shift M(e) is the graded
module with graded parts M(e)_d = M_{e + d}
- there is an exact functor from graded modules M over
the graded ring k[X_0, X_1, X_2] to quasi-coherent modules
on P^2_k. If \widetilde{M} is the sheaf associated to M
then we have
\widetilde{M}(D(F)) = (M[1/F])_0
- Applying the functor above to the short exact sequence
of graded modules we obtain the short exact sequence
0 → O_{P^2_k}(-d) → O_{P^2_k} → O_C → 0
- Taking the long exact cohomology sequence of this
sequence we obtain an isomorphism H^1(C, O_C) →
H^2(P^2_k, O_{P^2_k}(-d))
- Since the dimension of H^2(P^2_k, O_{P^2_k}(-d))
is (d - 1)(d - 2)/2 we conclude the genus of C
is (d - 1)(d - 2)/2.
- Lecture XVI: Unfortunately, I am blanking a bit on the
topics we discussed in this lecture. Please let me know if I am missing
some.
- We discussed the K\"unneth formula:
- If X and Y are affine varieties over a field k, then the
product XxY of X and Y, i.e., the fibre product of X and Y over Spec(k)
in the category of schemes, is corresponds to A ⊗_k B if X and Y
correspond to the finite type k-algebras A and B.
- If F and G are quasi-coherent modules over X and Y as above, then
pr_1^*F ⊗ pr_2^*G on XxY corresponds to the A ⊗_k B-module
M ⊗_k N if F and G correspond to M and N.
- The point above already proves the Kunneth formula for quasi-coherent
modules over affine varieties! Moreover, it is the starting point for
the proof in general, using Cech cohomology.
- Now let X and Y be general varieties and choose affine open
coverings X = ⋃ U_i with n terms and Y = ⋃ V_j with m
terms. Denote C(F) and C(G) the Cech complex for X and Y, F and G, and
these coverings.
Then we get an affine open covering XxY = ⋃ U_i x V_j
with nm terms. Moreover, using the affine case we see that the modules
Gamma(U_{i_0...i_p} x V_{j_0...j_p}, pr_1^*F ⊗ pr_2^*G)
which occur in the Cech complex C(F, G) of this affine open covering,
are tensor products of the terms in C(F) and C(G). This is good...
- It is *not* true that C(F, G) is the total complex of the double complex.
- A first way around this, is explained in the proof of
Tag 0BED.
Essentially, you carefully construct the map and then you
use induction on the degree of the cohomology (using an
intersting fact about cohomology of quasi-coherent modules
on varieties).
- What is true is that C(F) and C(G) are complexes
associated to simplicial k-vector spaces and that C(F, G)
is the complex associated to the tensor product of these
simplicial vector spaces in the category of simplicial
vector spaces!
- This fact combined with some simplicial arguments can be used
to give a second proof of the Kunneth formula, but I don't
have a good reference for this now.
- Fun fact, which is used in one of the proofs above:
Tag 0BDY
tells us that on a quasi-compact scheme with affine diagonal
taking cohomology of quasi-coherent modules is a universal
delta functor. More, precisely, every quasi-coherent module
has an embedding into a quasi-coherent module whose higher
cohomology groups are zero. (This is not a triviality because
we've defined cohomology using injective resolutions in the
category of all O_X-modules and "most" O_X-modules aren't
quasi-coherent.)
- Using Kunneth, we computed all cohomology groups of
all invertible modules on P^1xP^1. Please try to internalize
this, because it comes up a lot!
- Using this information, we discussed the genus of a
curve C lying on P^1xP^1. This means C is a closed subscheme
which is also a curve (a variety of dimension 1 and hence reduced
and irreducible):
- There is an exact sequence
0 → I → O_{P^1xP^1} → O_C → 0
- I = O_{P^1xP^1}(-a, -b) for some integers a, b
because I is an invertible module (as C is locally defined
by a single equation; sorry here we use a different notation
from the lecture)
- SheafHom(I, O_{P^1xP^1}) = O_{P^1xP^1}(a, b) has a nonzero
element and hence we conclude a, b are both ≥ 0
- if k is algebraically closed then a = 0 implies b = 1
and b = 0 implies a = 1 and in both cases the curve C is
isomorphic to P^1 (please think this through -- of course
this case is not so interesting)
- if neither a nor b is 0 then we conclude that H^0(C, O_C)
is equal to k and we conclude that the genus of C is
(a - 1)(b - 1)
- We discussed how finding a curve C of bidegree (a, b)
is the same as finding a bihomogeneous polynomial
F in k[X_0, X_1, Y_0, Y_1] of bidgree (a, b) which is moreover
irreducible (as a polynomial).
- To see whether C corresponding to F is smooth you can
dehomogenize the equation (in both sets of variables, each in
2 possible ways for a total of 4 affine pieces covering P^1xP^1)
and do the thing with derivatives we did before (the Jacobian
criterion for smoothness).
- In fact, it is true for every genus g ≥ 0 and for every
field there exists at least one curve (whose genus is defined)
of that genus over that field.
- Then there was a question as to how this works when you
are looking at projective curve in some projective space defined
by more than one equation. The answer is that it is tricky in general
to compute the genus. There are bounds on the genus in terms of
the degrees and number of equations, but there isn't a general
formula.
- If however, the curve is scheme theoretically cut out by n - 1
equations F_1,...,F_{n - 1} in P^n where the degree of F_i is d_i,
then there is a formula. To prove the formula you use exactness of
the Koszul complex (which I briefly discussed in the lecture -- ask
me again if you are interested). The formula says that
2g - 2 = d_1 d_2 ... d_{n - 1} (d_1 + ... + d_{n - 1} - n - 1)
where g is the genus.
- Lecture XVII:
We started this lecture with 20 questions where you guys had to
guess what scheme I had in mind. It turned out that my schemes
was the degree 5 Fermat surface over the field with 7 elements.
After this we discussed some other topics:
- When we say X : F = 0 where F is a homogeneous polynomial
of degree d in T_0, ..., T_n over a field or ring k then we mean
either of the following two equivalent things
- X is the unique closed subscheme of P^n_k which on each standard
affine piece D_+(T_i) = Spec(k[T_0/T_i, ..., T_n/T_i]) is given by
the zero scheme of the polynomial T_i^{-d}F, or
- X is the Proj of the graded ring k[T_0, ..., T_n]/(F)
- in the situation above, we say that X is a hypersurface
of degree d
- when we say X : F_1 = ... = F_m = 0 where F_1, ..., F_m are
homogeneous polynomials in T_0, ..., T_n over a field or ring k
then we mean the analogous thing to what we said above; we will sometimes
refer to this as saying that F_1, ..., F_m cut out X
scheme theoretically (as opposed to set
theoretically)
- Question: when can you parametrize a variety?
- Answer: this would mean the variety is rational!
- Fact: given a variety X of dimension n over a field k,
the following are equivalent
- there is a nonempty open U of affine n space A^n_k and
an open immersion U → X,
- X and P^n_k have isomorphic open subschemes,
- the function field k(X) of X is a purely inseparable extension of k
If this happens we say that X is rational over k
- Fact: given a variety X of dimension n over a field k,
the following are equivalent
- there is a nonempty open U of affine n space A^n_k and
a dominant morphism U → X,
- there is an embedding k(X) → k(t_1, ..., t_n) over k
If this happens we say that X is unirational over k
- Nontrivial facts: there are non-unirational varieties
and there are unirational varieties which are not rational
- A field extension K/k is
finitely generated as a field extension
if there is a finite subset E = {a_1, ..., a_n} ⊂ K
such that any subfield K' ⊂ K which contains both k and E
is equal to K
- Let K/k be a finitely generated field extension. Then there
is a variety X with k(X) = K. Namely, choose E = {a_1, ..., a_n} ⊂ K
as above and denote A ⊂ K the k-subalgebra generated by E.
Then X = Spec(A) works.
- Let K/k be a finitely generated field extension. Then there
is a projective variety X with k(X) = K. Namely, above we have
seen that there is an affine variety U with k(U) = K.
Then we can embed U into A^n_k for some n. Then we denote
X ⊂ P^n_k the (scheme theoretic) closure of U.
Then you check: (a) X is a variety, (b) U is open dense in X.
Hence k(U) = k(X) and the proof is done. Please work this out
and ask questions if you get stuck.
- Some references for the two points above:
Tag 0BXM
Tag 01RR
- After this general discussion we turned to a discussion for curves.
First we need to understand better how function fields correspond to
actual curves. Restricting to an algebraically closed ground field k
we have
- Let K/k be a finitely generated field extension of transcendence
degree 1. Then there is a unique (up to unique isomorphism) smooth
projective curve X over k with k(X) = K.
- In the situation above X is called the smooth projective model
of K.
- If K_1 → K_2 is a morphism of finitely generated field extensions
of k of transcendence degree 1, and if X_1 and X_2 denote their corresponding
smooth projective models, then there is a unique nonconstant morphism
X_2 → X_1 which induces the given map on function field.
This means that we have an anti-equivalence of categories
between finitely generated field extension of k of transcendence
degree 1 and smooth projective curves over k with nonconstant morphisms.
Reference: Tag 0BXX
- Fact: if X → Y is a nonconstant morphism of smooth projective curves
over k algebraically closed, then the genus of X is ≥ the genus of Y.
- Fact: if X is a smooth projective curves
over k algebraically closed of genus 0 then X is isomorphic to P^1_k.
- Conclusion from all of the above: if k is algebraically closed, then
a curve is unirational if and only if it is rational if and only if
the function field of X is isomorphic to k(t). Moreover if X is smooth
and projective then X is isomorphic to P^1_k.
- Question: how can we get closer to the cohomology we know
and love from our understanding of P^1 and curves as Riemann surfaces
and similarly for higher dimensional varieties?
- Answer: in order to discuss this in a first year algebraic geometry
course, the best thing is probably to talk about the module of differentials
and the de Rahm complex. Here are some topics you could ask about next time:
- how do we define Ω^1_{X/S}?
- what is the algebraic de Rham complex?
- what is coherent duality?
- why is the dualizing sheaf related to the modules of differentials?
- how can we use all of this to get a ``cohomology theory'' H^*
for smooth projective varieties over an algebraically closed field k
which for a smooth projective curve X gives betti numbers
1, 2g, 1, 0, 0, ... where g = g(X) is the genus?
- Lecture XIX:
We started this lecture with 20 questions where you guys had to
guess what scheme I had in mind. It turned out that my schemes
was the spectrum of the dual numbers over the complex numbers.
After this we discussed some other topics:
- Let k be a field. The dual numbers over k is the
k-algebra k[ε] where ε is a variable whose square
is zero.
- Let X be a scheme over a field k. Let x be a point of X whose
residue field is equal to k as a k-algebra. Then x is a closed point
of X, see exercises. A tangent vector at x is an element of
the k-vector space which is the k-linear dual to the k-vector space
m_x/(m_x)^2 where m_x ⊂ O_{X, x} is the maximal ideal.
- Fact: the set of all morphisms Spec(k[ε]) → X
over Spec(k) is in natural bijection with the set of pairs (x, θ)
where x is a closed point with residue field k and θ is a
tangent vector at x.
- This can be generalized, see
0B28
and please do some exercises in
029C.
- As you can see in the reference just given this geometric
notion of tangent vectors is related to an algebraic construction
of differentials.
- If R → A is a ring map, then we can define the module
of differentials Ω_{A/R} as the target of the universal
R-derivation d : A → Ω_{A/R}.
- I'm afraid now you'll actually have to read a bit of algebra
in order to understand what is going to follow next. Any good
book on algebra will carefully introduce d : A → Ω_{A/R}
and explain its properties. In the Stacks project you can read a bit
here: 00RM
- We proved that Ω_{A/R} if A = R[x_1, ..., x_n] is a free
A-module with basis d(x_1), ..., d(x_n).
- We indicated that Ω_{A/R} if A = R[x_1, ..., x_n]/(f_1, ..., f_m)
is the A-module with generators d(x_1), ..., d(x_n) and relations
∑ ∂_i(f_j)d(x_i) = 0.
- Fact: if A is a finite type algebra over a field k which is a domain
of dimension n then A is smooth over k if and only if Ω_{A/k}
is a finite locally free A-module of rank n.
- Given a ring map R → A the
de Rham complex of A over R
is the differential graded R-algebra
Ω_{A/R}^* whose terms are the exterior powers of
Ω_{A/R} and whose differential is given by
d(a_0 d(a_1) ∧ ... ∧ d(a_p)) =
d(a_0) ∧ d(a_1) ∧ ... ∧ d(a_p).
Please look this up in a good algebra book. In the Stacks project
you can look here
07HX
for the absolute case (namely where R = Z) and you can easily
generalize this to the general case (you can use the lemma following
the remark in the Stacks project to do this if you like).
- The de Rham cohomology H^*_{dR}(A/R) of A over R is the cohomology
of the de Rham complex of A over R.
- In the rest of the lecture we computed the de Rham cohomology
of A over R in a few cases:
- If A = R we get H^0_{dR}(A/R) = A and zero in all other degrees
- If k is a field of characteristic zero, then
H^0_{dR}(k[x]/k) = k and we get zero in all other degrees
- If k is a field of characteristic p, then
H^0_{dR}(k[x]/k) = k[x^p] and
H^1_{dR}(k[x]/k) = k[x^p]x^{p - 1}d(x) and
we get zero in all other degrees
- If k is a field of characteristic zero and A = k[x, x^{-1}]
then we get H^0_{dR}(A/k) = k and
H^1_{dR}(A/k) = k x^{-1}d(x) = k dlog(x) where dlog(x) is just a shorthand
for the element x^{-1}d(x) in Ω_{A/k}.
- Lecture XIX: The topic of this lecture was the relationship
between schemes of finite type over the complex numbers C and the
usual topology on C.
- Denote Sch/C the category of schemes of finite type over C
- Denote Top the category of topological spaces
- A map f : X → Y of Top will be called an open embedding
if f is a homeomorphism onto an open subset of Y
- A map f : X → Y of Top will be called an closed embedding
if f is a homeomorphism onto a closed subset of Y
- The goal of this lecture was to construct a functor
F : Sch/C → Top with the following properties:
- F(X) = X(C) functorially in X (identification as a set)
- F(A^n_C) = C^n with the standard topology
- F sends closed immersions to closed embeddings
- F sends open immersions to open embeddings
- It turns out that there is a unique functor F having these properties.
This functor is often denoted X ↦ X^{an} or simply X ↦ X(C).
- Using this topology, we can do some fun things. For example
- we can show that X is separated if and only if X(C) is Hausdorff
- we can show that X is proper over Spec(C) if and only if X(C)
is compact and Hausdorff
- we can consider the singular cohomology of the topological space
X(C) as a (contravariant) functor on Sch/C. Sometimes this is called
the Betti cohomology, so H^*_{Betti}(X) = H^*_{sing}(X(C), Z).
- Grothendieck proved that you can recover complexified Betti cohomology
H^*_{Betti}(X) ⊗_Z C as the de Rham cohomology H^*_{dR}(X/C)
for smooth varieties over C and in fact this was my motivation
for talking about the construction of the topology on X(C) in this lecture.
We will return to de Rham cohomology in the future.
- The construction of the functor F works more generally when you have
a field C which has a Hausdorff topology such that addition, substraction,
multiplication, and taking inverse are continuous (on their domains).
- First we prove a meta result: if we can construct a functor
F satisfying properties a, b, c, d on the category Aff/C of finite
type affine schemes over C, then it automatically extends uniquely
to the whole category Sch/C. I omit the proof; hint: first extend the
construction to separated objects of Sch/C by using that you get these
by glueing affine schemes along affine schemes and after that extend
to all objects of Sch/C by using that you get any X by glueing affine
opens along separated opens.
- Next, we prove that any family of polynomials f_1, ..., f_m in variables
x_1, ..., x_n defines a continuous map C^n → C^m by the properties of
the topology on C we mentioned above.
- For X in Aff/C choose a closed immersion X → A^n_C. Then
you show that X(C) ⊂ C^n is closed in the usual topology
because it is the vanishing set of some polynomials and {0}
is a closed subset of C.
- In the situation above, we can try to define the topology
on X(C) to be the one induced from C^n. Let's show this is
independent of the choice of closed immersion.
- For X in Aff/C suppose we have two closed immersions
X → A^n_C and X → A^m_C. Then we can consider the
product closed immersion X → A^{n + m}_C. Hence we
need only show that the topology τ induced from the map
X(C) → C^{n + m} and the topology τ' induced from the map
X(C) → C^n are the same. Clearly, the identity map
of X(C) is a continuous map (X(C), τ) → (X(C), τ').
To show that (X(C), τ') → (X(C), τ) is also
continuous, you can use that the additional coordinate functions
x_{n + 1}, ..., x_{n + m} on A^{n + m}_C restrict to polynomials
in x_1, ..., x_n on X because C[x_1, ..., x_n] surjects on to
H^0(X, O_X). Hence these functions (which are continuous in the
τ topology by fiat) are continuous in the τ' topology which
proves what we want. (This is slightly better than what I said
in the lecture.)
- OK, so now we have a well defined topology on X(C) for
any object X of Aff/C. An easy argument using suitable commutative
diagrams show that this topology is functorial in other words
a morphism X → Y of Aff/C is turned into a continuous map
X(C) → Y(C). Similarly, it is easy to show that property
b and c above hold.
- Suppose that X = Spec(A) and U = D(h) = Spec(A_h) is a principal
open. Property d says that U(C) should be identified with an open
subspace in X(C). This is more or less clear: the map h : X → A^1_C
turns into a continuous map h : X(C) → C. Then U(C) is the
inverse image of C - {0} which is open (as the topology on C is Hausdorff).
Thus U(C) → X(C) is a continuous bijection onto an open subset.
This isn't quite enough to show that it is a homeomorphism; to see
it you can use that U is a closed subscheme of X x A^1_C defined by
the vanishing of ht - 1 where t is the coordinate on A^1_C.
Then U(C) ⊂ X(C) x C is closed (by what we've already shown
about the topology so far) and you show that using t = h^{-1}
on the complement of (h = 0) in X(C) gives a continuous inverse.
(This is slightly better than in the lecture as we argue directly
on X and we don't use an embedding in affine n space.)
- Now there is a theoretical argument that shows in order to
esthablish d for all open immersions in Aff/C, it suffices to prove d for
the open immersions of principal opens. This can be a confusing and
long winded thing and is better done on a napkin than here in these notes.
- Lecture XX: The question that motivated this lecture was:
Have we proved in the lectures that Pic(P^1 x P^1) = Z ⊕ Z?
The answer is that we haven't proved it or rather we haven't discussed
it; see the end of this lecture for one possible proof.
Motivated by this question we discussed the following:
- Fact: given a variety X we have Pic(X x P^1) = Pic(X) ⊕ Z.
This is not actually that easy to show with the tools we have developed
so far (it is also not that hard). The case of X = P^1 is discussed
at the end.
- Let X and Y be schemes over a field k. Denote X x Y the fibre product
over Spec(k). Then pulling back by the projection maps X x Y → X and
X x Y → Y we obtain a map Pic(X) ⊕ Pic(Y) → Pic(X x Y).
- If in the situation above X has a k-rational point x_0 and
Y has a k-rational point y_0, then the inclusion morphisms
X = X x {y_0} → X x Y and Y = {x_0} x Y → X x Y
define a map Pic(X x Y) → Pic(X) ⊕ Pic(Y) which is inverse
to the map in the previous point. Hence in this case
Pic(X x Y) = Pic(X) ⊕ Pic(Y) ⊕ EXTRA
for some abelian group EXTRA.
- Here during the lecture we had a short discussion of how
Pic is a quadratic functor on the category of smooth projective
varieties and how the same is true for H^2( - , Z) on the category
of connected pointed (reasonable) topological spaces.
- The discussion above in particular applies with X and Y are varieties
and k is algebraically closed (because then we always have k-rational points).
- Warning: EXTRA can be nonzero even if X and Y are smooth projective
varieties over an algebraically closed field.
- Question: what is an example?
- Answer: Let X = Y = C be a smooth projective curve over an algebraically
closed field k of genus g_C > 0. Then EXTRA is nonzero.
- Consider the diagonal Δ ⊂ C x C. This is an
effective Cartier divisor, i.e., its sheaf of ideals I ⊂
O_{C x C} is an invertible module on C x C. Denote O_{C x C}(Δ)
the dual invertible module, i.e., the inverse in Pic(C x C) of I.
- Please read enough about (effective) Cartier divisors in order
to understand the discussion above and below about the relationship
between divisors on (smooth) varieties and their associated invertible
modules. For example Hartshorne has a section on this (and you don't
need to read all of it). In the Stacks project you can look at
(parts of)
01WQ
0C4S
0B3Q
- Claim: O_{C x C}(Δ) is an element of Pic(C x C) which is
not contained in the summand Pic(C) ⊕ Pic(C) constructed above
provided g_C > 0.
- To prove the claim choose a base point c_0 in C. (In other words,
let c_0 be a k-rational point on C.) Then using c_0 in the construction of
the inverse to the inclusion map Pic(C) ⊕ Pic(C) → Pic(C x C)
given above, we obtain: If the claim is wrong, then
O_{C x C}(Δ) must be isomorphic to
pr_1^*O_C(c_0) ⊗ pr_2^*O_C(c_0).
- The proof of the previous fact was done using a picture: namely
the diagonal Δ intersections the curve C x {c_0} in the point
(c_0, c_0). Hence the pullback of the divisor Δ on the surface
C x C via the inclusion C = C x {c_0} → C x C is the divisor
c_0 on C. This then implies the corresponding relation in the Picard group
(see for example Definition
01WV and Lemmas
02OO
and 0C4U).
- Sublemma: \dim H^0(C, O_C(c_0) = 1 if g_C > 0 and
\dim H^0(C, O_C(c_0) = 2 if g_C = 0.
- Assuming the sublemma we conclude using Kunneth that
H^0(C x C, pr_1^*O_C(c_0) ⊗ pr_2^*O_C(c_0)) is one
dimensional when g_C > 0. Thus there is a unique (up to scale)
nonvanishing section σ of this invertible module. Namely, σ
is the product of the pullback of the unique (up to scale) nonvanishing
section of O_C(c_0) via the two projections. The vanishing
locus of σ is therefore {c_0} x C ∪ C x {c_0}.
Since O_{C x C}(Δ) has a canonical section vanishing exactly
along Δ we conclude that the claim is true.
- Proof of the sublemma. A global section of O_C(c_0) is an
element f of k(C) wich has no poles except possibly at c_0 where the
pole order is at most 1. An example is f = 1. Now if the dimension of
H^0(C, O_C(c_0) is > 1, then we can find a nonconstant f with this
property. Such an f would correspond to a nonconstant morphism
f : C → P^1_k by the equivalence of categories we talked about
in Lecture XVII. The condition on the pole orders implies that
f^*(∞) = c_0 where here we are using pullbacks of divisors on curves
that we will discuss below. This then finally implies that f has
degree 1, in other words f is a birational maps, in other words,
C has genus 0 as desired.
- Discussion of facts about pullbacks of divisors via a
nonconstant morphism f : X → Y of smooth projective curves
over an algebraically closed field k; you should really
try to know these facts and look up their proofs in Hartshorne
for example.
- f^* : Div(Y) → Div(X) is defined by sending
D = ∑ n_i[y_i] to f^*D = ∑ n_if^*[y_i]
- f^*[y] = ∑ e_x [x] where the sum is over the
points x mapping to y
- e_x is the ramification index of f at x which
is defined as the ramification index of the map of dvrs
O_{Y, y} → O_{X, x}
- if A → B is an extension of dvrs then the
ramification index of B over A
is the integer e ≥ 0 such that π_A = (unit) π_B^e, see
09E4
- FACT (sum e_i f_i = n): for any point y of Y we have
∑ e_x = [k(X) : k(Y)] where the sum is over the points x in X
mapping to y.
- Definition: the degree of f : X → Y is [k(X) : k(Y)]
- We conclude that deg(f^*D) = deg(f) deg(D) for any divisor
D on Y
- Fact: f^*div_Y(g) = div_X(g) where g is an element of k(Y)
which we may also consider as an element of k(X)
- Cor: f^* induces a map f^* : Cl(Y) → Cl(X)
- Fact: f^* on Weil divisor class groups agrees with f^* on
Picard groups via the identification of those we've seen in a
previous lecture. All this is saying is the fact that pullback
of (effective) Cartier divisors and taking the associated invertible
modules commute which we saw above in a more general setting.
- Proof of Pic(P^1 x P^1) = Z ⊕ Z. We've seen that
Pic(P^1_k) = Z and hence the discussion above gives a direct
sum decomposition Pic(P^1 x P^1) = Z ⊕ Z ⊕ Extra.
Thus it suffices to show that Pic(P^1 x P^1) has
at most two generators. For this the easiest thing is to
prove the following two things
- Show that for a smooth variety X, such as P^1 x P^1, we
have Pic(X) = Cl(X) where Cl(X) is the Weil divisor class
group. This you prove in exactly the same way as we proved
this fact for nonsingular curves; see for example
0BE9
and use that the local rings of a smooth variety are regular
rings and in particular UFDs.
- Show that if X has an open subvariety U with Pic(U) = Cl(U) = 0
then Cl(X) is generated as an abelian group by the classes of
the irreducible components Z of the complement of U which have
codimension 1 in X. Namely, if D is a Weil divisor, then we
can write D ∩ U = div_U(f) for some f in k(U) = k(X)
and then we see that D - div_X(f) is a Weil divisor rationally
equivalent to D supported on X - U and hence D is rationally
equivalent to a Weil divisor which is a sum of the irreducible
components of the complement of U in X.
Now you apply this to the open U = A^1 x A^1 of P^1 x P^1.
The same argument proves more generally that
Pic(P^{n_1} x P^{n_2} x ... x P^{n_r}) = Z^r.
- Lecture XXI: In this lecture we circled back to the
de Rham complex.
- First there was a question: how does one recognize an
effective Cartier divisor?
- Answer: the easiest thing to remember is that if X is
a smooth variety, then any closed subvariety of codimension 1
is an effective Cartier divisor.
- The reason for the answer: local rings of a smooth variety
are UFDs and this is what locally gives you the single equation
for the codimension 1 subvariety...
- Example: the diagonal of a smooth projective curve over an
algebraically closed field is an effective Cartier divisor.
- The de Rham complex of a scheme over another scheme.
- Let X → S be a morphism of schemes.
- The module of differentials of X/S is a
quasi-coherent O_X-module Ω_{X/S} which comes
equipped with an S-derivation d_{X/S} : O_X → Ω_{X/S}
- This S-derivation is the universal S-derivation, see
01UM
if you're interested.
- You can construct Ω_{X/S} and d_{X/S} by using
affine locally the construction in algebra and then glueing.
If you want to do this you have to show the following two things:
- S^{-1} Ω_{A/R} = Ω_{S^{-1}A/R} if S is a multiplicative
subset of A, and
- if R → A factors as R → S^{-1} R → A for some
multiplicative subset S of R, then Ω_{A/R} = Ω_{A/S^{-1}R}
- Another option for constructing Ω_{X/S} is to
use Ω_{X/S} = Δ^*(I) where I ⊂ O_{X x_S X} is the ideal
sheaf of the diagonal (which is a closed immersion if X is
separated over S so let's assume that here; you can extend this easily
to the general case by working with a suitable open of X x_S X).
- Once you have constructed Ω_{X/S} and d_{X/S}
it is a simple matter to construct the
de Rham complex of X/S by taking exterior powers
and exterior derivation.
- We computed that for X = P^1_k and S = Spec(k) where k is
any ring, then Ω_{P^1_k/k} is isomorphic to O(-2).
We did this in two ways
- If x is a coordinate on P^1, then by the glueing method we have
d(x) is a generator of Ω_{P^1_k/k} over the affine open with
coordinate x but d(x) = -y^{-2}d(y) on the other affine open with
coordinate y = x^{-1}. Hence d(x) is a meromorphic section of the
invertible module Ω_{P^1_k/k} which has a pole of order exactly
2 along the point at infinity. Thus we get the result.
- Using the description Ω_{X/S} = Δ^*(I)
and using that O_{P^1 x P^1}(Δ) = O_{P^1 x P^1}(1, 1)
we get I = O_{P^1 x P^1}(-1, -1) and hence the pullback of
this givens O(-2). [Here we noted that the diagonal Δ
in P^1 x P^1 is the zero locus of the section X_0Y_1 - X_1Y_0
of O_{P^1 x P^1}(1, 1) to get the correct invertible module.]
- For a smooth projective curve E of genus 1 over an algebraically
closed field k we have that Ω_{E/k} is isomorphic to O_E.
We didn't prove this completely but we computed that
Ω_{A/k} is free with basis element (1/y)d(x) when
A = k[x, y]/(y^2 - x^3 + 1) and the characteristic of k
is not 2 or 3.
- Lecture XXII: In this lecture we tried to compute
the de Rham cohomology of a couple of smooth projective curves.
- For any morphism of schemes X → S there is a de Rham complex
Ω_{X/S}.
- For any topological space X and bounded below complex of abelian
sheaves F^* on X there is a way to define the cohomology of F^* on X:
- Choose an injective resolution F^* → I^* as defined
in Definition 013I
- Set H^n(X, F^*) = H^n(Γ(X, I^*)).
There is a way to do this for complexes of abelian sheaves which are not
bounded below, but we'll not discuss this in these lectures.
- Question: Shouldn't you resolve F^* by a complex
of complexes of injective sheaves?
- No, what was said above is the definition, please understand this first.
For example, read about it in
Section 013G.
- Yes, in some sense you are correct. Namely, suppose we choose for
each p an injective resolution F^p → I^{p, *}, then for each p we
may choose a map of complexes I^{p, *} → I^{p + 1, *} compatible
with the map F^p → F^{p + 1, *}. It turns out that you can do this
in such a way that the compositions
I^{p, *} → I^{p + 1, *} → I^{p + 2, *}
are zero (as maps of complexes). Then I^{*, *} is a double complex
(a complex of complexes as you were asking for above) and as our
injective resolution (as defined above) of F^* we can take
F^* → Tot(I^{*, *})
where Tot(I^{*, *}) is the totalization of the double complex: it is the
complex which in degree n has the direct sum of I^{p, q} with p + q = n.
- The discussion above immediately gives that there exists
a spectral sequence with E_1^{p, q} = H^q(X, F^p) with differentials
d_1^{p, q} : E_1^{p, q} → E_1^{p + 1, q} converging to
H^{p + q(X, F^*).
- Going back to the de Rham complex Ω_{X/S} on our scheme X
we define the de Rham cohomology of X/S H^n_{dR}(X/S)
as the cohomology of the complex Ω_{X/S} on X
- In the case of the de Rham complex, the spectral sequence above
is the Hodge to de Rham spectral sequence and it looks like
E_1^{p, q} = H^q(X, Ω_{X/S}^p) converging to H^n_{dR}(X/S).
- We worked out what this spectral sequence does for P^1_k/k
and we found that it degenerates at E_1 because the only nonzero
terms are E_1^{0, 0} = H^0(P^1, O) and E_1^{1, 1} = H^1(P^1, Ω^1)
which are both equal to k. Thus we see that H^n_{dR}(P^1/k) is k in
degrees 0, 2 and zero else.
- Let X be a smooth projective curve of genus 1 over an algebraically
closed field k and assume we have proven that Ω_{X/k} is isomorphic
to O_X (this is always true, but our arguments from last time didn't
completely show this). Then we found that the Hodge to de Rham spectral
sequence has nonvanishing terms E_1^{p, q} for (p, q) in
{(0, 0), (0, 1), (1, 0), (1, 1)}. The differential d_1^{0, 0} is
zero because the (exterior) derivative of a constant function is zero.
However, it is not completely obvious that the other differential
d : H^1(X, O_X) → H^1(X, Ω^1_{X/k}) is zero.
- More generally, for a smooth projective curve of genus g
over an algebraically closed field k, the Hodge to de Rham spectral
sequence has nonvanishing terms E_1^{p, q} for (p, q) in
{(0, 0), (0, 1), (1, 0), (1, 1)} having dimensions 1, g, ?, ?.
To see that we get betti numbers 1, 2g, 1 and Poincare duality
for de Rham cohomology (as we expect from the known picture over
the complex numbers) we discussed the following:
- There should be a duality on the de Rham complex itself
which should be a consequence of coherent duality on X.
- For a general smooth proper variety X/k of dimension n
this coherent duality should give a duality between
H^q(X, Ω^p_{X/k}) and H^{n - q}(X, Ω^{n - p}_{X/k}
- Once we have this for curves we find the Hodge to de Rham spectral
sequence has nonvanishing terms E_1^{p, q} for (p, q) in
{(0, 0), (0, 1), (1, 0), (1, 1)} having dimensions 1, g, g, 1.
Then we'll still need to show that the differential
d : H^1(X, O_X) → H^1(X, Ω^1_{X/k}) is zero.
- The vanishing of the differentials on the E_1 page of
the Hodge to de Rham spectral sequence does not always hold.
When we say ``degeneration of Hodge-to-de-Rham'' we are considering
the statement that it does hold. In characteristic zero (i.e., when
k contains the rational numbers), then it always does hold for smooth
and proper X. The first proof of this fact used Hodge theory. A
marvelous proof of this fact by Deligne and Illusie uses characteristic
p methods to prove it in characteristic zero (which is strange because
the result doesn't always hold in characteristic p).
- The Hodge to de Rham spectral sequence for a smooth affine variety
X = Spec(A) over k shows that there is a canonical isomorphism
H^n_{dR}(X/k) = H^n(Ω^*_{A/k}) thereby linking back the
global construction in this lecture to the algebraic one earlier in
the lectures. In particular, our earlier calculations show that
de Rham cohomology of the affine line in characteristic p is
infinite dimensional! It turns out that for affine singular varieties in
characteristic zero the de Rham cohomology can also be infinite dimensional.
- Recall that Grothendieck proved that for a smooth variety X
(not necessarily projective or proper) over the
complex numbers C the algebraic de Rham cohomology H^n_{dR}(X/C) is the
same as the singular cohomology with C coefficients of X(C)
endowed with the usal topology. In particular, H^n_{dR}(X/C)
is finite dimensional. The same is true for any smooth variety
over any field of characteristic zero.
- Lecture XXIII: In this lecture we talked about various and sundry.
- First we talked a bit more about injective resolutions as defined
in the lecture XXII. The question asked was: why given
a left exact functor F : A → B of abelian categories and
a quasi-isomorphism α: I^* → J^* of bounded below complexes
of injective objects of A, is it true that F(I^*) → F(J^*)
is a quasi-isomorphism?
- Answer 1: consider the cone C(α) on α
which fits into a triangle
I^* → J^* → C(α) → I^*[1], see
Section 014D.
Then you show: (a) the triangle gives rise to a long exact sequence
of cohomology objects in A, (b) the triangle
F(I^*) → F(J^*) → F(C(α)) → F(I^*)[1]
gives rise to a long exact sequence of cohomology objects in B,
(c) the cohomology objects of C(α)
are zero as α is a quasi-isomorphism,
(d) C(α) is a bounded below complex of injectives,
(e) F(C(α)) = C(F(α)).
Having said all of this it remains to show that for a
bounded below acyclic (all coh is 0) complex of injectives I^*
we have F(I^*) = 0. To see this last fact you decompose
any complex I^* like this into a sequence of split short exact
sequence for which the statement is obvious.
- Answer 2: Try to show that α: I^* → J^*
as above has an inverse up to homotopy. The the same
will be true for F(α) and we get what we want.
- Then we talked about blowing up. Please take a look at
- Harthorne's discussion of blow ups or Ravi's notes, or
Shavarevich, etc
- Affine blow up algebras as discussed in
Section 052P
- Blowing up as discussed in
Section 01OF
- Specifically, we tried to blow up the spectrum of
k[x, y, z]/(xy - z^2) in the ideal (x, z) and we found
that we got a nonsingular (smooth) scheme over k.
- I suggest doing some blowing ups yourself. You can
do standard blowing ups such as blowing up the spectrum
of k[x, y] in the ideal (x, y) but you can also do crazy
things such as blowing up the ideal generated by ε
in the spectrum of the dual numbers.
- Lectures XXIV and XXV: In these lectures we talked about
- the Euler characteristic of a coherent sheaf on a proper varierty
over an field,
- the relationship between Hilbert polynomials of finite graded modules
over graded polynomial rings and Hilbert polynomials of the associated
coherent sheaves on projective space (in terms of Euler characteristics),
- Riemann-Roch on smooth projective curves over algebraically closed
fields formulated as a relationship between the Euler characteristic
of an invertible sheaf, its degree, and the Euler characteristic of
the structure sheaf,
- Riemann-Roch on a smooth projective curve where we plug in
everything we now know about the genus, duality, etc,
- Riemann-Hurewitz for a nonconstant and separable morphism between
smooth projective curves over an algebraically closed field
where we worked out what the local structure of the morphism implies
about the local vanishing of the induced map on Ωs...
- That's all folks!
Homework:
- Due Thursday, Jan 31. Choose and do 2 exercises from
027A
and do
028P
028Q
028R
028W
- Due Thursday, Feb 7. Do
02CJ,
0E9D,
0E9E,
0E9F, and
give an example of a real valued function on the real line
which is a quotient of polynomials but not a polynomial.
I'm choosing these particular problems in the hope that these
will generate confusion and therefore questions in the lectures.
- Due Thursday, Feb 14. Prove the algebra statements (*)
and (**) from Lecture VI above. Do exercises
078S and
02DU
- Due Thursday, Feb 21. Do the following
- Let K/k be a finitely generated extension of fields.
Show that there exists a variety X over k such that
k(X) is isomorphic to K.
- Let A ⊂ B be an extension of domains such that
(i) B is a finitely generated A-algebra and
(ii) the incusion induces an isomorphism of fraction fields.
Show that there exists some nonzero f in A such that A_f = B_f.
- Let X and Y be varieties over the same ground field k.
Show that if X and Y are birational, i.e., if
k(X) and k(Y) are isomorphic as extensions of k,
then there exist nonempty affine open subschemes U ⊂ X and
V &subl; Y which are isomorphic as varieties over k.
(Hint: use the previous exercise twice.)
- Let k be your favorite algebraically closed field.
Find a surface X over k such that there does not exist
any nonconstant morphism from the affine line over k
into X. Here the affine line over k is just A^1_k = Spec(k[t])
as usual.
- Find a surjective morphism from A^1_k to P^1_k.
(This is one of my favorite questions; it'll be a little
bit hard to answer for you, because you'll have to think
or read about what it means to map into projective space.)
- Due Thursday, Feb 28. Do the following
- Let X be a scheme over a field k. Show that k-rationl points
of X correspond 1-to-1 with points x in X such that the map
from k to the residue field of x is an isomorphism. Moreover, show
that such points are closed points of X.
- Do exercise
0CYH
- Due Thursday, March 7. Do 5 of the exercises of
0D8P
- Due Thursday, March 14. Do 5 of the exercises of
0DAI
- Due Thursday, March 28. Explicitly find a smooth curve of genus 6
over the rational numbers. Explicitly find a smooth surface of degree 4
in P^3 over the field with 2 elements.
- Due Thursday, April 4.
- Let X be a scheme over a field k. Let x be a point of X whose residue
field is equal to k as a k-algebra. Show that x is a closed point.
- Do two exercises from
Section 029C
- Due Thursday, April 13.
- Do three exercised from
0293
- Do exercise 02EO
- The exercises for Thursday, April 18.
- Prepare a specific algebraic goemetry
question to ask during the lecture on Thursday.
- If you are an undergraduate: think about a topic to write about
for your final paper and email me about it. I would prefer the topic
to be very close to what we talked about during the lectures -- the
best is probably if you just carefully work out a bit of theory and/or
examples that came up. I intend to have a meeting
with you about this, so please suggest dates/times when you are free.
- If you are a graduate student and registered for this course:
please email me with dates/times where we can have our oral exam about
the material in this course. If you have questions about how this will
work, then please ask during the lectures.
References:
Algebraic Geometry, by Robin Hartshorne
springer link
Basic Algebraic Geometry I, II, by Igor Shavarevich
springer link
springer link
EGA, for example you can find this on
this page
Stacks project, see
Stacks project
Ravi's notes, see
download page
Mumford's Red book
springer link