- Fri., May. 5:
- A 1949 paper by André Weil gave evidence for the existence of
“topological cohomology” in algebraic geometry linked to the notion
of zeta function for a non-singular projective algebraic variety
defined over a finite field.
Let X be a scheme of finite type over Z. For each element x \in X
the residue field at x is the fraction field of an algebra of finite
type over Z. Thus, the residue field at a closed element x
is a field that is an algebra of finite type over Z, i.e., a finite
field. One defines the zeta function of X by
|
zeta_{X}(s) = PROD_{x closed in X}[ {1}/{1 - N(x)^{-s}} ]
|
where N(x) = |kappa(x)| is the number of elements of the
residue field of X at x. (Ignore questions of convergence for
now.) When X = Spec Z, zeta_{X}(s) is Riemann's zeta
function. When X is a scheme of finite type over F_{q}, each
residue field at a closed element is a finite extension field of
F_{q}, and, therefore, N(x) = q^{d(x)} where d(x) is the
extension degree. With t = q^{-s} one writes
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zeta_{X}(s) = Z_{X}(t) = PROD_{x closed}[ {1}/{1 - t^{d(x)}} ]
|
With the condition Z_{X}(0) = 1 the Z form of the zeta function is
determined by its logarithmic derivative
|
|
| SUM_{x closed}[ d(x) {t^{d(x)-1}}/{1 - t^{d(x)}} ] |
|
|
|
| {1}/{t} SUM_{r >= 1}[ SUM_{{x closed | d(x) = r}}[ r {t^{r}}/{1 - t^{r}} ]] |
|
|
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| {1}/{t} SUM_{r >= 1}[ r c_{r} {t^{r}}/{1 - t^{r}} ] |
|
|
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| {1}/{t} SUM_{r >= 1}[ r c_{r} SUM_{s >= 1}^{rs}[ t ]] |
|
|
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| {1}/{t} SUM_{nu >= 1}^{nu}[ SUM_{r divides nu}[ r c_{r} ] t ] |
|
|
|
| SUM_{nu >= 1}^{nu-1}[ N_{nu} t ] |
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where c_{r} denotes the number of closed elements in X with d(x) = r
and N_{nu} denotes the number of points of X
with values in the degree nu extension of F_{q}.
For a beginning example, when X = A^{n}, one has N_{nu} = q^{nnu},
and, therefore,
|
Z_{A^{n}}(t) = {1}/{1-q^{n}t} .
|
Of course, A^{n} is not a projective variety for n >= 1.
When F is a field, the set of F-valued points of P^{n} is
the disjoint union of A^{0}(F), A^{1}(F), …, A^{n}(F).
Therefore, Dlog Z_{P^{n}}(t) (over F_{q}) is the sum
of Dlog Z_{A^{j}}(t) for 0 <= j <= n. Hence,
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Z_{P^{n}}(t) = {1}/{(1 - t) (1 - q t) … (1 - q^{n} t)} .
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For P^{1} \times P^{1}, one has N_{nu} = (1 + q^{nu})^{2}, and, therefore
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Z_{(P^{1} \times P^{1})}(t) = {1}/{(1 - t)(1 - q t)^{2}(1 - q^{2} t)} .
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For curves of genus 1 defined over finite fields, the shape of its
Z function was established before the time of Weil's conjectures.
For example, in the case of the curve E given by the Weierstrass
equation y^{2} = x^{3} - 2 x over the field F_{5}, simply by counting
points to see that |E(F_{5})| = 10, it is a consequence of the
theoretical framework that Z(t) is the rational function
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Z_{E}(t) = {1 + 4 t + 5t^{2}}/{(1 - t) (1 - 5t)} .
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For each of these last examples P^{n}, P^{1} \times P^{1}, and E
one may observe that Z_{X}(t), relative to the field F_{q} is a
rational function in one variable and that:
the denominator is the product of polynomials whose degrees
are the classical topological Betti numbers of the base extension
X_{C} of X for even dimensions.
the numerator is the product of polynomials whose degrees
are the classical topological Betti numbers of the base extension
X_{C} of X for odd dimensions.
the polynomial factor corresponding to classical cohomology in
dimension j has the form of the characteristic polynomial of a
linear endomorphism varphi of the form det(1 - tvarphi) with
complex reciprocal roots all of absolute value q^{j/2}.
- Wed., May. 3:
- Beyond the theory of curves of genus 1 a good bit of what is involved
in the study of curves and of complete non-singular varieties in general
is studying the group Div(X)/Div_{ell}(X).
For curves one has
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Div_{ell}(X) <= Div_{0}(X) <= Div(X)
|
where the quotient for the second step is the discrete group Z when
Div_{0}(X) is defined as the group of divisors of degree 0.
It turns out that the quotient for the first step is a complete irreducible
group variety of dimension g, and, thus, one cannot study curves in depth
without studying varieties of higher dimension.
For varieties of dimension greater than 1, defining the degree of a
divisor as the sum of its coefficients will not lead in the right
direction. One would like a definition of Div_{0}(X) such
that the first step is a complete irreducible variety and the second
step a finitely-generated abelian group, but there is no hope with
these two conditions that the second step will always be cyclic since
for the case X = {P}_{k}^{1} \times {P}_{k}^{1} one will find that
Div(X)/Div_{ell}(X) ~= Z \times Z.
For the purpose of gaining insight about Div(X)/Div_{ell}(X)
in the theory of curves while at the same time beginning to understand
what might be required for defining Div_{0}(X) when
dim(X) > 1, consider what is available with transcendental methods
when k = C. Complex exponentiation provides the short exact sequence
of abelian sheaves for the classical (locally Euclidean) topology on X:
|
0 --> Z --> \cal{O}_{hol} \overset{e}{ --> }
\cal{O}_{hol}^{*} --> 0
|
where e(f) = e^{2pi i f}. In the long cohomology sequence the 0
stage splits off since H^{0}(X, \cal{O}_{hol}) ~= C. GAGA tells us that
coherent module cohomology matches, and although \cal{O}^{*} is
certainly not an \cal{O}-module, its H^{1} in both algebraic and
transcendental theories viewed through Czech theory classifies
isomorphism classes of invertible coherent modules. One has the
exact sequence:
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0 --> H^{1}(X, Z) --> H^{1}(X, \cal{O}_{hol}) --> H^{1}(X, \cal{O}_{hol}^{*}) --> H^{2}(X, Z) .
|
If dim(X) = 1, then H^{2}(X, Z) ~= Z, and one finds that the
last map in this sequence, a “connecting homomorphism”, sends the isomorphism
class of an invertible \cal{O}_{hol}-module to its degree. Therefore, remembering
that Div(X)/Div_{ell}(X) ~= H^{1}(X, \cal{O}^{*}),
one has
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H^{1}(X, \cal{O}_{hol})/H^{1}(X, Z) ~= Div_{0}(X)/Div_{ell}(X) ,
|
and, in fact, the left side is the quotient of a g-dimensional vector
space over C by a lattice. Thus, Div_{0}(X)/Div_{ell}(X)
is a g-dimensional complex torus; it is, moreover, a complete group
variety over C.
For dim(X) > 1 the kernel of the connecting homomorphism
will provide a correct notion of “degree 0”.
For working over an arbitrary algebraically closed field, one sees
that something is needed to replace classical cohomology. Because constant
sheaves are flasque in the Zariski topology, their Zariski-based cohomology
cannot be used.
- Mon., May. 1:
- Continuing with the discussion of the previous hour: If p, q, r
are any three points of X(k), then the triple sum p + q + r, like
any point of X(k) is characterized by the linear equivalence class
of the associated one point divisor. One has the relation
of linear equivalence
Therefore,
|
|
|
|
|
|
| + |
| + |
| = div(h) + 3 |
| for some h \in L(3 |
| ) |
|
|
|
|
| + |
| + |
| = div(s) for some s \in H^{0}(X, \cal{O}(3 |
| )) |
|
|
|
|
| + |
| + |
| = div(a x u^{3} + b y u^{3} + c u^{3}), some (a: b: c) \in \v{{P}_{k}^{2}} |
|
|
|
|
| + |
| + |
| = f^{-1}(D), D = div(a X + b Y + c Z) \in Div({P}_{k}^{2}) |
|
where f : X --> {P}_{k}^{2} is the projective embedding of X given by
the invertible \cal{O}-module \cal{O}(3 <o>). In other words, taking
multiplicities into consideration, three points sum to <o> in
the group law on X(k) if and only if the corresponding points of a
Weierstrass model in {P}_{k}^{2}, with o corresponding to the point on
the line at infinity, are collinear.
From this description of the group law on X(k), in view of the fact that
the third point of a cubic on the line through two given points (tangent
if the two points coincide) depends rationally on the coordinates of
the given points, it follows that
Addition X \times X --> X and negation X --> X
are morphisms of varieties over k.
If F is the field generated over the prime field by the
coefficients a_{0}, …, a_{6} of the Weierstrass equation, then
The Weierstrass equation defines a scheme X_{F} of finite type over
F whose base extension to k is X.
For each extension E of F the set X_{F}(E) is a group in a
functorial way.
X_{F}(k) ~= X(k).
- Fri., Apr. 28:
- Continuing with curves of genus 1, we wish to change notation so
that the projective embedding of the previous hour is given by the
very ample invertible sheaf \cal{O}(3<o>), o \in X(k).
This notational change notwithstanding, o is an arbitrary point.
Under the projective embedding given by \cal{O}(3<o>), one has
f(o) = (0: 0: 1), the unique point of f(X) on the line at infinity.
We wish to show that there is a unique commutative group law on the set
X(k) for which the map varphi : Div(X) --> X(k)
|
D = SUM_{p \in X(k)}[ n_{p} |
| ] :---> varphi(D) = SUM_{p \in X(k)}[ n_{p} p ] ,
|
which is tautologically a group homomorphism, has the property that
varphi(D_{1}) = varphi(D_{2}) whenever D_{1} EQUIV D_{2} (linear equivalence),
and further the property that o is the zero element in X(k).
(This is not the strongest statement of this type that can be made.)
Addition in X(k) is defined by observing that since for given p, q
\in X(k) the divisor <p> + <q> - <o> has degree 1, its
complete linear system consists of a single non-negative divisor of
degree 1, i.e., <r>, and this unique r \in X(k) is defined
to be p + q. Since the properties specified for varphi make this definition
necessary if, indeed, it defines a group.
It is straightforward to verify that the addition is associative,
that o is its identity, and that -p is given by the unique member
of the complete linear system |2<o> - <p>| . It is obvious
that this group law on X(k) is commutative and that varphi is surjective.
If Div_{0}(X) denotes the group of divisors of degree 0, then
since varphi(D) = varphi(D - (deg D)<o>), one sees that the restriction
varphi_{0} of varphi to Div_{0}(X) is a surjective homomorphism.
Let Div_{ell}(X) denote the group of divisors linearly equivalent
to zero. It is trivial that the map D :--> D - (deg D)<o>
defines a homomorphism Div(X) --> Div_{0}(X) which,
when followed with reduction provides a homomorphism
Div(X) --> Div_{0}(X)/Div_{ell}(X).
It is not difficult to verify that another homomorphism between this
latter pair of groups is given by
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D :--> |
| - |
| mod Div_{ell}(X) .
|
(That this is a homomorphism follows from reviewing the definition of
varphi(D_{1}) + varphi(D_{2}).) Since these two homomorphisms agree on divisors
of the form <p> – which generate the free abelian group
Div(X) –, one has for all D \in Div(X) that
We know that deg D depends only on the linear equivalence class
of D as the first consequence of the Riemann-Roch Theorem.
Since r \in X(k) is determined uniquely by the linear equivalence
class of <r>, this formula tells us that varphi(D) depends only
on the linear equivalence class of D. However, the formula also tells us
that the linear equivalence class of D depends only on varphi(D) and
deg(D). In particular, one has
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Div_{0}(X)/Div_{ell}(X) ~= X(k) .
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- Wed., Apr. 26:
- Suppose that X is a complete non-singular curve over an algebraically
closed field k of genus 1. The range of degrees where a divisor
D has H^{1}(\cal{O}(D)) ~= (0) is deg(D) >= 1, while we
have dim H^{1}(\cal{O}) = 1. For each a \in X(k) the invertible
module \cal{O}(2<a>) has no base point, and, therefore, defines a
morphism to {P}_{k}^{1}. One has a two step filtration of the 3-dimensional
linear subspace L(3<a>) of k(X):
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k = L(0) = L( |
| ) <= L(2 |
| ) <= L(3 |
| ) .
|
Choosing x \in L(2<a>)-L(0) and y \in L(3<a>) - L(2<a>)
one obtains a filtration-compatible basis {1, x, y} of L(3<a>),
and if u is a “rational section” of \cal{O}(<a>) with
div(u) = <a>, the morphism f : X --> {P}_{k}^{2} given by
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f = (Z: X: Y), Z = u^{3}, X = x u^{3}, Y = y u^{3}
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provides a projective embedding of X by the theorem of the last hour.
Extending the filtration inside k(X) by the L(m <a>), one sees
that {1, x, y, x^{2}, xy, x^{3}} is a filtration-compatible basis of
L(6 <a>). Since y^{2} \in L(6<a>) - L(5<a>), one has
a linear relation among monomials of degree 3
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Y^{2} Z + a_{1} X Y Z + a_{3} Y Z^{2} = a_{0} X^{3} + a_{2} X^{2} Z + a_{4} X Z^{2} + a_{6} Z^{3}
|
with a_{0} ≠ 0 that characterizes f(X) as a non-singular hypersurface
in {P}_{k}^{2}. One says that f(X) is in generalized Weierstrass form.
One regards Z = 0 as the “line at infinity” in {P}_{k}^{2}, while
one calls “affine” a point (X,Y) = (1: X: Y). The intersection of
f(X) with the line at infinity reduces to the equation a_{0} X^{3} = 0.
Therefore, the point (0: 0: 1) is the only point of f(X) on the line
at infinity, and as the point of intersection of the line at infinity
with f(X) it has multiplicity 3.
- Mon., Apr. 24:
- Continuing with the case of a complete normal
curve over an algebraically closed field k. When D is a divisor
with deg(D) >= 2g, then for each a \in X(k) one has
deg(D - <a>) >= 2g - 1, and, therefore, L(D -
<a>) is a hyperplane in L(D). Otherwise, said \cal{O}(D) has
no base point. A coordinate-free interpretation of the morphism
f : X --> {P}_{k}^{N}, where N = deg(D) - g, given by a basis of
H^{0}(X, \cal{O}(D)) is that f(a) is the hyperplane H^{0}(X, \cal{O}(D-<a>))
regarded as a point in the projective space of hyperplanes through the
origin in H^{0}(X, \cal{O}(D)). If, moreover, deg(D) >= 2g + 1,
then for a ≠ b in X(k) it follows that H^{0}(X, \cal{O}(D-<a>-<b>))
has codimension 2 in H^{0}(X, \cal{O}(D)) so that f(a) and f(b) must
be different points, i.e., f is injective. Since X is complete,
f(X) must be a closed subvariety of dimension 1 in {P}_{k}^{N}.
The fact that H^{0}(X, \cal{O}(D - 2<a>)) also has codimension 2 in
H^{0}(X, \cal{O}(D)) guarantees that d_{a}(f) : T_{a}(X) --> T_{f(a)}({P}_{k}^{N})
has rank 1 for each a, and, therefore, that f(X) is itself a
complete non-singular curve. Since morphisms of complete non-singular
curves are dual to the contravariant function field extensions, f must
be an isomorphism, i.e., \cal{O}(D) is very ample when
deg(D) >= 2 g + 1. As first example, when g = 0 and
D = <a>, the morphism f given by H^{0}(X, \cal{O}(<a>)) is
an isomorphism of X with {P}_{k}^{1}.
- Fri., Apr. 21:
- In the context of a complete normal variety X over an algebraically
closed field k an invertible \cal{O}_{X}-module \cal{L} is called very
ample if there is an integer N >= 0 and a closed immersion
f : X --> {P}_{k}^{N} such that \cal{L} ~= f^{*}\cal{O}_{{P}_{k}^{N}}(1). (Recall
the earlier description of the functor of points
over k of {P}_{k}^{N}.) If \cal{L} is very ample, then \cal{L}^{\otimesm}
is also very ample for each m >= 1. One says that \cal{L} is ample
if there exists m >= 1 such that \cal{L}^{\otimesm} is very ample.
Finally, if there is an integer N >= 0 and a morphism
f : X --> {P}_{k}^{N} such that \cal{L} ~= f^{*}\cal{O}_{{P}_{k}^{N}}(1), one
says that \cal{L} has no base point. For a particular value of N
if z_{0}, … z_{N} are homogeneous coordinates in {P}_{k}^{N}, hence,
a basis of H^{0}({P}_{k}^{N}, \cal{O}_{{P}_{k}^{N}}(1)), then the s_{j} = f^{*} z_{j}
are elements of H^{0}(X, \cal{L}) that do not vanish simultaneously at
any point of X(k). It follows that the members of any basis of
H^{0}(X, \cal{L}) also have no common zero, but it does not follow that
the {s_{j}} form a basis.
When dim(X) = 1, recall that for a divisor D of negative degree
one has dim_{k}(H^{0}(X, \cal{O}(D))) = 0.
If K is a canonical divisor and D a divisor with
|
deg(D) > deg(K) = 2 g - 2 ,
|
then K - D is a divisor of negative degree, and, consequently,
by Serre duality dim_{k} H^{1}(X, \cal{O}(D)) = 0
for any divisor D with deg(D) >= 2 g - 1.
When the genus g = 1, this means that dim_{k} H^{1}(X, \cal{O}(D)) = 0
for any divisor D of degree at least 1. The Riemann Roch formula
then implies that dim H^{0}(X, \cal{O}(D)) = deg(D).
In particular if D = <a> for a \in X(k), one sees that
L(<a>) >= L(0) ~= k while both have dimension 1.
Hence, there can be no f \in k(X)^{*} with only a single simple pole.
The same type of reasoning shows that k(X)^{*} contains an element whose
only pole is a double pole at a given point a \in X(k).
- Wed., Apr. 19:
- When A is a ring and B an A-algebra, the module Omega_{B/A} is
the B-module receiving an A-derivation from B that is initially
universal for derivations from B to B-modules. When
f : X --> Y is a morphism of schemes there is an \cal{O}_{X}-module
Omega_{X/Y} that globalizes the module of differentials from
commutative algebra. A morphism f : X --> Y of irreducible
varieties over an algebraically closed field k is called
smooth if (i) f is dominant, i.e., \overline{f(X)} = Y, and
(ii) Omega_{X/Y} is a locally-free \cal{O}_{X}-module of rank
dim(X) - dim(Y). A non-singular variety over
k is a variety X that is smooth over k. (An irreducible variety
of dimension 1 is non-singular if and only if it is normal.) When
X is a non-singular variety, one defines
{Omega}_{X}^{p} to be the p-th exterior power
^ ^{p}Omega_{X/k}. For n = dim(X) the top exterior
power omega_{X} = {Omega}_{X}^{n} is a locally-free
\cal{O}_{X}-module that is called the canonical \cal{O}_{X}-module.
A form of Serre duality, which could be the subject of an entire course,
is this:
-
Theorem.
If X is a complete non-singular variety of dimension n
and \cal{F} a coherent \cal{O}_{X}-module, then H^{p}(X, \cal{F}) and
{Ext}_{\cal{O}}^{n-p}(\cal{F}, omega_{X}) are dual vector spaces
over k.
An important special case is that when \cal{F} is a locally-free
\cal{O}-module. Then
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{Ext}_{\cal{O}}^{n-p}(\cal{F}, omega_{X}) ~= {Ext}_{\cal{O}}^{n-p}(\cal{O}, omega_{X} \otimes \cal{F}^{∨}) ~= H^{n-p}(X, omega_{X} \otimes \cal{F}^{∨})
|
where \cal{F}^{∨} denotes the \cal{O} dual of \cal{F}.
In the case of a complete normal curve a canonical divisor is any
divisor K for which \cal{O}(K) ~= omega_{X}. When \cal{F} = \cal{O}(D) for an
arbitary divisor D, the vector spaces H^{p}(X, \cal{O}(D)) and
H^{1-p}(X, \cal{O}(K - D)) have the same dimension for p = 0, 1.
In particular one has g = dim H^{1}(X, \cal{O}_{X}) =
dim H^{0}(X, omega_{X}), and application of the Riemann-Roch formula
to a canonical divisor leads to the conclusion that any canonical divisor
must have degree 2g - 2.
- Mon., Apr. 17:
- Continuing with the case of a complete normal curve X over an algebraically
closed field, some observations:
If H^{0}(X, \cal{O}(D)) ≠ (0), then deg(D) >= 0 since
D is linearly equivalent to a non-negative divisor div(f) + D
for some f \in L(D).
The set
|
|D| = {E \in Div(X) | E >= 0, E EQUIV D}
|
is called the complete linear system determined by D. It may
be bijectively identified with the projective space of lines through the
origin in the vector space L(D) ~= H^{0}(X, \cal{O}(D)). A linear
system is a projective subspace of a complete linear system. One has
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|D| = {div(s) | s \in H^{0}(X, \cal{O}(D))} .
|
Looking at the cohomology
sequence associated with the short exact sequence
|
0 --> \cal{O}(D- |
| ) --> \cal{O}(D) --> i_{*}i^{*}\cal{O}(D) --> 0 ,
|
one sees that in going from D-<a> to D either the dimension
of H^{0} goes up by 1 or the dimension of H^{1} goes down by 1 but
not both.
To go further with complete normal curves we want to talk about
Serre duality.
- Fri., Apr. 7:
- When X is a complete normal curve over an algebraically closed field
k, a \in X a closed point, <a> the corresponding divisor,
and i : {a} --> X the corresponding closed immersion of a
subvariety, one has the exact sequence of coherent \cal{O}-modules
|
0 --> \cal{I}_{ |
| } --> \cal{O} --> i_{*}\cal{O}_{ |
| } --> 0 ,
|
and, remembering that \cal{I}_{{a}} ~= \cal{O}(-<a>), then
tensoring this exact sequence with the invertible \cal{O}-module \cal{O}(D),
D an arbitary divisor on X, one obtains
|
0 --> \cal{O}(D- |
| ) --> \cal{O}(D) --> i_{*}i^{*}\cal{O}(D) --> 0 .
|
The third term above is a skyscraper that is rank 1 on
\cal{O}_{{a}}({a}) ~= k. The relation among Euler
characteristics given by the last short exact sequence reduces to
|
chi(X, D) = chi(X, D- |
| ) + 1
|
for every divisor D and every closed point a \in X, and, thus, the
observation that chi(X, D) - deg(D) is a constant depending
only on X where
|
deg(D) = SUM_{z}[ n_{z} ] when D = SUM_{z}[ n_{z} |
| ] .
|
This provides a substantial portion of the Riemann-Roch Theorem:
|
chi(X, D) = deg(D) + 1 - g
|
where g, the genus of X, is defined as
dim_{k} H^{1}(X, \cal{O}). As a corollary of this, together with the
observation that chi(X, D) depends only on \cal{O}(D), one sees that
deg(D) depends only on \cal{O}(D), and, therefore,
deg(div(f)) = 0 for each f \in k(X)^{*}, a result that
corresponds to the statement for compact Riemann surfaces that the number
of zeroes of a meromorphic function equals the number of its poles.
For an initial understanding of the genus of a complete normal curve,
consider the exact sequence of \cal{O}-modules
|
0 --> \cal{O} --> \underline{k(X)} --> \underline{k(X)}/\cal{O} --> 0
|
from which ensues the sequence of vector spaces over k
|
0 --> k --> k(X) --> H^{0}(X, \underline{k(X)}/\cal{O}) --> H^{1}(X, \cal{O}) --> 0
|
where the last 0 is H^{1} of the constant, hence flasque, sheaf
\underline{k(X)} and H^{0}(X, \underline{k(X)}/\cal{O}) is the vector
space of “principal part specifications”. Thus, g = 0 if and only
if every principal part specification is realized by an element of
k(X). Thereby it is clear that the genus of {P}_{k}^{1} is 0.
- Wed., Apr. 5:
- For D \in Div(X), X a normal variety, one defines
|
L(D) = {f \in k(X)^{*} | div(f) + D >= 0} cup |
| .
|
L(D) is an \cal{O}(X)-module that is isomorphic to the module of
global sections of \cal{O}(D). While a (regular) section of a
locally-free \cal{O}-module of rank 1 is not represented by a single
element of k(X)^{*}, it does have local pieces that are unique up to
multiplications from \cal{O}^{*} and, consequently, has a globally
well-defined divisor. If s_{f} ≠ 0 is the section of \cal{O}(D)
corresponding biuniquely with f \in L(D), one has div(s_{f}) =
div(f) + D. One sees that dim_{k} H^{0}(X, \cal{O}(D)) > 0
if and only if D is linearly equivalent to some non-negative divisor.
A non-negative divisor D determines an \cal{O}-ideal
\cal{I}_{D} that is locally the principal ideal generated by a local equation
for D. It follows that \cal{I}_{D} is a rank 1 locally-free \cal{O}-module,
and one sees easily that it is isomorphic to \cal{O}(-D).
When X is a complete variety over a field k and \cal{M} a coherent
\cal{O}-module the k-modules H^{q}(X,\cal{M}) are finite-dimensional over
k for all q. This is a consequence of the more general fact that
direct images and higher direct images of a coherent module under a
proper morphism are coherent (see the text).
One defines the Euler characteristic of a coherent \cal{O}-module by
|
chi(X, \cal{M}) = SUM_{q = 0}^{dim(X)}[ (-1)^{q} dim_{k} H^{q}(X, \cal{M}) ] .
|
When
|
0 --> \cal{M}^{′} --> \cal{M} --> \cal{M}^{″} --> 0
|
is an exact sequence of coherent \cal{O}-modules on X, one has
|
chi(X, \cal{M}) = chi(X, \cal{M}^{′}) + chi(X, \cal{M}^{″}) .
|
- Mon., Apr. 3:
- When X is a normal variety, the affine coordinate ring \cal{O}(U)
of an open affine subvariety U is the intersection of its localizations
at the prime ideals corresponding to the irreducible closed sets in
U of codimension 1. Hence \cal{O}(X)^{*} is the kernel of the
homomorphism div. Given a divisor D \in Div(X)
and an open covering {U_{i}} of X that principalizes D,
say, D|U_{i} = div_{U_{i}}(f_{i}), it follows from the computation
of the kernel of div on the open subvariety
U_{ij} = U_{i} cap U_{j} that f_{i} = u_{ij} f_{j} (all elements of k(X))
where u_{ij} \in \cal{O}(U_{ij})^{*}. The Cech 1-cocycle u_{ij} determines
an element \cal{O}(D) of the group {H}_{Cech}^{1}(X,\cal{O}^{*})
of locally-free \cal{O}-modules of rank 1, the map
D --> \cal{O}(D) is a group homomorphism, and the sequence
|
1 --> \cal{O}(X)^{*} --> k(X)^{*} --> Div(X) --> {H}_{Cech}^{1}(X,\cal{O}^{*}) --> 1
|
is exact. One says that two divisors D_{1} and D_{2} are linearly
equivalent (and one may write D_{1} EQUIV D_{2}) if D_{2} - D_{1} = div(f)
for some f \in k(X)^{*} or, otherwise stated, if \cal{O}(D_{1}) ~= \cal{O}(D_{2}).
- Fri., Mar. 31:
- For an irreducible variety X over an algebraically closed field k,
a divisor is an element of the free abelian group Div(X)
generated by the irreducible closed sets of codimension 1. When X
is normal, the local ring at each irreducible closed set Z of
codimension 1 is a principal valuation ring, and, therefore, each
element f ≠ 0 in the function field k(X) gives rise to a divisor
|
div(f) = SUM_{Z}[ ord_{Z}(f) ] ,
|
which is called a principal divisor. The map
div : k(X)^{*} --> Div(X) is a homomorphism of abelian
groups. Since an open set U in X is also a variety, the functor
U --> Div(U) defines an abelian sheaf \underline{Div}
on X that is easily seen to be flasque.
When X is normal and Z an irreducible closed set of codimension 1,
the divisor in an open neighborhood of Z of the unique prime in \cal{O}_{Z}
is the generating divisor corresponding to Z. Thus one sees that
each divisor on X is locally principal.
- Wed., Mar. 29:
- If f : X --> Y is an affine morphism of algebraic varieties over
an algebraically closed field k, then for each quasi-coherent
\cal{O}_{X}-module \cal{F} one has an isomorphism of H^{q}(X, \cal{F}) with
H^{q}(Y, f_{*}\cal{F}). Finite morphisms and closed immersions present
important special cases. To know the cohomology of every coherent
\cal{O}_{P}-module on each projective space P = {P}_{k}^{N} is to know the
cohomology of every coherent \cal{O}_{X}-module on every projective
variety X.
- Mon., Mar. 27:
- On a Noetherian space the cohomological functor H^{q} for abelian
sheaves vanishes when q > dim(X). The E_{2} spectral
sequence for composite functors is operative when application of the
first functor to an injective object in its domain yields an object
that is acyclic for the second functor. This applies to the direct
image functor followed by the global sections functor on abelian
sheaves since the direct image of an injective abelian sheaf is
flasque.
- Fri., Mar. 24:
- On a Noetherian space (descending chain condition for closed sets)
each of the sheaf cohomology functors H^{q} on the category of abelian
sheaves commutes with direct limits.
- Wed., Mar. 22:
- More on cohomology: Every abelian sheaf on a topological space X
may be regarded as a Z-module (sheaf of modules over the constant
sheaf Z). As base cohomology one uses the derived functors of the
global sections functor in the category of Z-modules. An abelian
sheaf is flasque if its restrictions between open sets are all
surjective. Every flasque sheaf is acyclic for cohomology, and every
injective \cal{A}-module, for any sheaf of rings \cal{A} on X, is flasque.
Consequently, sheaf cohomology in the category of \cal{A}-modules is
consistent with that in the category of Z-modules.
- Mon., Mar. 20:
- If f : (X,\cal{A}) --> (Y,\cal{B}) is a morphism of ringed spaces, for every
\cal{B}-module \cal{G} there is an \cal{A}-module pullback f^{*}(\cal{G}) which
at stalk level satisfies
|
f^{*}(\cal{G})_{x} = \cal{G}_{f(x)} \otimes_{\cal{B}_{f(x)}} \cal{A}_{x} .
|
For a morphism of affine schemes pullback of quasi-coherent modules on
the target is the same thing as base extension.
For P = {P}_{k}^{N}, k an algebraically closed field, the exact sequence
|
\cal{O}_{P}^{N+1} \overset{(x_{0}, …, x_{N})}{ ----> }
\cal{O}_{P}(1) --> 0
|
given by
|
(f_{0}, …, f_{N}) :--> f_{0} x_{0} + … + f_{N} x_{N}
|
spawns, via pullback, the functor of points
of {P}_{k}^{N} over k:
a morphism varphi : X --> {P}_{k}^{N} is “the same thing” as an invertible
\cal{O}_{X}-module \cal{L} and an N+1-tuple of sections s_{0}, … s_{N}
of \cal{L} that do not “vanish” simultaneously, i.e., that provide
the exact sequence
|
\cal{O}_{X}^{N+1} \overset{(s_{0}, …, s_{N})}{ ----> }
\cal{L} --> 0 ,
|
which is the varphi-pullback of the referenced exact sequence on {P}_{k}^{N}.
For a k-valued point x \in X(k) one has
|
varphi(x) = (s_{0}(x): s_{1}(x): … : s_{N}(x)) .
|
- Fri., Mar. 17:
- The isomorphism classes of locally-free \cal{A}-modules of rank 1 form
a group. The notion of an exact sequence of \cal{A}-modules.
\cal{A}-modules form an abelian category in which every object admits an
injective resolution. The global sections functor Gamma(\cal{M}) =
\cal{M}(X) is left exact. The q-th cohomology functor X :-->
H^{q}(X,\cal{M}) is defined as the q-th right derived functor of
Gamma. Sideline example: the short exact sequence
|
0 --> Z --> \cal{O}_{hol} \overset{e}{ --> }
{\cal{O}}_{hol}^{*} --> 0
|
of Z-modules in complex analytic geometry, where e(f) = e^{2pi i f}
is the complex exponential.
- Wed., Mar. 15:
- Homomorphisms of \cal{A}-modules when \cal{A} is a sheaf of rings on a
topological space. Locally-free \cal{A}-modules of rank r and
transition matrices relative to a trivializing covering. An
invertible \cal{A}-module is a locally-free \cal{A}-module of
rank 1.
- Mon., Mar. 13:
- Class cancelled.
- Fri., Mar. 10:
- Properties and significance of the \cal{O}_{P} modules
\cal{O}_{P}(d) on P = {P}_{k}^{n} for d \in Z
where k is an algebraically closed field.
- Wed., Mar. 8:
- The concept of sheaf of modules on a ringed space. Quasi-coherent
and coherent modules on a scheme. Examples.
- Mon., Mar. 6:
- If f : X --> Y is a morphism of schemes with Y separated, then
f is universally closed if every split base extension of f is
closed. Proper morphisms. Valuative criteria for separated morphisms
and proper morphisms.
- Fri., Mar. 3:
- Separated morphisms. If f : X --> Y is an S-morphism and Y is
separated over S, then the graph of f is closed in X \times Y
and f is separated if and only if X is separated over S.
Henceforth, an algebraic variety will be assumed to be separated over
its base field; consequently, all morphisms of varieties will be
separated. In a scheme that is separated over an affine base, the
intersection of any two open affines is affine.
- Wed., Mar. 1:
- If x is an element of X, the scheme underlying an irreducible
algebraic variety, the Krull dimension of the local ring
\cal{O}_{x} is the codimension of \overline{{x}} in X. When
X is normal, the local ring at an irreducible subvariety of
codimension 1 in X is a discrete valuation ring. The set of closed
points of a complete and normal irreducible algebraic curve correspond
biuniquely with the non-trivial discrete valuation rings in its
function field that contain the ground field, and the entire structure
of such a curve as a scheme may be recovered from its function field.
- Mon., Feb. 27:
- Finite morphisms — yet another class closed under composition and
base extension. The normalization of an irreducible variety. Universally
closed morphisms. Finite morphisms are universally closed.
- Fri., Feb. 17:
- Any base extension of a morphism of finite type is also a morphism of
finite type. Case in point: the fibre of a morphism f : X --> Y
of finite type over an element y \in Y is a scheme of finite type
over the residue field kappa(y). Over its image a morphism may be
viewed as providing a family of varieties, though not a well-behaved
one without assumptions on the morphism. The notion of affine
morphism: another class of morphisms that is closed under compostion
and base extension.
- Wed., Feb. 15:
- The join of two Cartesian squares is another. Cartesian squares
provide shelter for both the geometric notion of product and the
algebraic notion of base extension. The notion of base extension
of a morphism. Example: The action of Gal(\={k}/k)
on X_{\={k}} when X is a k-scheme (and \={k} is the
algebraic closure of the field k).
- Mon., Feb. 13:
- Detailed examination of the functor of points for
E = Spec(Z[x,y]/(F(x,y))) where
F(x, y) is the polynomial F(x, y) = y^{2} - (x - a)(x - b)(x - c),
particularly in relation to base extensions of the coordinate
ring. Existence and uniqueness of products in the category of
schemes over a given scheme.
- Fri., Feb. 10:
- The notion of morphism of a scheme over a “base scheme” globalizes
the notion of homomorphism for algebras over a base ring. If S is
a scheme, the functor
given by
|
T :---> Hom_{S}(T, X) = X(T)
|
is called the functor of points of X over S. X is
determined as an S-scheme by its functor of points. If X is the
scheme associated with a variety X_{0} over an algebraically closed field
k, then X(k) = X(Spec(k)) is the set underlying X_{0}. If
K is an extension field of k, a point xi \in
X(K) determines an
element x \in X (no longer called a “point”) that is called its
center and a k-algebra homomorphism from the residue field at
x to K. In the affine case X(K) is precisely the set of naive
points of X in K.
- Wed., Feb. 8:
- A morphism from a scheme to the affine scheme Spec(A) is dual
to a ring homomorphism from A to the ring of global sections of the
scheme's structure sheaf. The scheme associated with an affine
variety over an algebraically closed field is characterized as a
reduced scheme of finite type over (the spectrum of) the field.
- Mon., Feb. 6:
- The category of schemes. Locally closed subschemes. Morphisms;
schemes over a base scheme.
- Fri., Feb. 3:
- The category of affine schemes as (1) a fully faithful subcategory
of the category of local-ringed spaces and (2) as the opposite
category of the category of commutative rings.
- Wed., Feb. 1:
- The notion of an affine scheme as a topological space equipped with
a sheaf of rings; morphisms between affine schemes.
- Mon., Jan. 30:
- The sheaf of rings associated with the spectrum of a commutative ring;
the initial ring is the ring of global sections.
- Fri., Jan. 27:
- The spectrum of a commutative ring and its Zariski topology.
- Wed., Jan. 25:
- Presheaves and sheaves; examples.
- Mon., Jan. 23:
- Overview.