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ZQRCHPvpsetlengthheadheight0bpsetlengthheadsep0bpsetlengthtopmargin-36bpsetlengthtextheight704bpUniv at Albany: Math: W. F. Hammond: Courses: Math 825Topics in Algebraic Geometry (Math 825)<brk
/> Introduction to Schemes<brk
/> Outline with CommentsSpring Semester, 20061Note: If you found this document through a web search engine,
you may not be aware of its
href="http://math.albany.edu/math/pers/hammond/course/mat825s2006/"other
presentation formats&SecRef-1;OutlineFri.,May.5:A 1949 paper by Andre Weil gave evidence for the existence of
topological cohomology in algebraic geometry linked to the notion
of zeta function for a nonsingular projective algebraic variety
defined over a finite fieldLet X be a scheme of finite type over Z For each element x 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
X(s) x closed in X11 N(x)s
where N(x) (x) is the number of elements of the
residue field of X at x (Ignore questions of convergence for
now.) When X SpecZ, X(s) is Riemanns zeta
function When X is a scheme of finite type over Fq, each
residue field at a closed element is a finite extension field of
Fq, and, therefore, N(x) qd(x) where d(x) is the
extension degree With t qs one writes
X(s) ZX(t) x closed11 td(x)
With the condition ZX(0) 1 the Z form of the zeta function is
determined by its logarithmic derivative
ddtlog ZX(t) x closed d(x) td(x)11 td(x)1tr 1x closedd(x) r r tr1 tr1tr 1 r crtr1 tr1tr 1 r crs 1 trs1t 1r divides r cr t 1 N t1
where cr denotes the number of closed elements in X with d(x) r
and N denotes the number of iref="fieldpoint"points of X
with values in the degree extension of FqFor a beginning example, when X An, one has N qn,
and, therefore,
ZAn(t) 11qnt
Of course, An is not a projective variety for n 1When F is a field, the set of Fvalued points of Pn is
the disjoint union of A0(F), A1(F), , An(F)
Therefore, Dlog ZPn(t) (over Fq) is the sum
of Dlog ZAj(t) for 0 j n Hence,
ZPn(t) 1(1 t) (1 q t) (1 qn t)For P1P1, one has N (1 q)2, and, therefore
Z(P1P1)(t) 1(1 t)(1 q t)2(1 q2 t)For curves of genus 1 defined over finite fields, the shape of its
Z function was established before the time of Weils conjectures
For example, in the case of the curve E given by the Weierstrass
equation y2 x3 2 x over the field F5, simply by counting
points to see that E(F5) 10, it is a consequence of the
theoretical framework that Z(t) is the rational function
ZE(t) 1 4 t 5t2(1 t) (1 5t)For each of these last examples Pn, P1 P1, and E
one may observe that ZX(t), relative to the field Fq 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
XC of X for even dimensionsthe numerator is the product of polynomials whose degrees
are the classical topological Betti numbers of the base extension
XC of X for odd dimensionsthe polynomial factor corresponding to classical cohomology in
dimension j has the form of the characteristic polynomial of a
linear endomorphism of the form det(1 t) with
complex reciprocal roots all of absolute value qj2Wed.,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 nonsingular varieties in general
is studying the group Div(X)Div(X)
For curves one has
Div(X) Div0(X) Div(X)
where the quotient for the second step is the discrete group Z when
Div0(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 dimensionFor 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 Div0(X) such
that the first step is a complete irreducible variety and the second
step a finitelygenerated abelian group, but there is no hope with
these two conditions that the second step will always be cyclic since
for the case X Pk1Pk1 one will find that
Div(X)Div(X) ZZFor the purpose of gaining insight about Div(X)Div(X)
in the theory of curves while at the same time beginning to understand
what might be required for defining Div0(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 ZOholeOhol 0
where e(f) e2 i f In the long cohomology sequence the 0
stage splits off since H0(X, Ohol) C GAGA tells us that
coherent module cohomology matches, and although O is
certainly not an Omodule, its H1 in both algebraic and
transcendental theories viewed through Czech theory classifies
isomorphism classes of invertible coherent modules One has the
exact sequence:
0 H1(X, Z) H1(X, Ohol) H1(X, Ohol) H2(X, Z)
If dim(X) 1, then H2(X, Z) Z, and one finds that the
last map in this sequence, a connecting homomorphism, sends the isomorphism
class of an invertible Oholmodule to its degree Therefore, remembering
that Div(X)Div(X) H1(X, O),
one has
H1(X, Ohol)H1(X, Z) Div0(X)Div(X)
and, in fact, the left side is the quotient of a gdimensional vector
space over C by a lattice Thus, Div0(X)Div(X)
is a gdimensional complex torus; it is, moreover, a complete group
variety over CFor dim(X) 1 the kernel of the connecting homomorphism
will provide a correct notion of degree 0For 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 Zariskibased cohomology
cannot be usedMon.,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
p q rpqr 2 o
Therefore,
p q r o pqr 3 opqrdiv(h) 3 o for some h L(3o)pqrdiv(s)
for some s H0(X, O(3o))pqrdiv(a x u3 b y u3 c u3), some
(a: b: c) Pk2pqr f1(D),
D div(a X b Y c Z) Div(Pk2)
where f : X Pk2 is the projective embedding of X given by
the invertible Omodule 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 Pk2, with o corresponding to the point on
the line at infinity, are collinearFrom 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 X X and negation X X
are morphisms of varieties over kIf F is the field generated over the prime field by the
coefficients a0, , a6 of the Weierstrass equation, then
The Weierstrass equation defines a scheme XF of finite type over
F whose base extension to k is XFor each extension E of F the set XF(E) is a group in a
functorial wayXF(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 O(3o), o X(k)
This notational change notwithstanding, o is an arbitrary point
Under the projective embedding given by O(3o), 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 : Div(X) X(k) D p X(k) npp(D) p X(k) np p
which is tautologically a group homomorphism, has the property that
(D1) (D2) whenever D1 D2 (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
X(k) the divisor pqo has degree 1, its
complete linear system consists of a single nonnegative divisor of
degree 1, i.e., r, and this unique r X(k) is defined
to be p q Since pqor the properties specified for make this definition
necessary if, indeed, it defines a groupIt 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 2op It is obvious
that this group law on X(k) is commutative and that is surjective
If Div0(X) denotes the group of divisors of degree 0, then
since (D) (D (deg D)o), one sees that the restriction
0 of to Div0(X) is a surjective homomorphism
Let Div(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) Div0(X) which,
when followed with reduction provides a homomorphism
Div(X) Div0(X)Div(X)
It is not difficult to verify that another homomorphism between this
latter pair of groups is given by
D (D)oDiv(X)
(That this is a homomorphism follows from reviewing the definition of
(D1) (D2).) Since these two homomorphisms agree on divisors
of the form p which generate the free abelian group
Div(X), one has for all D Div(X) that
D (deg D)0(D)o
We know that deg D depends only on the linear equivalence class
of D as the first consequence of the iref="RRThm"RiemannRoch Theorem
Since r X(k) is determined uniquely by the linear equivalence
class of r, this formula tells us that (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 (D) and
deg(D) In particular, one has
Div0(X)Div(X) X(k) Wed.,Apr.26:Suppose that X is a complete nonsingular curve over an algebraically
closed field k of genus 1 The range of degrees where a divisor
D has H1(O(D)) (0) is deg(D) 1, while we
have dim H1(O) 1 For each a X(k) the invertible
module O(2a) has no base point, and, therefore, defines a
morphism to Pk1 One has a two step filtration of the 3dimensional
linear subspace L(3a) of k(X):
k L(0) L(a) L(2a) L(3a)
Choosing x L(2a)L(0) and y L(3a) L(2a)
one obtains a filtrationcompatible basis 1, x, y of L(3a),
and if u is a rational section of O(a) with
div(u) a, the morphism f : X Pk2 given by
f (Z: X: Y), Z u3, X x u3, Y y u3
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, x2, xy, x3 is a filtrationcompatible basis of
L(6 a) Since y2 L(6a) L(5a), one has
a linear relation among monomials of degree 3 Y2 Z a1 X Y Z a3 Y Z2
a0 X3 a2 X2 Z a4 X Z2 a6 Z3
with a0 0 that characterizes f(X) as a nonsingular hypersurface
in Pk2 One says that f(X) is in generalized Weierstrass form
One regards Z 0 as the line at infinity in Pk2, 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 a0 X3 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 3Mon.,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 X(k) one has
deg(D a) 2g 1, and, therefore, L(D a) is a hyperplane in L(D) Otherwise, said O(D) has
no base point A coordinatefree interpretation of the morphism
f : X PkN, where N deg(D) g, given by a basis of
H0(X, O(D)) is that f(a) is the hyperplane H0(X, O(Da))
regarded as a point in the projective space of hyperplanes through the
origin in H0(X, O(D)) If, moreover, deg(D) 2g 1,
then for a b in X(k) it follows that H0(X, O(Dab))
has codimension 2 in H0(X, 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 PkN
The fact that H0(X, O(D 2a)) also has codimension 2 in
H0(X, O(D)) guarantees that da(f) : Ta(X) Tf(a)(PkN)
has rank 1 for each a, and, therefore, that f(X) is itself a
complete nonsingular curve Since morphisms of complete nonsingular
curves are dual to the contravariant function field extensions, f must
be an isomorphism, i.e., 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 H0(X, O(a)) is
an isomorphism of X with Pk1Fri.,Apr.21:In the context of a complete normal variety X over an algebraically
closed field k an invertible OXmodule L is called very
ample if there is an integer N 0 and a closed immersion
f : X PkN such that L fOPkN(1) (Recall
the iref="funcptsprojsp"earlier description of the functor of points
over k of PkN.) If L is very ample, then Lm
is also very ample for each m 1 One says that L is ample
if there exists m 1 such that Lm is very ample
Finally, if there is an integer N 0 and a morphism
f : X PkN such that L fOPkN(1), one
says that Lhas no base point For a particular value of N
if z0, zN are homogeneous coordinates in PkN, hence,
a basis of H0(PkN, OPkN(1)), then the sjf zj
are elements of H0(X, L) that do not vanish simultaneously at
any point of X(k) It follows that the members of any basis of
H0(X, L) also have no common zero, but it does not follow that
the sj form a basisWhen dim(X) 1, recall that for a divisor D of negative degree
iref="nonnegdeg"one hasdimk(H0(X, 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 iref="serredimone"Serre dualitydimk H1(X, O(D)) 0
for any divisor D with deg(D) 2 g 1
When the genus g 1, this means that dimk H1(X, O(D)) 0
for any divisor D of degree at least 1 The Riemann Roch formula
then implies that dim H0(X, O(D)) deg(D)
In particular if D a for a X(k), one sees that
L(a) L(0) k while both have dimension 1
Hence, there can be no f 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 X(k)Wed.,Apr.19:When A is a ring and B an Aalgebra, the module BA is
the Bmodule receiving an Aderivation from B that is initially
universal for derivations from B to Bmodules When
f : X Y is a morphism of schemes there is an OXmodule
XY 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., f(X) Y, and
(ii) XY is a locallyfree OXmodule of rank
dim(X) dim(Y) A nonsingular variety over
k is a variety X that is smooth over k (An irreducible variety
of dimension 1 is nonsingular if and only if it is normal.) When
X is a nonsingular variety, one defines
Xp to be the pth exterior power
pXk For n dim(X) the top exterior
power XXn is a locallyfree
OXmodule that is called the canonicalOXmoduleA form of Serre duality, which could be the subject of an entire course,
is this:
AssertLabel-1AssertDefaultTheoremIf X is a complete nonsingular variety of dimension n
and F a coherent OXmodule, then Hp(X, F) and
ExtOnp(F, X) are dual vector spaces
over k
An important special case is that when F is a locallyfree
Omodule Then
ExtOnp(F, X) ExtOnp(O, XF)
Hnp(X, XF)
where F denotes the O dual of F
In the case of a complete normal curve a canonical divisor is any
divisor K for which O(K) X When FO(D) for an
arbitary divisor D, the vector spaces Hp(X, O(D)) and
H1p(X, O(K D)) have the same dimension for p 0, 1
In particular one has g dim H1(X, OX) dim H0(X, X), and application of the RiemannRoch formula
to a canonical divisor leads to the conclusion that any canonical divisor
must have degree 2g 2Mon.,Apr.17:Continuing with the case of a complete normal curve X over an algebraically
closed field, some observations:
If H0(X, O(D)) (0), then deg(D) 0 since
D is linearly equivalent to a nonnegative divisor div(f) D
for some f L(D)The set
DE Div(X)E 0,E 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) H0(X, O(D)) A linear
system is a projective subspace of a complete linear system. One has
Ddiv(s)s H0(X, O(D))Looking at the cohomology
sequence associated with the short exact sequence
0 O(Da) O(D) iiO(D) 0
one sees that in going from Da to D either the dimension
of H0 goes up by 1 or the dimension of H1 goes down by 1 but
not bothTo go further with complete normal curves we want to talk about
Serre dualityFri.,Apr.7:When X is a complete normal curve over an algebraically closed field
k, a 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 Omodules
0 IaO iOa 0
and, remembering that IaO(a), then
tensoring this exact sequence with the invertible Omodule O(D),
D an arbitary divisor on X, one obtains
0 O(Da) O(D) iiO(D) 0
The third term above is a skyscraper that is rank 1 on
Oa(a) k The relation among Euler
characteristics given by the last short exact sequence reduces to
(X, D) (X, Da) 1
for every divisor D and every closed point a X, and, thus, the
observation that (X, D) deg(D) is a constant depending
only on X where
deg(D) z nzwhen D z nzz
This provides a substantial portion of the RiemannRoch Theorem:
(X, D) deg(D) 1 g
where g, the genus of X, is defined as
dimk H1(X, O) As a corollary of this, together with the
observation that (X, D) depends only on O(D), one sees that
deg(D) depends only on O(D), and, therefore,
deg(div(f)) 0 for each f 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 polesFor an initial understanding of the genus of a complete normal curve,
consider the exact sequence of Omodules
0 Ok(X)k(X)O 0
from which ensues the sequence of vector spaces over k 0 k k(X) H0(X, k(X)O)
H1(X, O) 0
where the last 0 is H1 of the constant, hence flasque, sheaf
k(X) and H0(X, k(X)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 Pk1 is 0Wed.,Apr.5:For D Div(X), X a normal variety, one defines
L(D) f k(X)div(f) D 00L(D) is an O(X)module that is isomorphic to the module of
global sections of O(D) While a (regular) section of a
locallyfree Omodule 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 O and, consequently, has a globally
welldefined divisor If sf 0 is the section of O(D)
corresponding biuniquely with f L(D), one has div(sf) div(f) D One sees that dimk H0(X, O(D)) 0
if and only if D is linearly equivalent to some nonnegative divisorA nonnegative divisor D determines an Oideal
ID that is locally the principal ideal generated by a local equation
for D It follows that ID is a rank 1 locallyfree Omodule,
and one sees easily that it is isomorphic to O(D)When X is a complete variety over a field k and M a coherent
Omodule the kmodules Hq(X,M) are finitedimensional 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 Omodule by
(X, M) q0dim(X) (1)qdimk Hq(X, M)
When
0 MMM 0
is an exact sequence of coherent Omodules on X, one has
(X, M) (X, M) (X, M) Mon.,Apr.3:When X is a normal variety, the affine coordinate ring 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 O(X) is the kernel of the
homomorphism div Given a divisor D Div(X)
and an open covering Ui of X that principalizes D,
say, DUidivUi(fi), it follows from the computation
of the kernel of div on the open subvariety
Uij Ui Uj that fi uij fj (all elements of k(X))
where uijO(Uij) The Cech 1cocycle uij determines
an element O(D) of the group HCech1(X,O)
of locallyfree Omodules of rank 1, the map
D O(D) is a group homomorphism, and the sequence
1 O(X) k(X)Div(X)
HCech1(X,O) 1
is exact One says that two divisors D1 and D2 are linearly
equivalent (and one may write D1 D2) if D2 D1div(f)
for some f k(X) or, otherwise stated, if O(D1) O(D2)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) ZordZ(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 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 OZ
is the generating divisor corresponding to Z Thus one sees that
each divisor on X is locally principalWed.,Mar.29:If f : X Y is an affine morphism of algebraic varieties over
an algebraically closed field k, then for each quasicoherent
OXmodule F one has an isomorphism of Hq(X, F) with
Hq(Y, fF) Finite morphisms and closed immersions present
important special cases To know the cohomology of every coherent
OPmodule on each projective space P PkN is to know the
cohomology of every coherent OXmodule on every projective
variety XMon.,Mar.27:On a Noetherian space the cohomological functor Hq for abelian
sheaves vanishes when q dim(X) The E2 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
flasqueFri.,Mar.24:On a Noetherian space (descending chain condition for closed sets)
each of the sheaf cohomology functors Hq on the category of abelian
sheaves commutes with direct limitsWed.,Mar.22:More on cohomology: Every abelian sheaf on a topological space X
may be regarded as a Zmodule (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 Zmodules An abelian
sheaf is flasque if its restrictions between open sets are all
surjective Every flasque sheaf is acyclic for cohomology, and every
injective Amodule, for any sheaf of rings A on X, is flasque
Consequently, sheaf cohomology in the category of Amodules is
consistent with that in the category of ZmodulesMon.,Mar.20:If f : (X,A) (Y,B) is a morphism of ringed spaces, for every
Bmodule G there is an Amodule pullback f(G) which
at stalk level satisfies
f(G)xGf(x)Bf(x)Ax
For a morphism of affine schemes pullback of quasicoherent modules on
the target is the same thing as base extension
For P PkN, k an algebraically closed field, the exact sequence
OPN1(x0, , xN)OP(1) 0
given by
(f0, , fN) f0 x0 fN xN
spawns, via pullback, the functor of points
of PkN over k:
a morphism : X PkN is the same thing as an invertible
OXmodule L and an N1tuple of sections s0, sN
of L that do not vanish simultaneously, i.e., that provide
the exact sequence
OXN1(s0, , sN)L 0
which is the pullback of the referenced exact sequence on PkN
For a kvalued point x X(k) one has
(x) (s0(x): s1(x): : sN(x)) Fri.,Mar.17:The isomorphism classes of locallyfree Amodules of rank 1 form
a group The notion of an exact sequence of AmodulesAmodules form an abelian category in which every object admits an
injective resolution The global sections functor (M) M(X) is left exact The qth cohomology functor X
Hq(X,M) is defined as the qth right derived functor of
Sideline example: the short exact sequence
0 ZOholeOhol 0
of Zmodules in complex analytic geometry, where e(f) e2 i f
is the complex exponentialWed.,Mar.15:Homomorphisms of Amodules when A is a sheaf of rings on a
topological space Locallyfree Amodules of rank r and
transition matrices relative to a trivializing covering An
invertibleAmodule is a locallyfree Amodule of
rank 1Mon.,Mar.13:Class cancelledFri.,Mar.10:Properties and significance of the OP modules
OP(d) on P Pkn for d Z
where k is an algebraically closed fieldWed.,Mar.8:The concept of sheaf of modules on a ringed space Quasicoherent
and coherent modules on a scheme ExamplesMon.,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 morphismsFri.,Mar.3:Separated morphisms If f : X Y is an Smorphism and Y is
separated over S, then the graph of f is closed in X 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 affineWed.,Mar.1:If x is an element of X, the scheme underlying an irreducible
algebraic variety, the Krull dimension of the local ring
Ox is the codimension of 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 nontrivial 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 fieldMon.,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 closedFri.,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 Y is a scheme of finite type
over the residue field (y) Over its image a morphism may be
viewed as providing a family of varieties, though not a wellbehaved
one without assumptions on the morphism The notion of affine
morphism: another class of morphisms that is closed under compostion
and base extensionWed.,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(kk)
on Xk when X is a kscheme (and k is the
algebraic closure of the field k)Mon.,Feb.13:Detailed examination of the functor of points for
E SpecZx,yF(x,y) where
F(x, y) is the polynomial F(x, y) y2 (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 schemeFri.,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
SchemesSopSets
given by
T HomS(T, X) X(T)
is called the functor of points of X over SX is
determined as an Sscheme by its functor of points If X is the
scheme associated with a variety X0 over an algebraically closed field
k, then X(k) X(Spec(k)) is the set underlying X0 If
K is an extension field of k, a point
X(K) determines an
element x X (no longer called a point) that is called its
center and a kalgebra 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 KWed.,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
schemes 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 fieldMon.,Feb.6:The category of schemes Locally closed subschemes Morphisms;
schemes over a base schemeFri.,Feb.3:The category of affine schemes as (1) a fully faithful subcategory
of the category of localringed spaces and (2) as the opposite
category of the category of commutative ringsWed.,Feb.1:The notion of an affine scheme as a topological space equipped with
a sheaf of rings; morphisms between affine schemesMon.,Jan.30:The sheaf of rings associated with the spectrum of a commutative ring;
the initial ring is the ring of global sectionsFri.,Jan.27:The spectrum of a commutative ring and its Zariski topologyWed.,Jan.25:Presheaves and sheaves; examplesMon.,Jan.23:Overview&SecRef-2;CommentsThings Spotted on the WebWikipedia There are a number of ways to enterhref="http://en.wikipedia.org/wiki/AlgebraicGeometry"Algebraic Geometryhref="http://en.wikipedia.org/wiki/Scheme28mathematics29"Schemeshref="http://en.wikipedia.org/wiki/Special:Search?search=22algebraic+geometry22fulltext=fulltext"Search
Wikipedia for algebraic geometryNotes on Lectures by Hartshorne These are notes by William Stein of 1996 lectures given by
Robin Hartshorne at UC Berkeley: http://modular.ucsd.edu/AG.htmlHref="../"UPHref="/math/pers/hammond/"TOPHref="http://math.albany.edu/"Department