Introduction to Algebraic Varieties
∗
Nitin Nitsure
†CAAG lectures, July 2005
Contents
1 Introduction . . . 2
2 C∞ submanifolds of Rn . . . . 2
3 Holomorphic submanifolds of Cn . . . . 8
4 Some algebra . . . 13
5 Algebraic subvarieties ofCn . . . 18
6 Projective spaces . . . 24
7 Algebraic subvarieties of projective spaces . . . 26
8 Products, Dimension . . . 29
9 Tangent Space to a Variety . . . 34
10 Singular and Non-singular Varieties . . . 41
11 Abstract Manifolds and Varieties . . . 47
12 Some suggestions for further reading . . . 49
∗Course of 8 lectures at the NBHM Advanced Instructional School in Commutative Algebra
and Algebraic Geometry at IIT Mumbai in July 2005.
†School of Mathematics, Tata Institute of Fundamental Research, Mumbai 400 005. e-mail:
1
Introduction
Just as Euclidean Geometry is the study of Euclidean space and certain figures in it made from straight lines and circles, and Differential Geometry is the study of Rn
and its submanifolds, Algebraic Geometry in its classical form is the study of the affine spaceCnand the projective spacePnC, and their subspaces known as algebraic
varieties. These lectures are meant as a first introduction to the subject. They focus on setting up the basic definitions and explaining some elementary concepts about algebraic varieties. The treatment is linear, and many simple statements are left for the reader to prove as exercises.
The material here was delivered in a series of 8 lectures of 90 minutes each, to an audience consisting of mainly of PhD students together with some MSc students. Along with the material in the notes, a large number of examples were shown in the class and in the 4 discussion sessions which were part of the course.
A student who carefully studies these notes would be well prepared for studying any of the standard basic textbooks on algebraic geometry.
2
C
∞submanifolds of
R
n2.1 Linear coordinatesThe linear coordinates x1, . . . , xn on Rn are the
projec-tions from the product Rn to the individual factors R, that is,x
i :Rn→R is the i
th projection function.
2.2 Polynomial functionsAny polynomial f ∈ R[x1, . . . , xn] defines a function
f :Rn→R called a polynomial function.
2.3 ExerciseShow that two polynomials in R[x1, . . . , xn] are equal to each other
if and only if the corresponding functionsRn →Rare equal to each other.
2.4 Linear functionsThese are functions onRndefined by polynomials of degree
1. They have the form Piaixi+cwhere at least one of the ai is non-zero.
2.5 Change of linear coordinates Let A ∈ GLn(R) be an invertible n ×n
-matrix over R. Letb = (b1, . . . , bn)∈Rn. Let y
1, . . . , yn be linear functions defined
byyi =
P
Ai,jxj+bi. Then we say that theyi are a set of linear coordinates onRn.
As A is invertible, we can express eachxi as a linear functions of y1, . . . , yn.
2.6 Open subsetsWe giveRn the Euclidean topology. This makesRn the
prod-uct topological space of n copies of the topological space R, where R is given its
2.7 C∞ functions Let U
⊂ Rn be an open subset in Euclidean topology. A
function f : U → R is said to be a C∞ function if f is continuous and all partial
derivatives off of all orders exist and are continuous.
2.8 NoteBecause U is assumed to be open, the definition of a partial derivative makes sense for functions onU.
2.9 Exercise Give an example of open subsets U ⊂ V ⊂ Rn and a C∞ function
f :U →R, such that there exists no C∞ functiong :V
→R such that g|U =f.
2.10 ExerciseGive an example of nonempty open subsetsU ⊂V ⊂Rnand aC∞
function f : U → R, such that there exist two different C∞ function g, h : V → R
such thatg|U =h|U =f.
2.11 Exercise In the above two exercises, can we have polynomial functions in place of C∞ functions?
2.12 C∞ map from U ⊂ Rn to Rm Let y
1, . . . , ym be the coordinates on Rm.
For any set X, a map f : X → Rm is the same as an an ordered m-tuple of maps
fi :X →R, wherefi =yi◦f. If U is open inRn, a mapf = (f1, . . . , fm) :U →Rm
is called a C∞ map if each of the coordinate maps f
i :U →R isC∞.
2.13 C∞-Isomorphism Let U
⊂ Rn and V ⊂ Rm be open subsets. A C∞
-isomorphism f : U → V is a topological homeomorphism f of U onto V with inverse g :V → U such that the maps f and g are C∞ regarded as mapsU →Rm
and V →Rn. A C∞-isomorphism is also called a diffeomorphism.
2.14 Exercise If U is non-empty, then the existence of a C∞-isomorphism f :
U → V will imply m = n. Prove this by differentiating the composites g◦f and
f◦g and using the chain rule.
2.15 Note If U is non-empty, then the existence of even a homeomorphism f :
U → V will imply m = n. This is a famous result from algebraic topology called ‘invariance of domain’.
2.16 Coordinate chartA coordinate chart (U;u1, . . . , un) in Rn is a tuple
con-sisting of a non-empty open subset U ⊂ Rn together with a sequence of n C∞
functionsui :U →R such that the resulting map u:U →Rn is a C∞-isomorphism
2.17 Inverse function theoremLetU ⊂Rnbe an open subset and letf :U →
Rn be a C∞ map. Let P
∈ U such that the determinant of the n×n matrix (the
Jacobian) µ
∂fi
∂xj
¶
is non-zero at P. Then there exists an open neighbourhood V of P in U (that is,
P ∈V ⊂U with V open in Rn) such that the restriction f|
V is a C∞-isomorphism
of V onto an open subset W of Rn.
2.18 Exercise Let U ⊂Rn be an open subset and let f :U → Rn be a C∞ map
such that the Jacobian matrix (∂fi/∂xj)(P) is invertible for all P ∈U. Then show
that f is an open map. Give an example of such an f which is not injective.
2.19 Exercise Let U ⊂Rn be an open subset and let f :U → Rn be a C∞ map
such that (i) f is injective, and (ii) the Jacobian matrix (∂fi/∂xj)(P) is invertible
for all P ∈U. Then show thatf is aC∞-isomorphism of U onto an open subset of
Rn.
2.20 Cubical coordinate chartLet (U;u1, . . . , un) be a coordinate chart inRn,
such that the map u : U → Rn is a C∞-isomorphism of U onto an open subset
V ⊂Rn where V is of the form
V ={(b1, . . . , bn)∈Rn| −a < bi < a}
for some a > 0. The point P ∈U such that ui(P) = 0 for all i= 1, . . . , n is called
the centre of the cubical coordinate chart.
2.21 Exercise Letr and θ be the usual polar coordinates on R2. Let U ⊂R2 be
the subset where 1< r <3 and 0< θ <2π. Let u1 =r−2 and letu2 = (θ−π)/π.
Then show that (U;u1, u2) is a Cubical coordinate chart in R2. What is its centre?
2.22 Locally closed submanifolds Let d be an integer with 0 ≤ d ≤ n. Let
X ⊂ Rn be a locally closed subset of Rn (recall: a subset of a topological space is
called locally closed if it can be expressed as the intersection of an open subset and a closed subset of the space). Then X is called a locally closed submanifold of Rn
of dimensiond, where d≥0 is an integer, ifX is non-empty and each P ∈X is the centre of some cubical coordinate chart (U;u1, . . . , un) in Rn such that
X∩U ={Q|ui(Q) = 0 for all d+ 1≤i≤n}
If X is empty, it is called a locally closed submanifold of Rn of dimension −∞ (or
−1 in another convention).
If a locally closed submanifold X of Rn is closed in V where V ⊂Rn is open, then
2.23 Exercise If U ⊂ X is open in a locally closed submanifold X of Rn, then
show thatU itself is a locally closed submanifold ofRn. What is its dimension?
2.24 C∞-functions on submanifolds Let X be a locally closed submanifold of
Rn. Let f :X →R be a function. Then f is called aC∞-function on X if for each
P ∈ X there exists an open neighbourhood U in Rn together with a C∞-function
g :U →R such that
f|X∩U =g|X∩U
2.25 The R-algebra C∞(X) If X is a locally closed submanifold ofRn, then all
C∞-functions f : X → R form a commutative R-algebra C∞(X) under point-wise
operations.
2.26 Exercise IfX is empty, show that C∞(X) = 0.
2.27 The sheaf property Let U and V be open subsets of a locally closed C∞
submanifoldX ofRn, such thatU ⊂V ⊂X. Then for anyC∞-functionf :V →R,
the restriction f|U : U → R is again a C∞-function. Restriction of C∞-functions
defines an R-algebra homomorphism
C∞
(V)→ C∞
(U) :f 7→f|U
Given any locally closedC∞-submanifold V of Rn and any open covering (U
i)i∈I of
V, the following two properties are satisfied:
(1) SeparatednessIf f ∈ C∞(V) such that each restriction f
|Ui is 0, then f = 0.
(2) GluingIf for eachUi we are given aC∞-functionfi such that for any pair (i, j)
we have
fi|Ui∩Uj =fj|Ui∩Uj ∈ C
∞
(Ui∩Uj)
then there exists some f ∈ C∞(V) such that f
i = f|Ui for each i. (Such an f will
therefore be unique by (1)).
2.28 Exercise Let X ⊂ Rn be a locally closed submanifold of dimension d ≥ 0.
Show that a function f : X → R is C∞ if and only if for every cubical coordinate
chart (U;u1, . . . , un) inRnsuch thatX∩U ={Q|ui(Q) = 0 for all d+ 1≤i≤n},
f|X∩U is a C∞-function of the coordinates u1, . . . , ud.
2.29 The local ring of germsLet X ⊂Rn be a locally closed submanifold and
letP ∈X. Consider the setFP of all ordered pairs (U, f) whereU ⊂X is an open
neighbourhood of P in X, and f ∈ C∞(U). On the set F
P we put an equivalence
relation as follows. We say that pairs (U, f) and (V, g) are equivalent if there exists an open neighbourhood W of P in U ∩V such that f|W = g|W. Each equivalence
classes is called a germ of aC∞-function on X atP.
The set of all equivalence classes forms a R-algebra (where operations of addition
and multiplication are carried out point-wise on restrictions to common small neigh-bourhoods), denoted byC∞
X,P.
2.30 ExerciseVerify that the aboveR-algebra structure onC∞
X,P is well-defined.
2.31 ExerciseShow that we have a well-definedR-algebra homomorphism (called
the evaluation map)
C∞
X,P →R: (U, f)7→f(P)
Show thatC∞
X,P is a local ring, whose unique maximal ideal is the kernel mX,P of the evaluation mapC∞
X,P →R.
2.32 C∞-maps between submanifolds Let X
⊂ Rm and Y ⊂ Rn be locally
2.35 Closed submanifolds and implicit function theorem Let U ⊂ Rq be
open, and let f : U → Rn be a C∞-map such that for each point P
∈ U with
f(P) = 0, the rank of the n×q-matrix
µ
∂fi
∂xj
(P)
¶
1≤i≤n,1≤j≤q
isn. Then provided it is non-empty, the subset f−1(0) is a closed submanifold of U
of dimension q−n.
2.36 Exercise Prove the above statement as an application of the implicit func-tion theorem.
2.37 Real analytic category A function f :U →R, where U ⊂Rn is open, is
called a real analytic function if around each pointP = (a1, . . . , an)∈U it is given
by a convergent power series in then variablesx1−a1, . . . , xn−an. The concepts of
analytic isomorphism, analytic coordinate chart, cubical analytic coordinate chart, and locally closed analytic submanifold ofRnare defined by taking analytic functions
in place of C∞ functions in the various definitions above. The inverse and implicit
function theorems remain true for analytic functions in place of C∞ functions.
2.38 Exercise Show that if X ⊂ Rn is a closed subset then there exists a C∞
functionf :Rn→R which vanishes exactly on X.
2.39 ExerciseA similar property does not hold for analytic functions: show that ifU ⊂V ⊂Rnare non-empty open subsets whereV is connected, and iff :V →R
is an analytic function such thatf|U = 0 then f = 0.
2.40 Exercise: Spheres, classical matrix groups, surfaces
Show that the unit sphere Sn−1 ⊂ Rn, defined by x2
1 +. . .+x2n = 1, is a closed
analytic submanifold. What is its dimension?
Show that SLn(R) ⊂ Rn×n is a closed analytic submanifold, and determine its
dimension.
Show that for any integerg ≥0, there exists a closed C∞ submanifold of R3 which
is homeomorphic to a 2-sphere with g handles.
Does R3 have a closed C∞ submanifold homeomorphic to the real projective plane?
Does R3 have a closed C∞ submanifold homeomorphic to the mobius strip without
3
Holomorphic submanifolds of
C
nThe above notions about real analytic manifolds have their analogues in the theory of complex manifolds, where complex analytic functions (also known as holomorphic functions) of several complex variables replace the real analytic functions of the previous section.
3.1 Open subsets Let Cn have complex linear coordinates x1, . . . , xn. Let xi =
ui +
√
−1vi where ui and vi are the real and imaginary parts. We give Cn the
topology of R2n, under the bijection (x
1, . . . , xn) 7→ (u1, v1, . . . , un, vn). Open (or
closed) subsets in Cn mean open (or closed) subsets in the Euclidean topology on R2n.
3.2 Holomorphic functions and maps Let U ⊂ Cn be an open subset in
Eu-clidean topology. A function f : U → C is said to be holomorphic (or complex
analytic) if given any point P = (a1, . . . , an) ∈ U, the function f is given by a
convergent power series in then complex variables x1−a1, . . . , xn−an in a
neigh-bourhood ofP.
All holomorphic functions on an open set U form a C-algebra, which we denote
by H(U). If U ⊂ V, then restriction defines a C-algebra homomorphism H(V)→ H(U).
A map f = (f1, . . . , fm) :U →Cm is said to be holomorphic (or complex analytic)
if eachfi :U →C is analytic.
3.3 Holomorphic isomorphismLet U ⊂Cn and V ⊂ Cm be open subsets. A
holomorphic isomorphismf : U →V is a topological homeomorphism f of U onto
V with inverse g : V → U such that the maps f and g are holomorphic regarded as maps U → Cm and V → Cn. A holomorphic isomorphism is also called a a
bi-holomorphic map.
3.4 ExerciseIfU is non-empty, then the existence of a holomorphic isomorphism
f :U →V will imply m=n. Prove this by differentiating the composites g◦f and
f◦g and using the chain rule.
3.5 Holomorphic coordinate chartA holomorphic coordinate chart in Cnis a
tuple (U;u1, . . . , un) consisting of a non-empty open subset U ⊂ Cn together with
a sequence of n holomorphic functions ui : U → C such that the resulting map
u:U →Cn is a holomorphic isomorphism of U onto an open subset V ⊂Cn.
(the Jacobian) µ
∂fi
∂xj
¶
is non-zero at P. Then there exists an open neighbourhood V of P in U (that is,
P ∈ V ⊂ U with V open in Cn) such that the restriction f|V is a holomorphic
isomorphism ofV onto an open subset W of Cn.
3.7 ExerciseLetU ⊂Cnbe an open subset and letf :U →Cnbe a holomorphic
map such that the Jacobian matrix (∂fi/∂xj)(P) is invertible for allP ∈U. Then
show thatf is an open map. Give an example of such anf which is not injective.
3.8 Exercise Let U ⊂Cn be an open subset and let f :U → Cn be a
holomor-phic map such that (i) f is injective, and (ii) the Jacobian matrix (∂fi/∂xj)(P) is
invertible for allP ∈U. Then show thatf is a holomorphic isomorphism of U onto an open subset ofCn.
3.9 Cubical coordinate chart (polydisk)Let (U;u1, . . . , un) be a holomorphic
coordinate chart inCn, such that the mapu:U →Cnis a holomorphic isomorphism
of U onto an open subset V ⊂Cn whereV is of the form
V ={(b1, . . . , bn)∈Cn| |bi|< a}
for some a > 0. The point P ∈U such that ui(P) = 0 for all i= 1, . . . , n is called
the centre of the cubical coordinate chart.
3.10 Locally closed holomorphic submanifoldsLetdbe an integer with 0≤
d ≤ n. Let X ⊂ Cn be a locally closed subset of Cn. Then X is called a locally
closed holomorphic submanifold of Cn of (complex) dimensiond, where d≥0 is an
integer, ifX is non-empty and each P ∈X is the centre of some cubical coordinate chart (U;u1, . . . , un) in Cn such that
X∩U ={Q|ui(Q) = 0 for all d+ 1≤i≤n}
If X is empty, it is called a locally closed submanifold of Cn of dimension −∞ (or −1 in another convention).
If a locally closed submanifold X of Cn is closed in V where V ⊂Cn is open, then
X is called a closed submanifold of V.
3.11 Exercise If U ⊂ X is open in a locally closed submanifold X of Cn, then
3.12 Holomorphic functions on submanifoldsLetX be a locally closed sub-manifold of Cn. Let f : X → C be a function. Then f is called a holomorphic
function on X if for each P ∈ X there exists an open neighbourhood U in Cn
together with a holomorphic function g :U →C such that
f|X∩U =g|X∩U
3.13 The C-algebra H(X) If X is a locally closed submanifold of Cn, then all
holomorphic functions f : X → C form a commutative C-algebra H(X) under
point-wise operations. IfX is empty, we have H(X) = 0.
3.14 The sheaf property Let U and V be open subsets of a locally closed holomorphic submanifoldXofCn, such thatU ⊂V ⊂X. Then for any holomorphic
function f : V → C, the restriction f|U : U → C is again a holomorphic function.
Restriction of holomorphic functions defines anC-algebra homomorphism
H(V)→ H(U) :f 7→f|U
Given any locally closed holomorphic submanifold V of Cn and any open covering
(Ui)i∈I of V, the following two properties are satisfied:
(1) SeparatednessIf f ∈ H(V) such that each restriction f|Ui is 0, then f = 0.
(2) GluingIf for eachUi we are given a holomorphic function fi such that for any
pair (i, j) we have
fi|Ui∩Uj =fj|Ui∩Uj ∈ H(Ui∩Uj)
then there exists some f ∈ H(V) such that fi = f|Ui for each i. (Such an f will
therefore be unique by (1)).
3.15 Exercise Let X ⊂ Cn be a locally closed submanifold of dimension d ≥ 0.
Show that a functionf :X →Cis holomorphic if and only if for every cubical
holo-morphic coordinate chart (U;u1, . . . , un) in Cn such that X ∩U = {Q|ui(Q) =
0 for alld + 1 ≤ i ≤ n}, f|X∩U is a holomorphic function of the coordinates
u1, . . . , ud.
Show that it is enough to check the above condition for a family of such charts which covers X.
3.16 The local ring of germsLet X ⊂Cn be a locally closed submanifold and
letP ∈X. Consider the setFP of all ordered pairs (U, f) whereU ⊂X is an open
neighbourhood of P in X, and f ∈ H(U). On the set FP we put an equivalence
relation as follows. We say that pairs (U, f) and (V, g) are equivalent if there exists an open neighbourhood W of P in U ∩V such that f|W = g|W. Each equivalence
The set of all equivalence classes forms a C-algebra (where operations of addition
and multiplication are carried out point-wise on restrictions to common small neigh-bourhoods), denoted byHX,P.
3.17 ExerciseVerify that the above C-algebra structure onHX,P is well-defined.
3.18 ExerciseShow that we have a well-definedC-algebra homomorphism (called
the evaluation map)
HX,P →C: (U, f)7→f(P)
Show that HX,P is a local ring, whose unique maximal ideal is the kernel mX,P of
the evaluation mapHX,P →C.
3.19 Holomorphic maps between submanifoldsLetX ⊂CmandY ⊂Cnbe
locally closed holomorphic submanifolds. A mapf = (f1, . . . , fm) :X →Y is called
holomorphic if each fi is holomorphic, equivalently, if the composite X →Y ֒→Cn
is holomorphic.
3.20 Exercise: Pull-back of holomorphic functionsLetX and Y be locally closed holomorphic submanifolds of Cm and Cn, respectively. Let f : X → Y
be a holomorphic map. Let g : Y → C be a holomorphic function on Y. Then
show that the composite function g ◦f : X → C is holomorphic. The composite
g◦f is called the pull-back of g under f. This defines a C-algebra homomorphism
f# :H(Y)→ H(X) :g 7→g◦f.
3.21 ExerciseWithX and Y as above, let f1 and f2 be holomorphic maps from
X to Y. If f1#=f #
2 :H(Y)→ H(X), then show that f1 =f2.
3.22 Exercise: Composition of holomorphic mapsLetX,Y andZ be locally closed holomorphic submanifolds ofCm,Cn andCp respectively, and letf :X →Y
and g :Y →Z be holomorphic maps. The show that the compositeg◦f :X →Z
is a holomorphic map.
3.23 Implicit function theorem Let x1, . . . , xm be linear coordinates on Cm
and let y1, . . . , yn be linear coordinates on Cn. On the product Cm+n = Cm ×
Cn we get linear coordinates x
1, . . . , xm, y1, . . . , yn. Let U ⊂ Cm+n be an open
neighbourhood of the origin 0 = (0, . . . ,0) ∈ Cm+n. Let f = (f
1, . . . , fn) be an
n-tuple of holomorphic functions (same as a holomorphic map f : U → Cn), such
that
Let the followingn×n-matrix
µ
∂fi
∂yj
¶
1≤i,j≤n
be invertible at the point 0∈Cm+n. Then there exists real numbers a, b > 0 and a
holomorphic function g = (g1, . . . , gn) :Va→Wb where Va⊂Cm is the open subset
defined in terms of the coordinates by |xi| < a and Wb ⊂ Cn is the open subset
defined in terms of the coordinates by|yj|< b such that Va×Wb ⊂U and
f−1(0)
∩(Va×Wb) ={(x, g(x))|x∈Va}
3.24 Exercise Deduce the implicit function theorem from the inverse function theorem, and conversely.
3.25 Closed submanifolds and implicit function theorem Let U ⊂ Cq be
open, and let f : U → Cn be a holomorphic map such that for each point P ∈ U
with f(P) = 0, the rank of the n×q-matrix
µ
∂fi
∂xj
(P)
¶
1≤i≤n,1≤j≤q
is n. Then provided it is non-empty, the subset f−1(0) is a closed holomorphic
submanifold of U of dimension q−n.
3.26 Exercise Prove the above statement as an application of the implicit func-tion theorem.
3.27 Example: Non-singular algebraic curves in C2 Let f ∈ C[x, y] be an
irreducible polynomial of degree d≥1. Let Z(f)⊂C2 be defined as
Z(f) ={(a, b)∈C2|f(a, b) = 0}
Suppose that at any pointP = (a, b)∈Z(f), at least one of the partial derivatives
∂f /∂xand∂f /∂y is non-zero. Then by application of the implicit function theorem,
Z(f) is a closed submanifold of C2 of dimension 1.
3.28 Exercise Identifying C2 with R4, we can regard Z(f) as a closed subset of
3.29 Note on the general concept of locally closed submanifolds
There is another notion of a locally closed submanifold which is useful in differen-tial geometry and lie groups, which is more general than our definition. A locally closed submanifold of Rn according to this definition is a pair (X, φ) where X is
an abstract C∞-manifold (see the last section for definition), and φ : X
→ Rn is
a C∞-morphism that is injective and tangent-level injective. (Actually, one defines
an equivalence relation on such pairs (X, φ) by putting (X, φ)−(X′, φ′) if there
exists aC∞-isomorphism ψ :X
→ X′ such that φ =φ′
◦ψ, and then one defines a locally closed submanifold in the general sense as an equivalence class.) A standard example of such a more general concept is the so called skew line on a torus. For an elementary example, letX =R1, and let φ :X→R2 be defined by t7→(et,sint).
A similar more general notion exists for locally closed holomorphic submanifolds of
Cn or of more general complex manifolds.
We will not have occasion to use these general notions.
4
Some algebra
All rings are assumed to be commutative.
Integral ring extensions
4.1 Definition Let φ : A → B be a ring homomorphism. An element b ∈ B is said to beintegral elementoverA(or more precisely, integral over Awith respect to φ) if there exists some integer n ≥ 1 and elements a1, . . . , an ∈ A such that
bn+φ(a
1)bn−1 +. . .+φ(an) = 0 in B. We say that B is integral over A if each
b∈B is integral over A.
Note that b is integral over A w.r.t. φ if and only if b is integral over the subring
φ(A) ofB w.r.t. the inclusion φ(A)֒→B. For this reason, we can assume that φ is an inclusion giving a ring extension A⊂B for simplicity of notation.
4.2 LemmaLetA⊂B be a ring extension and b∈B. The following statements are equivalent:
(1) b is integral over A.
(2) TheA-subalgebra A[b]⊂B is finite over A.
(3) There exists a faithful A[b]-module M which is finite as an A-module. (Recall that a moduleM over a ring R is called faithful if no non-zero element ofR kills all of M.)
ProofClearly (1)⇒(2) ⇒(3). Now letM be a faithfulA[b]-module which is finite as anA-module, generated by v1, . . . , vn. Then there existn2 elementsci,j ∈A such
that bvi =
Pn
the element det(bI−C)∈A[b] annihilates eachvj hence annihilates M, where C is
the matrix (ci,j). Hence det(bI −C) = 0 asM is faithful overA[b]. As det(bI −C)
is a monic polynomial in b with coefficients in A, we see that (3) ⇒ (1).
4.3 LemmaLetA⊂B be rings, such thatB is finite overA. ThenB is integral overA.
ProofFor eachb∈B, note that B is a faithfulA[b]-moduleM which is finite as an
A-module.
4.4 Lemma Let A⊂ B be rings, such that B is finite-type and integral overA. ThenB is finite over A.
Proof Let B = A[b1, . . . , bn]. Then consider the tower of ring extensions A ⊂
A[b1]⊂A[b1, b2]⊂. . .⊂A[b1, b2, . . . , bn] =B, in which each step is finite.
4.5 LemmaLet A⊂B be rings. Then all elements of B which are integral over
A form anA-subalgebra ofB.
ProofIfb∈B is integral overA, then A[b] is finite over A. Ifc∈B is integral over
A then cis integral over A[b] and so A[b, c] is finite over A[b]. HenceA[b, c] is finite overA.
Nullstellensatz
4.6 TheoremLetA ⊂K be an integral ring extension. If K is a field, then A is a field.
Proof For any a∈A with a= 0, the element6 b = 1/a of K is integral overA. Let
bn+a
1bn−1+. . .+an= 0. Then multiplying by an−1, we getb =−(a1+a2a+. . .+
anan−1)∈A.
4.7 LemmaLetk be a field and k[x1, . . . , xr] be a polynomial ring inr variables
where r ≥ 1. If f ∈ k[x1, . . . , xr] then the localisation k[x1, . . . , xr,1/f] is not a
field.
ProofThere are infinitely many non-constant irreducible polynomials which are not scalar multiples of each other. Hence there is a non-constant irreducible polynomial
pwhich does not divide f. It follows that pis not invertible in k[x1, . . . , xr,1/f].
4.8 Theorem Letk be a field and A a finite-type k-algebra. If A is a field then
A is finite over k.
Proof We just have to show that A is algebraic over k. Let A = k[a1, . . . , an].
the ai, the first r generators a1, . . . , ar are algebraically independent over k and A
is algebraic over k(a1, . . . , ar). Each of ar+1, . . . , an satisfies a monic polynomial
fr+1(t), . . . , fn(t) in k(a1, . . . , ar)[t]. Let g(a1, . . . , ar)∈k[a1, . . . , ar] be the product
of all the denominators of the coefficients of thefr+1(t), . . . , fn(t) (written as rational
fractions in any chosen way). Then A is integral over k[a1, . . . , ar,1/g]. Hence
k[a1, . . . , ar,1/g] is a field, as it is given that A is a field. This is a contradiction,
unless r= 0.
(Another proof: By Noether Normalisation, there exist elements a1, . . . , ar ∈ A
which are algebraically independent overk such thatA is finite over the polynomial ringk[a1, . . . , ar]. Therefore asA is a field, this polynomial ring is also a field, which
showsr= 0.)
4.9 Theorem Let k be a field and φ : A → B a homomorphism between finite-typek-algebras. Then for any maximal ideal m ⊂B the inverse image φ−1(m) is a maximal ideal in A.
Proof B/m is algebraic hence integral over k, and k ⊂ A/φ−1(m) ⊂ B/m. Hence
B/mis integral over A/φ−1(m). Therefore A/φ−1(m) is a field.
4.10 Theorem Let k be a field and A a finite-type k-algebra. Let f ∈ A such that f lies in each maximal ideal of A. Then f is nilpotent.
Proof Suppose f is not nilpotent. Then Af is non-zero, so it has a maximal ideal
m ⊂ Af. We have a homomorphism A → Af of finite-type k-algebras, so the contraction mc ⊂A of m is maximal. As f lies in each maximal ideal of A, we get
f ∈mc and so f ∈m. Contradiction, as f is a unit in Af.
Algebraically closed base field
4.11 Corollary If k is an algebraically closed field, then any maximal ideal in a polynomial ringk[x1, . . . , xn] is of the form (x1−a1, . . . , xn−an) for somea1, . . . , an∈
k.
4.12 Corollary If k is an algebraically closed field and I ⊂ k[x1, . . . , xn] is an
ideal such thatI 6= (1), then the locus Z(I)⊂kn defined by I is non-empty.
4.13 Corollary If k is an algebraically closed field, I ⊂k[x1, . . . , xn] is an ideal,
and f ∈k[x1, . . . , xn] is a polynomial such thatf(a1, . . . , an) = 0 for each point
Prime decomposition
4.14 TheoremLetA be a noetherian ring. Then every radical idealI of Ais the intersection of a unique minimal finite set{p1, . . . ,pn} of prime ideals in A:
I =p1∩. . .∩pn
Proof We will first show that every radical ideal in Ais an intersection of a finite set of prime ideals. Suppose not, then by noetherian condition, there is a maximal elementIin the set of all radical ideals which cannot be so expressed as intersections. As such an I cannot be prime, there exist a, b ∈A−I such that ab ∈I. Then by using the fact thatI is radical, we see that
I = (I+ (a))∩(I+ (b))
Taking radicals, we get
I =pI+ (a)∩pI+ (b)
By maximality ofI, the radical idealspI+ (a) andpI+ (b) are finite intersections of prime ideals, hence so is I. If I =p1 ∩. . .∩pn then by discarding at most n of
the idealspi, we arrive at a minimal setS of primes which has intersection I (where
we say that a setS ={p1, . . . ,pn} which has intersection ∩p∈Sp=I is minimal if
there is no proper subsetS′ ⊂S which also has intersection I).
Next, we show the uniqueness of a minimal setS. Clearly,S is empty if and only if
I =A. So now we assume that cardinality of S is n≥1. Let S ={p1, . . . ,pn} and
T ={q1, . . . ,qm}be two minimal such sets, withn ≤m. Asp1∩. . .∩pn⊂q1, and
asq1 is prime, we must havepr ⊂q1 for some r. Similarly,qs ⊂pr for some s, and
so qs ⊂ q1. By minimality of T, no qj ∈T can contain another qk ∈ T, therefore
s= 1 andqs =q1. After re-indexing the pi we can assume that p1 =q1.
Having proved pi = qi for all 1 ≤ i ≤ n−1 (after possible re-indexing), consider
the inclusion p1∩. . .∩pn ⊂qn. Asqn is prime, we must have pk ⊂qn for somek.
If 1≤k≤n−1, then we would get the inclusionqk=pk ⊂qn which is impossible
by minimality of T. Hence k = n, and so pn ⊂ qn. Similarly, q1 ∩. . .∩qm ⊂ pn
and so by primality ofpn we haveqj ⊂pn, and hence qj ⊂qn, which shows j =n
by minimality of T. Hence pn =qn. Hence S ⊂T, so by minimality of T, S =T.
¤
(The above proposition also has a set-topological proof, which is simpler: as SpecA
More on integral extensions
4.15 Noether Normalisation TheoremLet k be a field, and let A be a finite-typekalgebra which is a domain. Then there exist finitely many elementsx1, . . . , xd
inA(whered≥0) that are algebraically independent overk, such thatAis integral over the subringk[x1, . . . , xd]⊂A.
We assume the student has seen a proof. In the lectures we will explain how to geometrically view this result in terms of choosing a ‘new coordinate projection which is finite’.
4.16 Going up Let A ⊂ B be an integral extension of rings. Then given any prime idealp ⊂A, there exists a prime idealq⊂B such thatp =q∩A.
4.17 Going downLet A⊂B be an integral extension of rings, such that B is a domain andA is a normal domain. Then given any prime ideal qin B and p′ inA
with p′
⊂q∩A, there exists a prime idealq′
⊂qin B with q′
∩A=p′
.
4.18 Note The conclusion of the above result (the going-down property) also holds for arbitrary flatA-algebras B.
Some dimension theory
4.19 Krull dimension Let A be a ring and p ⊂ A a prime ideal. The hight of pis by definition the largest integer n ≥0 such that there exists a chain of distinct prime ideals
p0 ⊂p1 ⊂. . .⊂pn=p
The Krull dimension of A is the supremum of the heights of all prime ideals ofA. The Krull dimension of a zero ring is the supremum of the empty set, which has value−∞ by convention.
4.20 Theorem Let k be a field, and let A be a finite-type k algebra which is a domain. Then the following holds.
(1) The Krull dimension ofA equals the transcendence degree over k of the field of fractions ofA.
4.21 Krull’s principal ideal theoremLetAbe a noetherian ring and letf ∈A
be neither a unit nor a zero divisor. Letp⊂Abe a minimal prime ideal containing
f. Then height of p is 1.
5
Algebraic subvarieties of
C
n5.1 Closed, open, locally closed subvarieties Let C[x1, . . . , xn] be the
poly-nomial ring in n variables over C. For any subset S ⊂C[x1, . . . , xn], let Z(S)⊂Cn
be the subset defined as the set of simultaneous solutions of all polynomials in S, that is,
Z(S) ={(a1, . . . , an)∈Cn|f(a1, . . . , an) = 0 for all f ∈S}
A subset X ⊂ Cn is called a closed subvariety of Cn if there exists some S ⊂ C[x1, . . . , xn] such that X = Z(S). A subset U ⊂ Cn is called an open subvariety
of Cn if its complement Cn−U is a closed subvariety of Cn. A subset Y ⊂ Cn is
called a locally closed subvariety if it can be expressed as the intersection of a closed subvariety of Cn with an open subvariety of Cn.
5.2 Exercise If S1 ⊂ S2 ⊂ C[x1, . . . , xn] then show that Z(S1) ⊃ Z(S2) (the
inclusion gets reversed). Is the converse true?
5.3 Exercise Let I = (S) ⊂ C[x1, . . . , xn] be the ideal generated by S. Then
show thatZ(S) = Z(I).
5.4 ExerciseLetI ⊂C[x1, . . . , xn] be an ideal, and let√I ⊂C[x1, . . . , xn] be its
radical (f ∈√I if and only if the powerfr is inI for somer ≥1 which may depend
onf). Show thatZ(I) =Z(√I).
5.5 Exercise If I and J are radical ideals (an ideal is called a radical ideal if it equals its own radical), then show thatI∩J is a radical ideal. IsI+J also a radical ideal?
5.6 Exercise Show that under partial order defined by inclusion (that is, I ≤ J
means I ⊂ J), all radical ideals in a ring form a lattice, with meet I ∩J and join
√
I+J. The minimum is the nil radical √0 and the maximum is (1).
5.7 Exercise IfI and J are ideals in C[x1, . . . , xn], show that
Z(I)∪Z(J) = Z(I∩J) IfSλ is any family of subsets of C[x1, . . . , xn], show that
Consequently, ifIλ are ideals inC[x1, . . . , xn] then show that
∩λZ(Iλ) = Z(
P
λIλ)
5.8 ExerciseShow that under partial order defined by inclusion (that is,X ≤Y
meansX ⊂Y), all closed subvarieties ofCnform a lattice, with meetX∩Y and join
X∪Y. The minimum is the empty set ∅=Z(1) and the maximum is Cn=Z(0).
5.9 Zariski topology via closed subvarieties A subset X ⊂ Cn is called a
closed subset in Zariski topology if X is a closed subvariety, that is, if X = Z(I) for some ideal I ⊂ C[x1, . . . , xn]. This specification of closed sets indeed defines a
topology on Cn, as ∅=Z(1), Cn =Z(0), Z(I)∪Z(J) =Z(I∩J), and ∩λZ(Iλ) =
Z(PλIλ) by an earlier exercise.
5.10 Exercise Show that the Euclidean topology on Cn for n ≥ 1 is finer than
the Zariski topology.
5.11 Exercise Show that any two non-empty Zariski-open subsets of Cn have a
non-empty intersection. (This property is expressed by saying thatCnis irreducible.)
5.12 Exercise Show that any non-empty Zariski-open subset of Cn is connected
in Euclidean topology.
5.13 Exercise What are all the Zariski-closed subsets ofC?
5.14 Exercise Show that the diagonal ∆ ={(a, b)∈ C2|a =b} is closed in C2
in Zariski topology. Conclude using the previous exercise that the Zariski topology onC2 is not the product of the Zariski topologies on the two factors C.
5.15 Exercise: Principal open subvarieties as basic open subsetsFor any
f ∈C[x1, . . . , xn], we denote the complementCn−Z(f) by the notationUf. This is
called as the principal open subvariety defined byf. Show that open subvarieties of the typeUf, as f varies over C[x1, . . . , xn], form a basis of open sets for the Zariski
topology. Moreover, show that any Zariski open set is a union of finitely many sets of the formUf.
5.16 Exercise: Noetherianness Show that Cn under Zariski topology is a
noetherian topological space, that is, any decreasing sequence of closed subsets is finite. Deduce that every subset of Cn with induced topology is noetherian. In
5.17 Exercise: Quasi-compactnessShow that any noetherian topological spa-ceX is quasi-compact, that is, every Zariski open cover of X has a finite subcover. In particular, locally closed subvariety X ⊂ Cn is quasi-compact, that is, every
Zariski open cover of X has a finite subcover.
5.18 Irreducibility A non-empty topological space X is said to be reducible if there exist non-empty open subsetsU ⊂ X and V ⊂X such that U ∩V =∅. The empty topological space∅is also defined to be reducible. A locally closed subvariety
X ⊂Cn is called irreducible if it is not reducible in the induced Zariski topology on
X.
5.19 Irreducibility in terms of closed subsetsShow that a non-empty topo-logical space X is irreducible if and only if for any proper closed subsets Y and Z
of X, the unionY ∪Z is also a proper subset ofX.
5.20 Exercise: Irreducible decomposition Show that if X is a noetherian topological space, then X can be uniquely written as a minimal finite union of irreducible closed subsets. Use this to give another proof that any radical ideal in a noetherian ring is uniquely a minimal finite intersection of primes.
5.21 Theorem There is an inclusion-reversing bijective correspondence between the set of all radical ideals in C[x1, . . . , xn] and the set of all closed subvarieties of Cn given by I 7→Z(I), with inverse given by
X 7→IX ={f ∈C[x1, . . . , xn]|f(a1, . . . , an) = 0 for all (a1, . . . , an)∈X}
Under this correspondence, the set of all maximal ideals inC[x1, . . . , xn] is in
bijec-tion with the set of all points of Cn. The set of all prime ideals in C[x
1, . . . , xn] is
in bijection with the set of all irreducible closed subvarieties in Cn.
5.22 Corollary Let X be a closed subvariety of Cn, defined by a radical ideal
J. There is an inclusion-reversing bijective correspondence between the set of all closed subvarieties Y of X and the set of all radical ideals in the quotient ring
R=C[x1, . . . , xn]/J, given (in a well-defined manner) by
Y 7→IY ={f ∈R|f∼(a1, . . . , an) = 0 for all (a1, . . . , an)∈Y }
where f∼
∈ C[x1, . . . , xn] is any polynomial that lies over f ∈ C[x1, . . . , xn]/J.
5.23 Regular functions on an open subvariety A function φ : V → C on
an open subvariety V ⊂ Cn is called a regular function if given any principal open
Uf ⊂V, the restriction φ|Uf is of the form g/f
r where g ∈C[x
1, . . . , xn], and r is a
non-negative integer.
5.24 Exercise Show that if (Ufi)i=1,...,m is an open cover of an open subvariety
V ⊂Cn, and ifφ:V →Cis a function such that each restrictionφ|
Ufi is of the form
gi/firi wheregi ∈C[x1, . . . , xn], and ri is a non-negative integer, then φ is regular.
5.25 ExerciseShow that every regular function is continuous in Zariski topology. Is the converse true?
5.26 Exercise Show that every regular function is holomorphic. Is the converse true?
5.27 Exercise Show that all regular functions on an open subvariety V ⊂ Cn
form a C-algebra (which we denote by O(V)) under point-wise operations. Show
that O(Cn) =C[x
1, . . . , xn]. For any principal open Uf ⊂Cn, show that
O(Uf) = C[x1, . . . , xn,1/f]
which is the localisation of the polynomial ring by inverting f. Is this also true for
f = 0?
5.28 Exercise LetP ∈C2. What are the regular functions on C2− {P}? What
are the holomorphic functions on C2− {P}?
5.29 Exercise If X ⊂Cn is a closed or an open subvariety of Cn, show that the C-algebra O(X) is of finite type over C.
5.30 Caution For an arbitrary locally closed subvariety X ⊂Cn, the C-algebra O(X) need not be of finite type over C. Example due to Nagata: See ‘Lectures on
the 14 th problem of Hilbert’ by Nagata, TIFR lecture notes, 1965.
5.31 Regular functions on a locally closed subvariety Let X ⊂ Cn be a
locally closed subvariety. A function φ : X → C is called a regular function if for
each P ∈ X, there exists a Zariski open neighbourhood U ⊂ Cn together with a
regular function onψ :U →C such thatφ|X∩U =ψ|X∩U.
5.32 Exercise Show that the regular functions on any locally closed subvariety
5.33 Exercise Show that for any closed subvariety X ⊂Cn, the restriction map
C[x1, . . . , xn]→ O(X) :f 7→f|X
is a surjectiveC-algebra homomorphism. IfX =Z(I) whereI =√I ⊂C[x1, . . . , xn]
is any radical ideal, conclude that the restriction mapC[x1, . . . , xn]→ O(X) induces
aC-algebra isomorphism C[x1, . . . , xn]/I → O(X).
5.34 ExerciseShow that the only subsets of Cn which are both open and closed
in Zariski topology are ∅ and Cn. Show that O(Cn) is the same whether Cn is
regarded as an open subset or a closed subset ofCn. Also, a similar statement holds
forO(∅).
5.35 The local ring OX,P of germs LetX ⊂ Cn be a locally closed subvariety
and let P ∈ X. Consider the set FP of all ordered pairs (U, f) where U ⊂ X is a
Zariski open neighbourhood of P in X, and f ∈ O(U). On the set FP we put an
equivalence relation as follows. We say that pairs (U, f) and (V, g) are equivalent if there exists an open neighbourhood W of P in U ∩V such that f|W =g|W. Each
equivalence classes is called a germ of a regular function onX at P.
The set of all equivalence classes forms a C-algebra (where operations of addition
and multiplication are carried out point-wise on restrictions to common small neigh-bourhoods), denoted byOX,P.
5.36 ExerciseVerify that the aboveC-algebra structure on OX,P is well-defined.
5.37 ExerciseShow that we have a well-definedC-algebra homomorphism (called
the evaluation map)
OX,P →C: (U, f)7→f(P)
Show that OX,P is a local ring, whose unique maximal ideal is the kernel mX,P of
the evaluation mapOX,P →C.
5.38 The ring OX,P as localisation Let X be a closed subvariety of Cn,
de-fined by radical ideal I. Let P ∈ X be defined by maximal ideal m ⊂ O(X) =
C[x1, . . . , xn]/I. Then show that we have a canonical isomorphism of C-algebras O(X)m→ OX,P.
Deduce that if X = Z∩Uf where Z is closed in Cn and Uf is a principal open in
Cn, then again we have a canonical isomorphism of C-algebras O(X)
m→ OX,P.
5.39 Regular morphisms Let X ⊂ Cm and Y ⊂ Cn be locally closed
5.40 Regular morphisms to C Show that a regular morphism f : X → C is
the same as a regular functionf, that is, an element ofO(X).
5.41 Regular morphisms to Cn Show that a regular morphism f : X → Cn
is a map f = (f1, . . . , fn) : X → Cn such that each of the component maps fi =
πi◦f :X → C is a regular function on X, that is, each fi is in O(X). If Y ⊂ Cn
is a locally closed subvariety, show that a regular morphismf :X →Y is the same as a regular morphismf :X →Cn whose image lies inY.
5.42 Exercise Show that for any closed subvariety X ⊂ Cm, a map X → Cn is
a regular morphism if and only if there exist polynomialsg1, . . . , gn∈C[x1, . . . , xm]
such that fi = gi|X. In other words, when X is a closed subvariety, every regular
morphism f : X → Cn admits a prolongation to a regular morphism g : Cm → Cn
(withg|X =f).
5.43 ExerciseIf X ⊂Cm and Y ⊂Cn are Zariski closed subsets and f :X →Y
a regular morphism, does there exist a regular morphism g : Cm → Y such that
the restriction g|X equals f : X → Y? (Hint: Consider the special case where
m=n= 1, X =Y and f = idX.)
5.44 Exercise: Pull-back of regular functions Let X ⊂ Cm and Y ⊂ Cn be
locally closed subvarieties, and let f : X → Y be a regular morphism. Show that
φ7→φ◦f defines a C-algebra homomorphismf#:O(Y)→ O(X).
5.45 TheoremLetX ⊂Cm be a locally closed subvariety andY ⊂Cnbe a closed
subvariety. Then associating the pull-back C-algebra homomorphismf#:O(Y)→
O(X) to a regular morphism f :X →Y defines a bijective correspondence between the set of all regular morphismsX →Y and the set of allC-algebra homomorphisms O(Y)→ O(X).
5.46 Exercise Suppose that in the hypothesis of the above theorem, instead of
Y being closed, we take Y to be merely locally closed. Then does the conclusion of the theorem remain valid? (Hint: TakeY = C2− {P}). Is it true that f 7→f# is
injective even whenY ⊂Cn is locally closed?
5.47 Exercise: Composition of regular morphisms Let X ⊂ Cm, Y ⊂ Cn
and Z ⊂ Cp be Zariski locally closed, and let f : X → Y and g : Y → Z be
regular morphisms. Then show that the composite g ◦ f : X → Z is again a regular morphism. Moreover, show that the composite f#◦ g# of the pull-back
homomorphism (g ◦f)# for the composite, that is, f#◦g# = (g◦f)# : O(Z) →
O(X).
5.48 The category of quasi-affine varieties We define a category QAV as follows. The objects of QAV are all locally closed subvarieties of Cn, as n varies
over all non-negative integers. We call them as quasi-affine varieties. The morphisms inQAV are regular morphisms as defined above.
5.49 Affine variety A quasi-affine varietyX ⊂Cn is said to be an affine variety
if there exists some closed subvariety Z ⊂Cm such that X is isomorphic to Z.
5.50 Note Similarly, a quasi-projective variety (which will be defined later) is said to be an affine variety if there exists some closed subvarietyZ ⊂Cm such that
X is isomorphic to Z.
5.51 Exercise If a locally closed subvariety X ⊂ Cn can be expressed as an
intersectionZ∩Uf where Z ⊂Cn is Zariski closed andUf =Cn−Z(f) is principal
open, then show thatX is affine.
5.52 Existence of an affine open cover Deduce from the above exercise that given any locally closed subvariety X ⊂ Cn, there exist an open cover of X by
subsetsVi ⊂X such that each Vi is an affine variety.
5.53 Equivalence of categoriesShow that we have a contravariant functor from the category of all affine varieties (regarded as a full subcategory of all quasi-affine varieties) to the category of all finite-type reduced C-algebras, which associates to
any affine variety X the C-algebra O(X), and associates to any regular morphism
f : X → Y the pull-back homomorphism f# : O(Y) → O(X). Show that this
functor is fully faithful and essentially surjective, and therefore an (anti-)equivalence of categories.
6
Projective spaces
Let V be a vector space over C. The set of points of the corresponding projective
spaceP(V) is the set of all 1-dimensional vector subspaces L⊂V. In particular, if
V = 0 then P(V) is empty.
ForV =Cn+1wheren≥0 is an integer, the corresponding projective spaceP(Cn+1)
is denoted byPn, and it is called then-dimensional complex projective space.
The spaceP1is called as the complex projective line, andP2 is called as the complex
Any pointL⊂Cn+1 ofPn is represented by a vector 06=v ∈L. If v = (a
0, . . . , an),
then we call then-tuple (a0, . . . , an) as the homogeneous coordinates ofL. Note that
not allai can be zero. Given any (a0, . . . , an)∈Cn+1−0, we denote byha0, . . . , ani
the corresponding pointL of Pn. With this notation, we have
ha0, . . . , ani=hλa0, . . . , λani for all λ6= 0
Conversely, if ha0, . . . , ani = hb0, . . . , bni ∈ Pn, then there exists some λ 6= 0 such
that (a0, . . . , an) =λ(b0, . . . , bn)∈Cn+1.
We give Pn the quotient topology induced from the topology on Cn+1 − {0}. If
we give Cn+1− {0} the Zariski topology, then we call the corresponding quotient
topology onPn as the Zariski topology onPn. If we givenCn+1− {0}the Euclidean
topology, then we call the corresponding quotient topology onPn as the Euclidean
topology onPn.
6.1 Exercise Show that the Euclidean topology on Pn is finer than the Zariski
topology onPn. Show thatPnis Hausdorff in the Euclidean topology, but forn
≥1 Pn is not Hausdorff in the Zariski topology.
6.2 ExerciseShow that the map φ:C→P1 :z 7→ hz,1i gives an open inclusion
of C in P1, with P1 −φ(C) equal to the singleton set {h1,0i}. This extra point
was regarded by Riemann as the ‘point at infinity’ to be added toC, to obtain the
‘Riemann sphere’, which we now identify with P1.
6.3 Exercise Show that we have a surjective map from the unit sphere S2n+1 ⊂
Cn+1 to the projective space Pn. Deduce that Pn is compact in the Euclidean
topology.
6.4 Projective linear subspaces If W ⊂ V is a vector subspace, then we get an inclusion P(W) ֒→ P(V). For various linear subspaces W of Cn+1, show that
the corresponding subsetsP(W)⊂Pn are closed in Zariski topology.
6.5 Open cover by affine spacesLetf :V →Cbe a non-zero linear functional.
LetK = ker(f), and let v ∈V with f(v)6= 0. This gives an injective map
φ:K →Pn :w
7→ hv+wi
When V =Cn+1, show that any such map is open in both the Zariski topology and
Euclidean topology (whereK becomes isomorphic to Cn on choosing a linear basis,
and so has a well-defined Zariski as well as Euclidean topology). We have
Takingf to be the linear functionalxi for i= 0, . . . , n, show that the corresponding
φ(K) define an open cover of Pn byn+ 1 copies of Cn. When f =x
i, v =ei, and
K is given the basis (e0, . . . , ei−1, ei+1, . . . , en) show that the corresponding map φ
(which we denote by ψi is given by
ψi :Cn→Pn: (z0, . . . , zi−1, zi+1, . . . , zn)7→ hz0, . . . , zi−1,1, zi+1, . . . , zni
6.6 Exercise Show that any two projective linear subspaces P(V) ⊂ Pn and
P(W)⊂Pn have intersectionP(V ∩W). In particular, any two distinct projective
lines inP2 intersect in exactly a single point.
7
Algebraic subvarieties of projective spaces
7.1 Closed subsets Z(S)⊂PnLetS ⊂C[x
0, . . . , xn] be a set, all whose elements
are homogeneous polynomials. Then it defines a subsetZ(S) ⊂Pn, which consists
of all pointsha0, . . . , ani such thatf(a0, . . . , an) = 0 for allf ∈S. This condition is
well-defined, as
f(λa0, . . . , λan) = λdf(a0, . . . , an)
for any homogeneous polynomial of degreed. Under the quotient map
q:Cn+1− {0} →Pn
the inverse image ofZ(S)⊂Pnis the subsetZ(S)∩(Cn+1−{0}), whereZ(S)⊂Cn+1
denotes the Zariski closed subset defined by S. Hence by definition of quotient topology,Z(S) is Zariski closed in Pn. Similarly, Z(S) is Euclidean closed in Pn.
7.2 ExerciseLetX ⊂Cn+1 be a closed subvariety, which is conical (that is, ifv ∈
X then λv∈X for all λ∈C). Then show that there exists a set S ⊂C[x0, . . . , xn]
of homogeneous polynomials, such that X = Z(S). (Hint: If f vanishes over X, show that each homogeneous component of f also vanishes over X).
7.3 Homogeneous ideals An ideal I ⊂ C[x0, . . . , xn] is called homogeneous if
for any f ∈ I, all the homogeneous components of f are in I. Deduce from the above exercise that a subset X ⊂ Pn is Zariski closed if and only if there exists a
homogeneous idealI ⊂C[x0, . . . , xn] such thatX =Z(I).
7.4 Projective variety A projective variety is a Zariski closed subset ofPn.
7.5 Homogeneous Nullstellensatz Let I ⊂ C[x0, . . . , xn] be a homogeneous
ideal. Letf ∈C[x0, . . . , xn] be a homogeneous polynomial of degree≥1. Show that
7.6 Exercise: When is Z(I) empty Let I ⊂ S = C[x0, . . . , xn] be a
homo-geneous ideal. Show that Z(I) is empty if and only if S+ ⊂
√
I, where S+ ⊂
C[x0, . . . , xn] is the ideal (x0, . . . , xn).
7.7 Theorem There is an inclusion-reversing bijective correspondence between the set HRI of all homogeneous radical ideals in S = C[x0, . . . , xn] which do not
contain the ideal S+, and the set of all non-empty closed subvarieties of Pn, given
byI 7→Z(I), with inverse given by
X 7→IX ={f ∈C[x0, . . . , xn]|f(a0, . . . , an) = 0 for all ha0, . . . , ani ∈X}
Under this correspondence, the set of all maximal elements of HRI is in bijection with the set of all points of Pn. The set of all prime ideals in HRI is in bijection
with the set of all irreducible closed subvarieties in Pn.
7.8 Affine coneXb over a projective variety X ⊂PnGiven any closed subset
X ⊂Pn defined by a homogeneous radical idealI ⊂C[x
0, . . . , xn], the closed subset
b
X =Z(I)⊂Cn+1 is called the affine cone overX. The ringC[x0, . . . , xn]/I =O(Xb)
is called the homogeneous coordinate ring ofX ⊂Pn.
7.9 Exercise: NoetheriannessShow thatPnunder Zariski topology is a
noethe-rian topological space, that is, any decreasing sequence of closed subsets is finite. Deduce that any locally closed subvariety of Pn is noetherian. In particular, any
locally closed subvarietyX ⊂Pnis quasi-compact, that is, every Zariski open cover
ofX has a finite subcover. Moreover,X can be uniquely written as a minimal finite union of irreducible closed subsets of X.
7.10 Principal open subsets Let f ∈C[x0, . . . , xn] be a homogeneous
polyno-mial. The Zariski open subsetUf =Pn−Z(f) is called the principal open subset
defined by f. Show that principal open subsets form a basis of open sets for the Zariski topology onPn.
7.11 Regular functions on an open subvariety A function φ : V → C on
an open subvarietyV ⊂ Pn is called a regular function if given any principal open
Uf ⊂ V, the restriction φ|Uf is of the form g/fr where g ∈ C[x1, . . . , xn] is a
homogeneous polynomial andris a non-negative integer such that deg(g) =rdeg(f) (so that the rational functiong/fr is homogeneous of degree 0).
7.12 Exercise Show that if (Ufi)i=1,...,m is a principal open cover of an open
subvariety V ⊂Pn, and if φ :V → C is a function such that each restriction φ| Ufi
is of the above form gi/firi where gi ∈ C[x1, . . . , xn] is a homogeneous polynomial
