Introduction to Algebraic Varieties
Introduction to Algebraic Varieties ∗
Nitin Nitsure † CAAG lectures, July 2005
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 n C submanifolds of R ............................. 2
3 Holomorphic submanifolds of C n ........................ 8
4 Some algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5 Algebraic subvarieties of C n . . . . . . . . . . . . . . . . . . . . . . . . . . 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: nitsure@math.tifr.res.in
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 R n
and its submanifolds, Algebraic Geometry in its classical form is the study of the affine space C n n and the projective space P
C , 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 n C submanifolds of R
2.1 n Linear coordinates The linear coordinates x
1 ,...,x n on R are the projec-
i :R → R is the i th projection function.
tions from the product R n to the individual factors R, that is, x
2.2 Polynomial functions Any polynomial f ∈ R[x 1 ,...,x n ] defines a function f:R n → R called a polynomial function.
2.3 Exercise Show that two polynomials in R[x 1 ,...,x n ] are equal to each other if and only if the corresponding functions R n → R are equal to each other.
2.4 n Linear functions These are functions on R defined by polynomials of degree
1. They have the form i a i x i + c where at least one of the a i is non-zero.
2.5 Change of linear coordinates Let A ∈ GL n (R) be an invertible n × n- matrix over R. Let b = (b n
be linear functions defined P by y n
1 ,...,b n ) ∈R . Let y 1 ,...,y n
A i,j x j +b i . Then we say that the y i are a set of linear coordinates on R . As A is invertible, we can express each x i as a linear functions of y 1 ,...,y n .
2.6 n Open subsets We give R the Euclidean topology. This makes R the prod- uct topological space of n copies of the topological space R, where R is given its
usual topology induced by the metric d(x, y) = |x − y|.
be an open subset in Euclidean topology. A function f : U
2.7 n C functions Let U ⊂R
→ R is said to be a C ∞ function if f is continuous and all partial derivatives of f of all orders exist and are continuous.
2.8 Note Because U is assumed to be open, the definition of a partial derivative makes sense for functions on U .
2.9 ∞ Exercise Give an example of open subsets U ⊂V⊂R and a C function f:U ∞ → R, such that there exists no C function g : V → R such that g|
U =f.
2.10 ∞ Exercise Give an example of nonempty open subsets U ⊂V⊂R and a C function f : U ∞ → R, such that there exist two different C function g, h : V →R such that g | U =h | U =f.
2.11 Exercise In the above two exercises, can we have polynomial functions in place of ∞ C functions?
be the coordinates on R . For any set X, a map f : X m →R is the same as an an ordered m-tuple of maps
2.12 m C map from U ⊂R to R Let y
1 ,...,y m
i :X → R, where f i =y i ◦ f. If U is open in R , a map f = (f 1 ,...,f m ):U →R
is called a ∞ C map if each of the coordinate maps f
i :U → R is C .
be open subsets. A ∞ C - isomorphism f : U → V is a topological homeomorphism f of U onto V with
2.13 m C -Isomorphism Let U ⊂R and V ⊂R
inverse g : V m → U such that the maps f and g are C regarded as maps U →R
and V ∞ →R .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’.
1 ,...,u n ) in R is a tuple con- sisting of a non-empty open subset U n together with a sequence of n ∞ ⊂R C
2.16 n Coordinate chart A coordinate chart (U ; u
functions u ∞
i :U → R such that the resulting map u : U → R is a C -isomorphism of U onto an open subset V n ⊂R .
2.17 n Inverse function theorem Let U ⊂R be an open subset and let f : U → R n
be a ∞ C map. Let P ∈ U such that the determinant of the n × n matrix (the Jacobian)
∂f i ∂x j
is non-zero at P . Then there exists an open neighbourhood V of P in U (that is, P
V is a ∞ C -isomorphism of V onto an open subset W of R n .
∈ V ⊂ U with V open in R n ) such that the restriction f |
be a ∞ C map such that the Jacobian matrix (∂f i /∂x j )(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.
2.18 n Exercise Let U ⊂R be an open subset and let f : U →R
2.19 ∞ Exercise Let U ⊂R be an open subset and let f : U →R be a C map such that (i) f is injective, and (ii) the Jacobian matrix (∂f i /∂x j )(P ) is invertible
for all P ∞ ∈ U. Then show that f is a C -isomorphism of U onto an open subset of R n .
2.20 n Cubical coordinate chart Let (U ; u
1 ,...,u n ) be a coordinate chart in R ,
such that the map u : U ∞ →R is a C -isomorphism of U onto an open subset
V n ⊂R where V is of the form
V= n { (b
1 ,...,b n ) ∈R |−a<b i <a }
for some a > 0. The point P ∈ U such that u i (P ) = 0 for all i = 1, . . . , n is called the centre of the cubical coordinate chart.
2 2.21 2 Exercise Let r and θ be the usual polar coordinates on R . Let U ⊂R be the subset where 1 < r < 3 and 0 < θ < 2π. Let u 1 =r − 2 and let u 2 = (θ − π)/π.
Then show that (U ; u 2
1 ,u 2 ) is a Cubical coordinate chart in R . What is its centre?
2.22 Locally closed submanifolds Let d be an integer with 0 ≤ d ≤ n. Let
X n ⊂R be a locally closed subset of R (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 R n of dimension d, where d ≥ 0 is an integer, if X is non-empty and each P ∈ X is the
centre of some cubical coordinate chart (U ; u n
1 ,...,u n ) in R such that
X ∩U={Q|u i (Q) = 0 for all d + 1 ≤i≤n} If X is empty, it is called a locally closed submanifold of R n of dimension −∞ (or −1 in another convention).
If a locally closed submanifold X of R n is closed in V where V ⊂R is open, then
X is called a closed submanifold of V .
2.23 n Exercise If U ⊂ X is open in a locally closed submanifold X of R , then show that U itself is a locally closed submanifold of R n . What is its dimension?
2.24 C ∞ -functions on submanifolds Let X be a locally closed submanifold of R n
. Let f : X ∞ → R be a function. Then f is called a C -function on X if for each P n together with a ∞ ∈ X there exists an open neighbourhood U in R C -function
g:U → R such that
f | X∩U =g | X∩U
2.25 n The R-algebra ∞ C (X) If X is a locally closed submanifold of R , then all
C ∞ -functions f : X → R form a commutative R-algebra C (X) under point-wise operations.
2.26 Exercise If X is empty, show that ∞ C (X) = 0.
2.27 ∞ The sheaf property Let U and V be open subsets of a locally closed C
submanifold X of R ∞ , such that U ⊂ V ⊂ X. Then for any C -function f : 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|
Given any locally closed n C -submanifold V of R and any open covering (U
i ) i∈I of
V , the following two properties are satisfied: (1) Separatedness If f ∞ ∈C (V ) such that each restriction f |
U i is 0, then f = 0. (2) Gluing If for each U ∞
i we are given a C -function f i such that for any pair (i, j) we have
i | U i ∩ U j =f j | U i ∩ U j ∈C (U i ∩U j )
then there exists some f ∞ ∈C (V ) such that f
i =f | U i for each i. (Such an f will therefore be unique by (1)).
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
2.28 n Exercise Let X ⊂R
chart (U ; u n
1 ,...,u n ) in R such that X ∩U={Q|u i (Q) = 0 for all d + 1 ≤ i ≤ n },
X∩U is a C -function of the coordinates u 1 ,...,u d .
Show that it is enough to check the above condition for a family of such charts which covers X.
be a locally closed submanifold and let P ∈ X. Consider the set F P of all ordered pairs (U, f ) where U ⊂ X is an open neighbourhood of P in X, and f ∞ ∈C (U ). On the set F
2.29 n The local ring of germs Let X ⊂R
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 a ∞ C -function on X at P .
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 by ∞ C X,P .
2.30 ∞ Exercise Verify that the above R-algebra structure on C X,P is well-defined.
2.31 Exercise Show that we have a well-defined R-algebra homomorphism (called the evaluation map)
X,P → R : (U, f) 7→ f(P )
Show that ∞ C X,P is a local ring, whose unique maximal ideal is the kernel m X,P of
the evaluation map ∞ C X,P → R.
2.32 n C -maps between submanifolds Let X ⊂R and Y ⊂R
be locally
closed submanifolds. A map f = (f 1 ,...,f m ):X
→ Y is called C ∞ if each f i is ∞ C ,
equivalently, if the composite X ∞ → Y ֒→ R is C .
2.33 m Implicit function theorem Let x
1 ,...,x m be linear coordinates on R and
be linear coordinates on R . On the product R =R ×R we get linear coordinates x m+n
let y n
1 ,...,y n
m+n m
1 ,...,x m ,y 1 ,...,y n . Let U ⊂R
be an open neighbourhood
of the origin 0 = (0, . . . , 0) ∞ ∈R . Let f = (f
m+n
1 ,...,f n ) be an n-tuple of C
functions (same as a n C -map f : U →R ), such that
f (0) = 0
Let the following n × n-matrix
∂f i ∂y j 1≤ i,j≤n
be invertible at the point 0 m+n ∈R . Then there exists real numbers a, b > 0 and a
1 ,...,g n ):V a →W b where V a ⊂R is the open subset defined in terms of the coordinates by n −a < x
C m -function g = (g
i < a and W b ⊂R is the open subset defined in terms of the coordinates by −b < y j < b such that V a ×W b ⊂ U and
− f 1 (0) ∩ (V
a ×W b )= { (x, g(x)) | x ∈ V a }
2.34 Exercise Deduce the implicit function theorem from the inverse function theorem, and conversely.
2.35 q Closed submanifolds and implicit function theorem Let U ⊂R be open, and let f : U n
be a →R ∞ C -map such that for each point P ∈ U with
f (P ) = 0, the rank of the n × q-matrix
∂f i (P )
∂x j
1≤ i≤n, 1≤j≤q
− is n. 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 n Real analytic category A function f : U → R, where U ⊂ R is open, is called a real analytic function if around each point P = (a 1 ,...,a n ) ∈ U it is given
by a convergent power series in the n variables x 1 −a 1 ,...,x n −a n . The concepts of analytic isomorphism, analytic coordinate chart, cubical analytic coordinate chart,
and locally closed analytic submanifold of R n are 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 ⊂R is a closed subset then there exists a C function f : R n → R which vanishes exactly on X.
2.39 Exercise A similar property does not hold for analytic functions: show that if U n ⊂V⊂R are non-empty open subsets where V is connected, and if f : V →R
is an analytic function such that f | U = 0 then f = 0.
2.40 Exercise: Spheres, classical matrix groups, surfaces
2 Show that the unit sphere S 2 ⊂R , defined by x
1 +...+x n = 1, is a closed analytic submanifold. What is its dimension? Show that SL n×n
n−1
n (R) ⊂R is a closed analytic submanifold, and determine its dimension.
Show that for any integer g 3 ≥ 0, there exists a closed C submanifold of R which is homeomorphic to a 2-sphere with g handles.
3 Does R ∞ have a closed C submanifold homeomorphic to the real projective plane? Does R 3 have a closed ∞ C submanifold homeomorphic to the mobius strip without
boundary?
3 n Holomorphic submanifolds of C
The 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.
1 ,...,x n . Let x i √ = u n
3.1 n Open subsets Let C have complex linear coordinates x
i + −1v i where u i and v i are the real and imaginary parts. We give C the
2 topology of R n , under the bijection (x
1 ,...,x n ) 7→ (u 1 ,v 1 ,...,u n ,v n ). Open (or closed) subsets in C n mean open (or closed) subsets in the Euclidean topology on
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 = (a 1 ,...,a n ) ∈ U, the function f is given by a convergent power series in the n complex variables x 1 −a 1 ,...,x n −a n in a neigh- bourhood of P .
3.2 n Holomorphic functions and maps Let U ⊂C
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).
1 ,...,f m ):U →C is said to be holomorphic (or complex analytic) if each f i :U → C is analytic.
A map f = (f m
be open subsets. A holomorphic isomorphism f : U → V is a topological homeomorphism f of U onto
3.3 m Holomorphic isomorphism Let U ⊂C and V ⊂C
V with inverse g : V → U such that the maps f and g are holomorphic regarded
as maps U n →C and V →C . A holomorphic isomorphism is also called a a bi-holomorphic map.
3.4 Exercise If U 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 n Holomorphic coordinate chart A holomorphic coordinate chart in C is a tuple (U ; u n
1 ,...,u n ) consisting of a non-empty open subset U ⊂C together with
a sequence of n holomorphic functions u i :U → C such that the resulting map
u:U n →C is a holomorphic isomorphism of U onto an open subset V ⊂C .
3.6 n Inverse function theorem Let U ⊂C
be an open subset and let f : U →
C n be a holomorphic map. Let P ∈ U such that the determinant of the n×n matrix
(the Jacobian)
∂f i ∂x j
is non-zero at P . Then there exists an open neighbourhood V of P in U (that is, P n ∈ V ⊂ U with V open in C ) such that the restriction f |
V is a holomorphic isomorphism of V onto an open subset W of C n .
3.7 n Exercise Let U ⊂C be an open subset and let f : U →C be a holomorphic map such that the Jacobian matrix (∂f i /∂x j )(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.
3.8 n Exercise Let U ⊂C be an open subset and let f : U →C be a holomor- phic map such that (i) f is injective, and (ii) the Jacobian matrix (∂f i /∂x j )(P ) is
invertible for all P ∈ U. Then show that f is a holomorphic isomorphism of U onto an open subset of C n .
3.9 Cubical coordinate chart (polydisk) Let (U ; u 1 ,...,u n ) be a holomorphic
coordinate chart in C n , such that the map u : U →C is a holomorphic isomorphism of U onto an open subset V n ⊂C where V is of the form
V= n { (b
1 ,...,b n ) ∈C | |b i |<a}
for some a > 0. The point P ∈ U such that u i (P ) = 0 for all i = 1, . . . , n is called the centre of the cubical coordinate chart.
3.10 Locally closed holomorphic submanifolds Let d be an integer with 0 ≤
d n ≤ n. Let X ⊂ C be a locally closed subset of C . Then X is called a locally closed holomorphic submanifold of C n of (complex) dimension d, where d ≥ 0 is an integer, if X is non-empty and each P ∈ X is the centre of some cubical coordinate chart (U ; u n
1 ,...,u n ) in C such that
X ∩U={Q|u i (Q) = 0 for all d + 1 ≤i≤n} If X is empty, it is called a locally closed submanifold of C n of dimension −∞ (or −1 in another convention).
If a locally closed submanifold X of C n is closed in V where V ⊂C is open, then
X is called a closed submanifold of V .
3.11 n Exercise If U ⊂ X is open in a locally closed submanifold X of C , then show that U itself is a locally closed submanifold of C n . What is its dimension?
3.12 Holomorphic functions on submanifolds Let X be a locally closed sub- manifold of C n . Let f : X → C be a function. Then f is called a holomorphic
function on X if for each P n ∈ X there exists an open neighbourhood U in C together with a holomorphic function g : U → C such that
f | X∩U =g | X∩U
3.13 n The C-algebra H(X) If X is a locally closed submanifold of C , then all holomorphic functions f : X → C form a commutative C-algebra H(X) under
point-wise operations. If X is empty, we have H(X) = 0.
3.14 The sheaf property Let U and V be open subsets of a locally closed holomorphic submanifold X of C n , such that U ⊂ 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 an C-algebra homomorphism
H(V ) → H(U) : f 7→ f| U
Given any locally closed holomorphic submanifold V of C n and any open covering (U i ) i∈I of V , the following two properties are satisfied: (1) Separatedness If f ∈ H(V ) such that each restriction f| U i is 0, then f = 0. (2) Gluing If for each U i we are given a holomorphic function f i such that for any
pair (i, j) we have
f i | U i ∩ U j =f j | U i ∩ U j ∈ H(U i ∩U j )
then there exists some f ∈ H(V ) such that f i =f | U i for each i. (Such an f will therefore be unique by (1)).
be a locally closed submanifold of dimension d ≥ 0. Show that a function f : X → C is holomorphic if and only if for every cubical holo- morphic coordinate chart (U ; u n
3.15 n Exercise Let X ⊂C
1 ,...,u n ) in C such that X ∩U={Q|u i (Q) =
0 for all d + 1 ≤ i ≤ n }, f| X∩U is a holomorphic function of the coordinates u 1 ,...,u d .
Show that it is enough to check the above condition for a family of such charts which covers X.
be a locally closed submanifold and let P ∈ X. Consider the set F P of all ordered pairs (U, f ) where U ⊂ X is an open neighbourhood of P in X, and f ∈ H(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 a holomorphic function on X at P .
3.16 n The local ring of germs Let X ⊂C
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 by H X,P .
3.17 Exercise Verify that the above C-algebra structure on H X,P is well-defined.
3.18 Exercise Show that we have a well-defined C-algebra homomorphism (called the evaluation map)
H X,P → C : (U, f) 7→ f(P )
Show that H X,P is a local ring, whose unique maximal ideal is the kernel m X,P of
the evaluation map H X,P → C.
3.19 Holomorphic maps between submanifolds Let X ⊂C m and Y ⊂C n be locally closed holomorphic submanifolds. A map f = (f 1 ,...,f m ):X → Y is called
holomorphic if each f n
i is holomorphic, equivalently, if the composite X → Y ֒→ C is holomorphic.
3.20 Exercise: Pull-back of holomorphic functions Let X and Y be locally
closed holomorphic submanifolds of C n and C , 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 Exercise With X and Y as above, let f 1 and f 2 be holomorphic maps from
X to Y . If f #
1 =f 2 : H(Y ) → H(X), then show that f 1 =f 2 .
3.22 Exercise: Composition of holomorphic maps Let X, Y and Z be locally
closed holomorphic submanifolds of C p ,C and C respectively, and let f : X →Y and g : Y → Z be holomorphic maps. The show that the composite g ◦ f : X → Z
is a holomorphic map.
3.23 m Implicit function theorem Let x
1 ,...,x m
be linear coordinates on C
and let y m
1 ,...,y n
be linear coordinates on C . On the product C =C ×
m+n
be an open neighbourhood of the origin 0 = (0, . . . , 0) m+n ∈C . Let f = (f
C n we get linear coordinates x m+n
1 ,...,x m ,y 1 ,...,y n . Let U ⊂C
1 ,...,f n ) be an n-tuple of holomorphic functions (same as a holomorphic map f : U n →C ), such
that
f (0) = 0
Let the following n × n-matrix
∂f i ∂y j 1≤ i,j≤n
be invertible at the point 0 m+n ∈C . Then there exists real numbers a, b > 0 and a holomorphic function g = (g m
1 ,...,g n ):V a →W b where V a ⊂C is the open subset defined in terms of the coordinates by n |x
i | < a and W b ⊂C is the open subset defined in terms of the coordinates by |y j | < b such that V a ×W b ⊂ U and
− f 1 (0) ∩ (V
a ×W b )= { (x, g(x)) | x ∈ V a }
3.24 Exercise Deduce the implicit function theorem from the inverse function theorem, and conversely.
3.25 q Closed submanifolds and implicit function theorem Let U ⊂C be open, and let f : U n →C
be a holomorphic map such that for each point P ∈U with f (P ) = 0, the rank of the n × q-matrix
∂f i (P )
∂x j
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 2 Example: Non-singular algebraic curves in C Let f ∈ C[x, y] be an irreducible polynomial of degree d 2 ≥ 1. Let Z(f) ⊂ C
be defined as
Z(f ) = 2 { (a, b) ∈ C | f(a, b) = 0 }
Suppose that at any point P = (a, b) ∈ Z(f), at least one of the partial derivatives ∂f /∂x and ∂f /∂y is non-zero. Then by application of the implicit function theorem,
Z(f ) is a closed submanifold of C 2 of dimension 1.
2 3.28 4 Exercise Identifying C with R , we can regard Z(f ) as a closed subset of R 4 ∞
. Is it a 4 C submanifold of R , and if so of what dimension?
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 R n according to this definition is a pair (X, φ) where X is
an abstract n C -manifold (see the last section for definition), and φ : X →R 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 a ′ C -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
1 2 elementary example, let X = R t , and let φ : X →R be defined by t 7→ (e , sin t).
A similar more general notion exists for locally closed holomorphic submanifolds of
C n 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 be integral element over A (or more precisely, integral over A with respect to φ) if there exists some integer n ≥ 1 and elements a 1 ,...,a n ∈ A such that
b n−1 + φ(a
1 )b + . . . + φ(a n ) = 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) of B 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 Lemma Let A ⊂ B be a ring extension and b ∈ B. The following statements are equivalent:
(1) b is integral over A. (2) The A-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 module M over a ring R is called faithful if no non-zero element of R kills all of M .)
Proof Clearly (1) ⇒ (2) ⇒ (3). Now let M be a faithful A[b]-module which is finite as an A-module, generated by v 2
1 ,...,v n . Then there exist n elements c i,j P n
∈ A such that bv i = j=1 c i,j v j for each i = 1, . . . , n. Hence by the determinant argument, ∈ A such that bv i = j=1 c i,j v j for each i = 1, . . . , n. Hence by the determinant argument,
4.3 Lemma Let A ⊂ B be rings, such that B is finite over A. Then B is integral over A.
Proof For each b ∈ B, note that B is a faithful A[b]-module M which is finite as an A-module.
4.4 Lemma Let A ⊂ B be rings, such that B is finite-type and integral over A. Then B is finite over A.
Proof Let B = A[b 1 ,...,b n ]. Then consider the tower of ring extensions A ⊂ A[b 1 ] ⊂ A[b 1 ,b 2 ] ⊂ . . . ⊂ A[b 1 ,b 2 ,...,b n ] = B, in which each step is finite.
4.5 Lemma Let A ⊂ B be rings. Then all elements of B which are integral over
A form an A-subalgebra of B. Proof If b ∈ B is integral over A, then A[b] is finite over A. If c ∈ B is integral over
A then c is integral over A[b] and so A[b, c] is finite over A[b]. Hence A[b, c] is finite over A.
Nullstellensatz
4.6 Theorem Let A ⊂ K be an integral ring extension. If K is a field, then A is
a field. Proof For any a ∈ A with a 6= 0, the element b = 1/a of K is integral over A. Let
b n +a 1 b n−1 +...+a n = 0. Then multiplying by a n−1 , we get b = −(a 1 +a 2 a+...+
a n−1 n a ) ∈ A.
4.7 Lemma Let k be a field and k[x 1 ,...,x r ] be a polynomial ring in r variables where r ≥ 1. If f ∈ k[x 1 ,...,x r ] then the localisation k[x 1 ,...,x r , 1/f ] is not a field.
Proof There are infinitely many non-constant irreducible polynomials which are not scalar multiples of each other. Hence there is a non-constant irreducible polynomial
p which does not divide f . It follows that p is not invertible in k[x 1 ,...,x r , 1/f ].
4.8 Theorem Let k 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[a 1 ,...,a n ].
If A is not algebraic, there exists an integer r ≥ 1 such that after re-indexing If A is not algebraic, there exists an integer r ≥ 1 such that after re-indexing
f r+1 (t), . . . , f n (t) in k(a 1 ,...,a r )[t]. Let g(a 1 ,...,a r ) ∈ k[a 1 ,...,a r ] be the product of all the denominators of the coefficients of the f r+1 (t), . . . , f n (t) (written as rational fractions in any chosen way). Then A is integral over k[a 1 ,...,a r , 1/g]. Hence k[a 1 ,...,a r , 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 a 1 ,...,a r ∈A which are algebraically independent over k such that A is finite over the polynomial ring k[a 1 ,...,a r ]. Therefore as A is a field, this polynomial ring is also a field, which shows r = 0.)
4.9 Theorem Let k be a field and φ : A → B a homomorphism between finite- − type k-algebras. Then for any maximal ideal m 1 ⊂ B the inverse image φ (m) is a
maximal ideal in A. − Proof B/m is algebraic hence integral over k, and k 1 ⊂ A/φ (m) ⊂ B/m. Hence
B/m is 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 A f is non-zero, so it has a maximal ideal m ⊂A f . We have a homomorphism A →A f of finite-type k-algebras, so the contraction m c ⊂ A of m is maximal. As f lies in each maximal ideal of A, we get
f c ∈m and so f ∈ m. Contradiction, as f is a unit in A
Algebraically closed base field
4.11 Corollary If k is an algebraically closed field, then any maximal ideal in a polynomial ring k[x 1 ,...,x n ] is of the form (x 1 −a 1 ,...,x n −a n ) for some a 1 ,...,a n ∈ k.
4.12 Corollary If k is an algebraically closed field and I ⊂ k[x 1 ,...,x n ] is an ideal such that I n 6= (1), then the locus Z(I) ⊂ k defined by I is non-empty.
4.13 Corollary If k is an algebraically closed field, I ⊂ k[x 1 ,...,x n ] is an ideal, and f ∈ k[x 1 ,...,x n ] is a polynomial such that f (a 1 ,...,a n ) = 0 for each point
(a m
1 ,...,a n ) ∈ Z(I) ⊂ k , then f ∈ I for some integer m ≥ 1.
Prime decomposition
4.14 Theorem Let A be a noetherian ring. Then every radical ideal I of A is the intersection of a unique minimal finite set {p 1 ,...,p n } of prime ideals in A:
I=p 1 ∩...∩p n
Proof We will first show that every radical ideal in A is an intersection of a finite set of prime ideals. Suppose not, then by noetherian condition, there is a maximal element I in 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 that I is radical, we see that
I = (I + (a)) ∩ (I + (b))
Taking radicals, we get
By maximality of I, the radical ideals
I + (b) are finite intersections of prime ideals, hence so is I. If I = p 1 ∩...∩p n then by discarding at most n of the ideals p i , we arrive at a minimal set S of primes which has intersection I (where we say that a set S = {p 1 ,...,p n } which has intersection ∩ p∈ S p = I is minimal if there is no proper subset S ′ ⊂ S which also has intersection I).
I + (a) and
Next, we show the uniqueness of a minimal set S. Clearly, S is empty if and only if
I = A. So now we assume that cardinality of S is n ≥ 1. Let S = {p 1 ,...,p n } and T= {q 1 ,...,q m } be two minimal such sets, with n ≤ m. As p 1 ∩...∩p n ⊂q 1 , and as q 1 is prime, we must have p r ⊂q 1 for some r. Similarly, q s ⊂p r for some s, and so q s ⊂q 1 . By minimality of T , no q j ∈ T can contain another q k ∈ T , therefore s = 1 and q s =q 1 . After re-indexing the p i we can assume that p 1 =q 1 .
Having proved p i =q i for all 1 ≤ i ≤ n − 1 (after possible re-indexing), consider the inclusion p 1 ∩...∩p n ⊂q n . As q n is prime, we must have p k ⊂q n for some k. If 1 ≤ k ≤ n − 1, then we would get the inclusion q k =p k ⊂q n which is impossible by minimality of T . Hence k = n, and so p n ⊂q n . Similarly, q 1 ∩...∩q m ⊂p n and so by primality of p n we have q j ⊂p n , and hence q j ⊂q n , which shows j = n by minimality of T . Hence p n =q n . Hence S ⊂ T , so by minimality of T , S = T . ¤
(The above proposition also has a set-topological proof, which is simpler: as Spec A is noetherian, it follows that any closed set is a finite union of irreducible closed subsets in a unique way.)
More on integral extensions
4.15 Noether Normalisation Theorem Let k be a field, and let A be a finite- type k algebra which is a domain. Then there exist finitely many elements x 1 ,...,x d in A (where d ≥ 0) that are algebraically independent over k, such that A is integral
over the subring k[x 1 ,...,x d ] ⊂ 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 ideal p ⊂ A, there exists a prime ideal q ⊂ B such that p = q ∩ A.
4.17 Going down Let A ⊂ B be an integral extension of rings, such that B is a domain and A is a normal domain. Then given any prime ideal q in B and p ′ in A
with p ′
⊂ q ∩ A, there exists a prime ideal q ′ ⊂ q in B with q ∩A=p .
4.18 Note The conclusion of the above result (the going-down property) also holds for arbitrary flat A-algebras B.
Some dimension theory
4.19 Krull dimension Let A be a ring and p ⊂ A a prime ideal. The hight of p is by definition the largest integer n ≥ 0 such that there exists a chain of distinct prime ideals
p 0 ⊂p 1 ⊂...⊂p n =p
The Krull dimension of A is the supremum of the heights of all prime ideals of A. 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 of A equals the transcendence degree over k of the field of fractions of A.
(2) If p ⊂ A is a prime ideal, then dim(A/p) + ht(p) = dim(A) where ht(p) denotes the hight and dim denotes the Krull dimension.
4.21 Krull’s principal ideal theorem Let A be a noetherian ring and let f ∈A
be neither a unit nor a zero divisor. Let p ⊂ A be a minimal prime ideal containing
f . Then height of p is 1.
5 n Algebraic subvarieties of C
5.1 Closed, open, locally closed subvarieties Let C[x 1 ,...,x n ] be the poly- nomial ring in n variables over C. For any subset S n ⊂ C[x
1 ,...,x n ], let Z(S) ⊂C
be the subset defined as the set of simultaneous solutions of all polynomials in S, that is,
Z(S) = n {(a
1 ,...,a n ) ∈C | f(a 1 ,...,a n ) = 0 for all f ∈S}
A subset X n ⊂C is called a closed subvariety of C if there exists some S ⊂
C [x n
1 ,...,x n ] such that X = Z(S). A subset U ⊂C is called an open subvariety
of C n if its complement C − U is a closed subvariety of C . A subset Y ⊂C is called a locally closed subvariety if it can be expressed as the intersection of a closed
subvariety of C n with an open subvariety of C .
5.2 Exercise If S 1 ⊂S 2 ⊂ C[x 1 ,...,x n ] then show that Z(S 1 ) ⊃ Z(S 2 ) (the inclusion gets reversed). Is the converse true?
5.3 Exercise Let I = (S) ⊂ C[x 1 ,...,x n ] be the ideal generated by S. Then show that Z(S) = Z(I).
5.4 Exercise Let I ⊂ C[x 1 ,...,x n ] be an ideal, and let I ⊂ C[x 1 ,...,x n √ ] be its radical (f r ∈ I if and only if the power f is in I for some r √ ≥ 1 which may depend
on f ). Show that Z(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 that I ∩ J is a radical ideal. Is I + 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 If I and J are ideals in C[x 1 ,...,x n ], show that
Z(I) ∪ Z(J) = Z(I ∩ J) If S λ is any family of subsets of C[x 1 ,...,x n ], show that ∩ λ Z(S λ ) = Z( ∪ λ S λ )
Consequently, if I λ are ideals in C[x 1 ,...,x n ] then show that P
∩ λ Z(I λ ) = Z( λ I λ )
5.8 Exercise Show that under partial order defined by inclusion (that is, X ≤Y means X n ⊂ Y ), all closed subvarieties of C form a lattice, with meet X ∩Y and join
X n ∪ Y . The minimum is the empty set ∅ = Z(1) and the maximum is C = Z(0).
5.9 Zariski topology via closed subvarieties A subset X ⊂C n 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[x 1 ,...,x n ]. This specification of closed sets indeed defines a
topology on C n , as P
∅ = Z(1), C = Z(0), Z(I) ∪ Z(J) = Z(I ∩ J), and ∩ λ Z(I λ )= Z( λ I λ ) by an earlier exercise.
5.10 n Exercise Show that the Euclidean topology on C for n ≥ 1 is finer than the Zariski topology.
5.11 n Exercise Show that any two non-empty Zariski-open subsets of C have a non-empty intersection. (This property is expressed by saying that C n is irreducible.)
5.12 n Exercise Show that any non-empty Zariski-open subset of C is connected in Euclidean topology.
5.13 Exercise What are all the Zariski-closed subsets of C?
2 5.14 2 Exercise Show that the diagonal ∆ = { (a, b) ∈ C | a = b } is closed in C in Zariski topology. Conclude using the previous exercise that the Zariski topology
on C 2 is not the product of the Zariski topologies on the two factors C.
5.15 Exercise: Principal open subvarieties as basic open subsets For any
1 ,...,x n ], we denote the complement C − Z(f) by the notation U f . This is called as the principal open subvariety defined by f . Show that open subvarieties of the type U f , as f varies over C[x 1 ,...,x n ], 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 form U f .
f n ∈ C[x
5.16 n Exercise: Noetherianness Show that C under Zariski topology is a noetherian topological space, that is, any decreasing sequence of closed subsets is
finite. Deduce that every subset of C n with induced topology is noetherian. In particular, any locally closed subvariety of C n is noetherian.
5.17 Exercise: Quasi-compactness Show that any noetherian topological spa-
ce X is quasi-compact, that is, every Zariski open cover of X has a finite subcover. In particular, locally closed subvariety X n ⊂C 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 subsets U ⊂ X and V ⊂ X such that U ∩ V = ∅. The empty topological space ∅ is also defined to be reducible. A locally closed subvariety
X n ⊂C is called irreducible if it is not reducible in the induced Zariski topology on
X.
5.19 Irreducibility in terms of closed subsets Show 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 union Y ∪ Z is also a proper subset of X.
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[x 1 ,...,x n ] and the set of all closed subvarieties of
C n given by I 7→ Z(I), with inverse given by
X 7→ I X = { f ∈ C[x 1 ,...,x n ] | f(a 1 ,...,a n ) = 0 for all (a 1 ,...,a n ) ∈X} Under this correspondence, the set of all maximal ideals in C[x 1 ,...,x n ] is in bijec-
tion with the set of all points of C n . The set of all prime ideals in C[x
1 ,...,x n ] is in bijection with the set of all irreducible closed subvarieties in C n .
5.22 n Corollary Let X be a closed subvariety of C , 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[x 1 ,...,x n ]/J, given (in a well-defined manner) by
Y ∼ 7→ I Y = {f∈R|f (a 1 ,...,a n ) = 0 for all (a 1 ,...,a n ) ∈Y} where f ∼ ∈ C[x
1 ,...,x n ] is any polynomial that lies over f ∈ C[x 1 ,...,x n ]/J. Under this correspondence, the set of all maximal ideals in R is in bijection with the set of all points of X. The set of all prime ideals in R is in bijection with the set of all irreducible closed subvarieties in X.
5.23 Regular functions on an open subvariety A function φ : V → C on an open subvariety V n ⊂C is called a regular function if given any principal open
f ⊂ V , the restriction φ| U f is of the form g/f where g ∈ C[x 1 ,...,x n ], and r is a non-negative integer.
5.24 Exercise Show that if (U f i ) i=1,...,m is an open cover of an open subvariety
V n ⊂C , and if φ : V → C is a function such that each restriction φ|
fi is of the form
g i i /f
i where g i ∈ C[x 1 ,...,x n ], and r i is a non-negative integer, then φ is regular.
5.25 Exercise Show 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 n Exercise Show that all regular functions on an open subvariety V ⊂C form a C-algebra (which we denote by O(V )) under point-wise operations. Show
that n O(C ) = C[x
1 ,...,x n ]. For any principal open U f ⊂C , show that
O(U f ) = C[x 1 ,...,x n , 1/f ]
which is the localisation of the polynomial ring by inverting f . Is this also true for
f = 0?
2 5.28 2 Exercise Let P ∈C . What are the regular functions on C − {P }? What
are the holomorphic functions on C 2 − {P }?
5.29 n Exercise If X ⊂C is a closed or an open subvariety of C , show that the
C -algebra O(X) is of finite type over C.
5.30 n Caution For an arbitrary locally closed subvariety X ⊂C , 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.
be a locally closed subvariety. A function φ : X → C is called a regular function if for each P n ∈ X, there exists a Zariski open neighbourhood U ⊂ C together with a regular function on ψ : U → C such that φ| X∩U =ψ | X∩U .
5.31 n Regular functions on a locally closed subvariety Let X ⊂C
5.32 Exercise Show that the regular functions on any locally closed subvariety
X n ⊂C form an algebra (which we denote by O(X)) over C.
5.33 n Exercise Show that for any closed subvariety X ⊂C , the restriction map
C [x 1 ,...,x n ] → O(X) : f 7→ f| X
√ is a surjective C-algebra homomorphism. If X = Z(I) where I = I ⊂ C[x 1 ,...,x n ]
is any radical ideal, conclude that the restriction map C[x 1 ,...,x n ] → O(X) induces
a C-algebra isomorphism C[x 1 ,...,x n ]/I → O(X).
5.34 n Exercise Show that the only subsets of C which are both open and closed
in Zariski topology are n ∅ and C . Show that O(C ) is the same whether C is regarded as an open subset or a closed subset of C n . Also, a similar statement holds for O(∅).
be a locally closed subvariety and let P ∈ X. Consider the set F P 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 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 a regular function on X at P .
5.35 n The local ring O
X,P of germs Let X ⊂C
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 by O X,P .
5.36 Exercise Verify that the above C-algebra structure on O X,P is well-defined.
5.37 Exercise Show that we have a well-defined C-algebra homomorphism (called the evaluation map)
O X,P → C : (U, f) 7→ f(P )
Show that O X,P is a local ring, whose unique maximal ideal is the kernel m X,P of the evaluation map O X,P → C.
5.38 n The ring O X,P as localisation Let X be a closed subvariety of C , de- fined by radical ideal I. Let P ∈ X be defined by maximal ideal m ⊂ O(X) =
C [x 1 ,...,x n ]/I. Then show that we have a canonical isomorphism of C-algebras O(X) m →O X,P . Deduce that if X = Z n ∩U
f where Z is closed in C and U f is a principal open in
C n , then again we have a canonical isomorphism of C-algebras O(X) m →O X,P .
be locally closed subvari- eties. A regular morphism f : X → Y is a continuous map in Zariski topology, such that for any open subset V ⊂ Y and any regular function φ ∈ O(V ), the composite − 1 − map φ 1 ◦f:f (V ) → C is a regular function on f (V ).
5.39 n Regular morphisms Let X ⊂C and Y ⊂C
5.40 Regular morphisms to C Show that a regular morphism f : X → C is the same as a regular function f , that is, an element of O(X).
→C n is a map f = (f n
5.41 Regular morphisms to C n Show that a regular morphism f : X
1 ,...,f n ):X →C such that each of the component maps f i = π n
i ◦ f : X → C is a regular function on X, that is, each f i is in O(X). If Y ⊂ C is a locally closed subvariety, show that a regular morphism f : X → Y is the same as a regular morphism f : X n →C whose image lies in Y .
5.42 n Exercise Show that for any closed subvariety X ⊂C , a map X →C is
a regular morphism if and only if there exist polynomials g 1 ,...,g n ∈ C[x 1 ,...,x m ] such that f i =g i | X . In other words, when X is a closed subvariety, every regular
morphism f : X n →C admits a prolongation to a regular morphism g : C →C (with g | X = f ).
5.43 n Exercise If X ⊂C and Y ⊂C are Zariski closed subsets and f : X →Y
a regular morphism, does there exist a regular morphism g : C m → Y such that the restriction g | X equals f : X → Y ? (Hint: Consider the special case where
m = n = 1, X = Y and f = id X .)
5.44 n Exercise: Pull-back of regular functions Let X ⊂C and Y ⊂C be locally closed subvarieties, and let f : X → Y be a regular morphism. Show that
φ # 7→ φ ◦ f defines a C-algebra homomorphism f : O(Y ) → O(X).
5.45 n Theorem Let X ⊂C be a locally closed subvariety and Y ⊂C be a closed subvariety. Then associating the pull-back C-algebra homomorphism f # : O(Y ) →
O(X) to a regular morphism f : X → Y defines a bijective correspondence between the set of all regular morphisms X → Y and the set of all C-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
2 the theorem remain valid? (Hint: Take Y = C # − {P }). Is it true that f 7→ f is injective even when Y n ⊂C is locally closed?
5.47 n Exercise: Composition of regular morphisms Let X ⊂C ,Y ⊂C and Z p ⊂C
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
homomorphisms f # : O(Y ) → O(X) and g : O(Z) → O(Y ) equals the pull-back homomorphisms f # : O(Y ) → O(X) and g : O(Z) → O(Y ) equals the pull-back
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 C n , as n varies
over all non-negative integers. We call them as quasi-affine varieties. The morphisms in QAV are regular morphisms as defined above.
5.49 n Affine variety A quasi-affine variety X ⊂C is said to be an affine variety if there exists some closed subvariety Z m ⊂C 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 subvariety Z m ⊂C such that
X is isomorphic to Z.
5.51 n Exercise If a locally closed subvariety X ⊂C can be expressed as an
intersection Z n ∩U
f where Z ⊂C is Zariski closed and U f =C − Z(f) is principal open, then show that X is affine.
5.52 Existence of an affine open cover Deduce from the above exercise that given any locally closed subvariety X n ⊂C , there exist an open cover of X by
subsets V i ⊂ X such that each V i is an affine variety.
5.53 Equivalence of categories Show 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 space P(V ) is the set of all 1-dimensional vector subspaces L ⊂ V . In particular, if
V = 0 then P(V ) is empty.
For V = C n+1 where n ≥ 0 is an integer, the corresponding projective space P(C ) is denoted by P n , and it is called the n-dimensional complex projective space.
n+1
The space P 2 is called as the complex projective line, and P is called as the complex
projective plane. The points of P n+1 are 1-dimensional vector subspaces L ⊂C .
Any point L n ⊂C of P is represented by a vector 0 6= v ∈ L. If v = (a
0 ,...,a n ), then we call the n-tuple (a 0 ,...,a n ) as the homogeneous coordinates of L. Note that not all a n+1
n+1
− 0, we denote by ha 0 ,...,a n i the corresponding point L of P n . With this notation, we have
i can be zero. Given any (a 0 ,...,a n ) ∈C
ha 0 ,...,a n i = hλa 0 , . . . , λa n i for all λ 6= 0
Conversely, if n ha
0 ,...,a n i = hb 0 ,...,b n i∈P , then there exists some λ 6= 0 such that (a n+1
0 ,...,a n ) = λ(b 0 ,...,b n ) ∈C .