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 of Cn . . . . . . . . . . . . . . . . . . . . . . . . . . 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

1 Introduction

Just as Euclidean Geometry is the study of Euclidean space and certain figures in itmade 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 theaffine space C

n and the projective space PnC, and their subspaces known as algebraic

varieties. These lectures are meant as a first introduction to the subject. They focuson setting up the basic definitions and explaining some elementary concepts aboutalgebraic varieties. The treatment is linear, and many simple statements are left forthe reader to prove as exercises.

The material here was delivered in a series of 8 lectures of 90 minutes each, to anaudience 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 theclass 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 anyof the standard basic textbooks on algebraic geometry.

2 C∞ submanifolds of Rn

2.1 Linear coordinates The linear coordinates x1, . . . , xn on Rn are the projec-

tions from the product Rn to the individual factors R, that is, xi : R

n → R is the ith projection function.

2.2 Polynomial functions Any polynomial f ∈ R[x1, . . . , xn] defines a functionf : R

n → R called a polynomial function.

2.3 Exercise Show that two polynomials in R[x1, . . . , xn] are equal to each otherif and only if the corresponding functions R

n → R are equal to each other.

2.4 Linear functions These are functions on Rn defined by polynomials of degree

1. They have the form∑

i aixi + c where 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. Let b = (b1, . . . , bn) ∈ R

n. Let y1, . . . , yn be linear functions definedby yi =

∑Ai,jxj + bi. Then we say that the yi are a set of linear coordinates on R

n.As A is invertible, we can express each xi as a linear functions of y1, . . . , yn.

2.6 Open subsets We give Rn the Euclidean topology. This makes R

n the prod-uct topological space of n copies of the topological space R, where R is given itsusual topology induced by the metric d(x, y) = |x − y|.

2

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 partialderivatives of f of all orders exist and are continuous.

2.8 Note Because U is assumed to be open, the definition of a partial derivativemakes sense for functions on U .

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∞ function g : V → R such that g|U = f .

2.10 Exercise Give an example of nonempty open subsets U ⊂ V ⊂ Rn 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 inplace of C∞ functions?

2.12 C∞ map from U ⊂ Rn to R

m Let y1, . . . , 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, where fi = yi ◦ f . If U is open in Rn, a map f = (f1, . . . , fm) : U → R

m

is called a C∞ map if each of the coordinate maps fi : U → R is C∞.

2.13 C∞-Isomorphism Let U ⊂ Rn and V ⊂ R

m be open subsets. A C∞-isomorphism f : U → V is a topological homeomorphism f of U onto V withinverse g : V → U such that the maps f and g are C∞ regarded as maps U → R

m

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 andf ◦ 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 chart A 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∞

functions ui : U → R such that the resulting map u : U → Rn is a C∞-isomorphism

of U onto an open subset V ⊂ Rn.

3

2.17 Inverse function theorem Let U ⊂ Rn be an open subset and let f : U →

Rn be a C∞ map. Let P ∈ U such that the determinant of the n × n matrix (the

Jacobian) (∂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 R

n) such that the restriction f |V is a C∞-isomorphismof V onto an open subset W of R

n.

2.18 Exercise Let U ⊂ Rn be an open subset and let f : U → R

n be a C∞ mapsuch that the Jacobian matrix (∂fi/∂xj)(P ) is invertible for all P ∈ U . Then showthat 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 → R

n be a C∞ mapsuch that (i) f is injective, and (ii) the Jacobian matrix (∂fi/∂xj)(P ) is invertiblefor all P ∈ U . Then show that f is a C∞-isomorphism of U onto an open subset ofR

n.

2.20 Cubical coordinate chart Let (U ; u1, . . . , un) be a coordinate chart in Rn,

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 calledthe centre of the cubical coordinate chart.

2.21 Exercise Let r and θ be the usual polar coordinates on R2. Let U ⊂ R

2 bethe subset where 1 < r < 3 and 0 < θ < 2π. Let u1 = r − 2 and let u2 = (θ − π)/π.Then show that (U ; u1, u2) is a Cubical coordinate chart in R

2. What is its centre?

2.22 Locally closed submanifolds Let d be an integer with 0 ≤ d ≤ n. LetX ⊂ R

n 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 anda 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 thecentre of some cubical coordinate chart (U ; u1, . . . , un) in R

n 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 R

n of dimension −∞ (or−1 in another convention).

If a locally closed submanifold X of Rn is closed in V where V ⊂ R

n is open, thenX is called a closed submanifold of V .

4

2.23 Exercise If U ⊂ X is open in a locally closed submanifold X of Rn, then

show that U itself is a locally closed submanifold of Rn. What is its dimension?

2.24 C∞-functions on submanifolds Let X be a locally closed submanifold ofR

n. Let f : X → R be a function. Then f is called a C∞-function on X if for eachP ∈ X there exists an open neighbourhood U in R

n together with a C∞-functiong : U → R such that

f |X∩U = g|X∩U

2.25 The R-algebra C∞(X) If X is a locally closed submanifold of Rn, then all

C∞-functions f : X → R form a commutative R-algebra C∞(X) under point-wiseoperations.

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 Rn, 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∞-functionsdefines an R-algebra homomorphism

C∞(V ) → C∞(U) : f 7→ f |U

Given any locally closed C∞-submanifold V of Rn and any open covering (Ui)i∈I of

V , the following two properties are satisfied:

(1) Separatedness If f ∈ C∞(V ) such that each restriction f |Uiis 0, then f = 0.

(2) Gluing If for each Ui we are given a C∞-function fi 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 fi = f |Uifor 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 coordinatechart (U ; u1, . . . , un) in R

n such that X ∩U = {Q |ui(Q) = 0 for all d+1 ≤ i ≤ n },f |X∩U is a C∞-function of the coordinates u1, . . . , ud.

Show that it is enough to check the above condition for a family of such charts whichcovers X.

5

2.29 The local ring of germs Let X ⊂ Rn be a locally closed submanifold and

let P ∈ X. Consider the set FP of all ordered pairs (U, f) where U ⊂ X is an openneighbourhood of P in X, and f ∈ C∞(U). On the set FP we put an equivalencerelation as follows. We say that pairs (U, f) and (V, g) are equivalent if there existsan open neighbourhood W of P in U ∩ V such that f |W = g|W . Each equivalenceclasses 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 additionand 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 (calledthe evaluation map)

C∞X,P → R : (U, f) 7→ f(P )

Show that C∞X,P is a local ring, whose unique maximal ideal is the kernel mX,P of

the evaluation map C∞X,P → R.

2.32 C∞-maps between submanifolds Let X ⊂ Rm and Y ⊂ R

n be locallyclosed submanifolds. A map f = (f1, . . . , fm) : X → Y is called C∞ if each fi is C∞,equivalently, if the composite X → Y ↪→ R

n is C∞.

2.33 Implicit function theorem Let x1, . . . , xm be linear coordinates on Rm and

let y1, . . . , yn be linear coordinates on Rn. On the product R

m+n = Rm ×R

n we getlinear coordinates x1, . . . , xm, y1, . . . , yn. Let U ⊂ R

m+n be an open neighbourhoodof the origin 0 = (0, . . . , 0) ∈ R

m+n. Let f = (f1, . . . , fn) be an n-tuple of C∞

functions (same as a C∞-map f : U → Rn), such that

f(0) = 0

Let the following n × n-matrix(

∂fi

∂yj

)

1≤i,j≤n

be invertible at the point 0 ∈ Rm+n. Then there exists real numbers a, b > 0 and a

C∞-function g = (g1, . . . , gn) : Va → Wb where Va ⊂ Rm is the open subset defined

in terms of the coordinates by −a < xi < a and Wb ⊂ Rn is the open subset defined

in terms of the coordinates by −b < yj < b such that Va × Wb ⊂ U and

f−1(0) ∩ (Va × Wb) = { (x, g(x)) |x ∈ Va }

2.34 Exercise Deduce the implicit function theorem from the inverse functiontheorem, and conversely.

6

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

(∂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 submanifold of Uof 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 point P = (a1, . . . , an) ∈ U it is givenby a convergent power series in the n variables x1−a1, . . . , xn −an. The concepts ofanalytic isomorphism, analytic coordinate chart, cubical analytic coordinate chart,and locally closed analytic submanifold of R

n are defined by taking analytic functionsin place of C∞ functions in the various definitions above. The inverse and implicitfunction 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∞

function f : Rn → R which vanishes exactly on X.

2.39 Exercise A similar property does not hold for analytic functions: show thatif U ⊂ V ⊂ R

n 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

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 integer g ≥ 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

boundary?

7

3 Holomorphic submanifolds of Cn

The above notions about real analytic manifolds have their analogues in the theory ofcomplex manifolds, where complex analytic functions (also known as holomorphicfunctions) of several complex variables replace the real analytic functions of theprevious 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 C

n thetopology of R

2n, under the bijection (x1, . . . , xn) 7→ (u1, v1, . . . , un, vn). Open (orclosed) subsets in C

n mean open (or closed) subsets in the Euclidean topology onR

2n.

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 complexanalytic) if given any point P = (a1, . . . , an) ∈ U , the function f is given by aconvergent power series in the n complex variables x1 − a1, . . . , xn − an in a neigh-bourhood of P .

All holomorphic functions on an open set U form a C-algebra, which we denoteby 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 each fi : U → C is analytic.

3.3 Holomorphic isomorphism Let U ⊂ Cn and V ⊂ C

m be open subsets. Aholomorphic isomorphism f : U → V is a topological homeomorphism f of U ontoV with inverse g : V → U such that the maps f and g are holomorphic regardedas maps U → C

m and V → Cn. 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 isomorphismf : U → V will imply m = n. Prove this by differentiating the composites g ◦ f andf ◦ g and using the chain rule.

3.5 Holomorphic coordinate chart A holomorphic coordinate chart in Cn is 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 mapu : U → C

n is a holomorphic isomorphism of U onto an open subset V ⊂ Cn.

3.6 Inverse function theorem Let U ⊂ Cn be an open subset and let f : U →

Cn be a holomorphic map. Let P ∈ U such that the determinant of the n×n matrix

8

(the Jacobian) (∂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 C

n) such that the restriction f |V is a holomorphicisomorphism of V onto an open subset W of C

n.

3.7 Exercise Let U ⊂ Cn be an open subset and let f : U → C

n be a holomorphicmap such that the Jacobian matrix (∂fi/∂xj)(P ) is invertible for all P ∈ U . Thenshow that f is an open map. Give an example of such an f which is not injective.

3.8 Exercise Let U ⊂ Cn be an open subset and let f : U → C

n be a holomor-phic map such that (i) f is injective, and (ii) the Jacobian matrix (∂fi/∂xj)(P ) isinvertible for all P ∈ U . Then show that f is a holomorphic isomorphism of U ontoan open subset of C

n.

3.9 Cubical coordinate chart (polydisk) Let (U ; u1, . . . , un) be a holomorphiccoordinate chart in C

n, such that the map u : U → Cn is a holomorphic isomorphism

of U onto an open subset V ⊂ Cn where V 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 calledthe centre of the cubical coordinate chart.

3.10 Locally closed holomorphic submanifolds Let d be an integer with 0 ≤d ≤ n. Let X ⊂ C

n be a locally closed subset of Cn. Then X is called a locally

closed holomorphic submanifold of Cn 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 coordinatechart (U ; u1, . . . , un) in C

n 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 ⊂ C

n is open, thenX is called a closed submanifold of V .

3.11 Exercise If U ⊂ X is open in a locally closed submanifold X of Cn, then

show that U itself is a locally closed submanifold of Cn. What is its dimension?

9

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 holomorphicfunction on X if for each P ∈ X there exists an open neighbourhood U in C

n

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) underpoint-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 closedholomorphic submanifold X of C

n, such that U ⊂ V ⊂ X. Then for any holomorphicfunction 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 Cn and any open covering

(Ui)i∈I of V , the following two properties are satisfied:

(1) Separatedness If f ∈ H(V ) such that each restriction f |Uiis 0, then f = 0.

(2) Gluing If for each Ui we are given a holomorphic function fi such that for anypair (i, j) we have

fi|Ui∩Uj= fj|Ui∩Uj

∈ H(Ui ∩ Uj)

then there exists some f ∈ H(V ) such that fi = f |Uifor 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 function f : X → C is holomorphic if and only if for every cubical holo-morphic coordinate chart (U ; u1, . . . , un) in C

n such that X ∩ U = {Q |ui(Q) =0 for all d + 1 ≤ i ≤ n }, f |X∩U is a holomorphic function of the coordinatesu1, . . . , ud.

Show that it is enough to check the above condition for a family of such charts whichcovers X.

3.16 The local ring of germs Let X ⊂ Cn be a locally closed submanifold and

let P ∈ X. Consider the set FP of all ordered pairs (U, f) where U ⊂ X is an openneighbourhood of P in X, and f ∈ H(U). On the set FP we put an equivalencerelation as follows. We say that pairs (U, f) and (V, g) are equivalent if there existsan open neighbourhood W of P in U ∩ V such that f |W = g|W . Each equivalenceclasses is called a germ of a holomorphic function on X at P .

10

The set of all equivalence classes forms a C-algebra (where operations of additionand multiplication are carried out point-wise on restrictions to common small neigh-bourhoods), denoted by HX,P .

3.17 Exercise Verify that the above C-algebra structure on HX,P is well-defined.

3.18 Exercise Show that we have a well-defined C-algebra homomorphism (calledthe 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 ofthe evaluation map HX,P → C.

3.19 Holomorphic maps between submanifolds Let X ⊂ Cm and Y ⊂ C

n belocally closed holomorphic submanifolds. A map f = (f1, . . . , fm) : X → Y is calledholomorphic if each fi is holomorphic, equivalently, if the composite X → Y ↪→ C

n

is holomorphic.

3.20 Exercise: Pull-back of holomorphic functions Let X and Y be locallyclosed holomorphic submanifolds of C

m and Cn, respectively. Let f : X → Y

be a holomorphic map. Let g : Y → C be a holomorphic function on Y . Thenshow that the composite function g ◦ f : X → C is holomorphic. The compositeg ◦ f is called the pull-back of g under f . This defines a C-algebra homomorphismf# : H(Y ) → H(X) : g 7→ g ◦ f .

3.21 Exercise With X and Y as above, let f1 and f2 be holomorphic maps fromX to Y . If f#

1 = f#2 : H(Y ) → H(X), then show that f1 = f2.

3.22 Exercise: Composition of holomorphic maps Let X, Y and Z be locallyclosed holomorphic submanifolds of C

m, Cn and C

p respectively, and let f : X → Yand g : Y → Z be holomorphic maps. The show that the composite g ◦ f : X → Zis 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 C

m+n = Cm ×

Cn we get linear coordinates x1, . . . , xm, y1, . . . , yn. Let U ⊂ C

m+n be an openneighbourhood of the origin 0 = (0, . . . , 0) ∈ C

m+n. Let f = (f1, . . . , fn) be ann-tuple of holomorphic functions (same as a holomorphic map f : U → C

n), suchthat

f(0) = 0

11

Let the following n × n-matrix

(∂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 functiontheorem, 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

(∂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 holomorphicsubmanifold 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 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.

3.28 Exercise Identifying C2 with R

4, we can regard Z(f) as a closed subset ofR

4. Is it a C∞ submanifold of R4, and if so of what dimension?

12

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 locallyclosed submanifold of R

n according to this definition is a pair (X,φ) where X isan abstract C∞-manifold (see the last section for definition), and φ : X → R

n isa C∞-morphism that is injective and tangent-level injective. (Actually, one definesan equivalence relation on such pairs (X,φ) by putting (X,φ) − (X ′, φ′) if thereexists a C∞-isomorphism ψ : X → X ′ such that φ = φ′ ◦ ψ, and then one defines alocally closed submanifold in the general sense as an equivalence class.) A standardexample of such a more general concept is the so called skew line on a torus. For anelementary example, let X = R

1, and let φ : X → R2 be defined by t 7→ (et, sin t).

A similar more general notion exists for locally closed holomorphic submanifolds ofC

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 issaid to be integral element over A (or more precisely, integral over A with respectto φ) if there exists some integer n ≥ 1 and elements a1, . . . , an ∈ A such thatbn + φ(a1)b

n−1 + . . . + φ(an) = 0 in B. We say that B is integral over A if eachb ∈ 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 subringφ(A) of B w.r.t. the inclusion φ(A) ↪→ B. For this reason, we can assume that φ isan 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 statementsare 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. (Recallthat a module M over a ring R is called faithful if no non-zero element of R kills allof M .)

Proof Clearly (1) ⇒ (2) ⇒ (3). Now let M be a faithful A[b]-module which is finiteas an A-module, generated by v1, . . . , vn. Then there exist n2 elements ci,j ∈ A suchthat bvi =

∑n

j=1 ci,jvj for each i = 1, . . . , n. Hence by the determinant argument,

13

the element det(bI −C) ∈ A[b] annihilates each vj hence annihilates M , where C isthe matrix (ci,j). Hence det(bI − C) = 0 as M is faithful over A[b]. As det(bI − C)is a monic polynomial in b with coefficients in A, we see that (3) ⇒ (1).

4.3 Lemma Let A ⊂ B be rings, such that B is finite over A. Then B is integralover A.

Proof For each b ∈ B, note that B is a faithful A[b]-module M which is finite as anA-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[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 Lemma Let A ⊂ B be rings. Then all elements of B which are integral overA 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 overA then c is integral over A[b] and so A[b, c] is finite over A[b]. Hence A[b, c] is finiteover A.

Nullstellensatz

4.6 Theorem Let A ⊂ K be an integral ring extension. If K is a field, then A isa field.

Proof For any a ∈ A with a 6= 0, the element b = 1/a of K is integral over A. Letbn + a1b

n−1 + . . . + an = 0. Then multiplying by an−1, we get b = −(a1 + a2a + . . . +ana

n−1) ∈ A.

4.7 Lemma Let k be a field and k[x1, . . . , xr] be a polynomial ring in r variableswhere r ≥ 1. If f ∈ k[x1, . . . , xr] then the localisation k[x1, . . . , xr, 1/f ] is not afield.

Proof There are infinitely many non-constant irreducible polynomials which are notscalar multiples of each other. Hence there is a non-constant irreducible polynomialp which does not divide f . It follows that p is not invertible in k[x1, . . . , xr, 1/f ].

4.8 Theorem Let k be a field and A a finite-type k-algebra. If A is a field thenA is finite over k.

Proof We just have to show that A is algebraic over k. Let A = k[a1, . . . , an].If A is not algebraic, there exists an integer r ≥ 1 such that after re-indexing

14

the ai, the first r generators a1, . . . , ar are algebraically independent over k and Ais algebraic over k(a1, . . . , ar). Each of ar+1, . . . , an satisfies a monic polynomialfr+1(t), . . . , fn(t) in k(a1, . . . , ar)[t]. Let g(a1, . . . , ar) ∈ k[a1, . . . , ar] be the productof all the denominators of the coefficients of the fr+1(t), . . . , fn(t) (written as rationalfractions in any chosen way). Then A is integral over k[a1, . . . , ar, 1/g]. Hencek[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 ∈ Awhich are algebraically independent over k such that A is finite over the polynomialring k[a1, . . . , ar]. Therefore as A is a field, this polynomial ring is also a field, whichshows 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 ⊂ B the inverse image φ−1(m) is amaximal ideal in A.

Proof B/m is algebraic hence integral over k, and k ⊂ A/φ−1(m) ⊂ B/m. HenceB/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 suchthat 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 idealm ⊂ Af . We have a homomorphism A → Af of finite-type k-algebras, so thecontraction m

c ⊂ A of m is maximal. As f lies in each maximal ideal of A, we getf ∈ m

c 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 apolynomial ring k[x1, . . . , xn] is of the form (x1−a1, . . . , xn−an) for some a1, . . . , an ∈k.

4.12 Corollary If k is an algebraically closed field and I ⊂ k[x1, . . . , xn] is anideal such that I 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 that f(a1, . . . , an) = 0 for each point(a1, . . . , an) ∈ Z(I) ⊂ kn, then fm ∈ I for some integer m ≥ 1.

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Prime decomposition

4.14 Theorem Let A be a noetherian ring. Then every radical ideal I of A is theintersection 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 A is an intersection of a finiteset of prime ideals. Suppose not, then by noetherian condition, there is a maximalelement 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 byusing the fact that I is radical, we see that

I = (I + (a)) ∩ (I + (b))

Taking radicals, we getI =

√I + (a) ∩

√I + (b)

By maximality of I, the radical ideals√

I + (a) and√

I + (b) are finite intersectionsof prime ideals, hence so is I. If I = p1 ∩ . . . ∩ pn then by discarding at most n ofthe ideals pi, we arrive at a minimal set S of primes which has intersection I (wherewe say that a set S = {p1, . . . ,pn} which has intersection ∩p∈Sp = I is minimal ifthere is no proper subset S ′ ⊂ S which also has intersection I).

Next, we show the uniqueness of a minimal set S. Clearly, S is empty if and only ifI = A. So now we assume that cardinality of S is n ≥ 1. Let S = {p1, . . . ,pn} andT = {q1, . . . ,qm} be two minimal such sets, with n ≤ m. As p1∩ . . .∩pn ⊂ q1, andas q1 is prime, we must have pr ⊂ q1 for some r. Similarly, qs ⊂ pr for some s, andso qs ⊂ q1. By minimality of T , no qj ∈ T can contain another qk ∈ T , therefores = 1 and qs = 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), considerthe inclusion p1 ∩ . . .∩pn ⊂ qn. As qn is prime, we must have pk ⊂ qn for some k.If 1 ≤ k ≤ n− 1, then we would get the inclusion qk = pk ⊂ qn which is impossibleby minimality of T . Hence k = n, and so pn ⊂ qn. Similarly, q1 ∩ . . . ∩ qm ⊂ pn

and so by primality of pn we have qj ⊂ pn, and hence qj ⊂ qn, which shows j = nby 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 Spec Ais noetherian, it follows that any closed set is a finite union of irreducible closedsubsets in a unique way.)

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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 x1, . . . , xd

in A (where d ≥ 0) that are algebraically independent over k, such that A is integralover the subring k[x1, . . . , xd] ⊂ A.

We assume the student has seen a proof. In the lectures we will explain how togeometrically view this result in terms of choosing a ‘new coordinate projectionwhich is finite’.

4.16 Going up Let A ⊂ B be an integral extension of rings. Then given anyprime 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 adomain and A is a normal domain. Then given any prime ideal q in B and p′ in Awith 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) alsoholds 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 ofp is by definition the largest integer n ≥ 0 such that there exists a chain of distinctprime ideals

p0 ⊂ p1 ⊂ . . . ⊂ pn = 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 hasvalue −∞ by convention.

4.20 Theorem Let k be a field, and let A be a finite-type k algebra which is adomain. Then the following holds.

(1) The Krull dimension of A equals the transcendence degree over k of the field offractions of A.

(2) If p ⊂ A is a prime ideal, then dim(A/p)+ht(p) = dim(A) where ht(p) denotesthe hight and dim denotes the Krull dimension.

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4.21 Krull’s principal ideal theorem Let A be a noetherian ring and let f ∈ Abe neither a unit nor a zero divisor. Let p ⊂ A be a minimal prime ideal containingf . Then height of p is 1.

5 Algebraic subvarieties of Cn

5.1 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) ⊂ C

n

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 C

n if there exists some S ⊂C[x1, . . . , xn] such that X = Z(S). A subset U ⊂ C

n is called an open subvarietyof C

n if its complement Cn − U is a closed subvariety of C

n. A subset Y ⊂ Cn is

called a locally closed subvariety if it can be expressed as the intersection of a closedsubvariety of C

n with an open subvariety of Cn.

5.2 Exercise If S1 ⊂ S2 ⊂ C[x1, . . . , xn] then show that Z(S1) ⊃ Z(S2) (theinclusion gets reversed). Is the converse true?

5.3 Exercise Let I = (S) ⊂ C[x1, . . . , xn] be the ideal generated by S. Thenshow that Z(S) = Z(I).

5.4 Exercise Let I ⊂ C[x1, . . . , xn] be an ideal, and let√

I ⊂ C[x1, . . . , xn] be itsradical (f ∈

√I if and only if the power f r 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 itequals its own radical), then show that I∩J is a radical ideal. Is I +J also a radicalideal?

5.6 Exercise Show that under partial order defined by inclusion (that is, I ≤ Jmeans 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[x1, . . . , xn], show that

Z(I) ∪ Z(J) = Z(I ∩ J)

If Sλ is any family of subsets of C[x1, . . . , xn], show that

∩λZ(Sλ) = Z(∪λSλ)

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Consequently, if Iλ are ideals in C[x1, . . . , xn] then show that

∩λZ(Iλ) = Z(∑

λIλ)

5.8 Exercise Show that under partial order defined by inclusion (that is, X ≤ Ymeans X ⊂ Y ), all closed subvarieties of C

n form a lattice, with meet X∩Y and joinX ∪ Y . The minimum is the empty set ∅ = Z(1) and the maximum is C

n = 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 atopology on C

n, as ∅ = Z(1), Cn = Z(0), Z(I) ∪ Z(J) = Z(I ∩ J), and ∩λZ(Iλ) =

Z(∑

λIλ) 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 that Cn is 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 of C?

5.14 Exercise Show that the diagonal ∆ = { (a, b) ∈ C2 | a = b } is closed in C

2

in Zariski topology. Conclude using the previous exercise that the Zariski topologyon 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 anyf ∈ C[x1, . . . , xn], we denote the complement C

n −Z(f) by the notation Uf . This iscalled as the principal open subvariety defined by f . Show that open subvarieties ofthe type Uf , as f varies over C[x1, . . . , xn], form a basis of open sets for the Zariskitopology. Moreover, show that any Zariski open set is a union of finitely many setsof the form Uf .

5.16 Exercise: Noetherianness Show that Cn under Zariski topology is a

noetherian topological space, that is, any decreasing sequence of closed subsets isfinite. Deduce that every subset of C

n with induced topology is noetherian. Inparticular, any locally closed subvariety of C

n is noetherian.

19

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 ⊂ C

n is quasi-compact, that is, everyZariski open cover of X has a finite subcover.

5.18 Irreducibility A non-empty topological space X is said to be reducible ifthere exist non-empty open subsets U ⊂ X and V ⊂ X such that U ∩ V = ∅. Theempty topological space ∅ is also defined to be reducible. A locally closed subvarietyX ⊂ C

n is called irreducible if it is not reducible in the induced Zariski topology onX.

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 Zof X, the union Y ∪ Z is also a proper subset of X.

5.20 Exercise: Irreducible decomposition Show that if X is a noetheriantopological space, then X can be uniquely written as a minimal finite union ofirreducible closed subsets. Use this to give another proof that any radical ideal in anoetherian ring is uniquely a minimal finite intersection of primes.

5.21 Theorem There is an inclusion-reversing bijective correspondence betweenthe set of all radical ideals in C[x1, . . . , xn] and the set of all closed subvarieties ofC

n 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 in C[x1, . . . , xn] is in bijec-tion with the set of all points of C

n. The set of all prime ideals in C[x1, . . . , xn] isin bijection with the set of all irreducible closed subvarieties in C

n.

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 allclosed subvarieties Y of X and the set of all radical ideals in the quotient ringR = 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 .Under this correspondence, the set of all maximal ideals in R is in bijection withthe set of all points of X. The set of all prime ideals in R is in bijection with theset of all irreducible closed subvarieties in X.

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5.23 Regular functions on an open subvariety A function φ : V → C onan open subvariety V ⊂ C

n is called a regular function if given any principal openUf ⊂ V , the restriction φ|Uf

is of the form g/f r where g ∈ C[x1, . . . , xn], and r is anon-negative integer.

5.24 Exercise Show that if (Ufi)i=1,...,m is an open cover of an open subvariety

V ⊂ Cn, and if φ : V → C is a function such that each restriction φ|Ufi

is of the formgi/f

ri

i where gi ∈ C[x1, . . . , xn], and ri 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 conversetrue?

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. Showthat O(Cn) = C[x1, . . . , xn]. For any principal open Uf ⊂ C

n, show that

O(Uf ) = C[x1, . . . , xn, 1/f ]

which is the localisation of the polynomial ring by inverting f . Is this also true forf = 0?

5.28 Exercise Let P ∈ C2. What are the regular functions on C

2 − {P}? Whatare the holomorphic functions on C

2 − {P}?

5.29 Exercise If X ⊂ Cn is a closed or an open subvariety of C

n, show that theC-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 onthe 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 foreach P ∈ X, there exists a Zariski open neighbourhood U ⊂ C

n together with aregular function on ψ : U → C such that φ|X∩U = ψ|X∩U .

5.32 Exercise Show that the regular functions on any locally closed subvarietyX ⊂ C

n form an algebra (which we denote by O(X)) over C.

21

5.33 Exercise Show that for any closed subvariety X ⊂ Cn, the restriction map

C[x1, . . . , xn] → O(X) : f 7→ f |Xis a surjective C-algebra homomorphism. If X = Z(I) where I =

√I ⊂ C[x1, . . . , xn]

is any radical ideal, conclude that the restriction map C[x1, . . . , xn] → O(X) inducesa C-algebra isomorphism C[x1, . . . , xn]/I → O(X).

5.34 Exercise Show 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 C

n isregarded as an open subset or a closed subset of C

n. Also, a similar statement holdsfor O(∅).

5.35 The local ring OX,P of germs Let X ⊂ Cn be a locally closed subvariety

and let P ∈ X. Consider the set FP of all ordered pairs (U, f) where U ⊂ X is aZariski open neighbourhood of P in X, and f ∈ O(U). On the set FP we put anequivalence relation as follows. We say that pairs (U, f) and (V, g) are equivalent ifthere exists an open neighbourhood W of P in U ∩ V such that f |W = g|W . Eachequivalence classes is called a germ of a regular function on X at P .

The set of all equivalence classes forms a C-algebra (where operations of additionand multiplication are carried out point-wise on restrictions to common small neigh-bourhoods), denoted by OX,P .

5.36 Exercise Verify that the above C-algebra structure on OX,P is well-defined.

5.37 Exercise Show that we have a well-defined C-algebra homomorphism (calledthe 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 ofthe evaluation map OX,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-algebrasO(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 ⊂ C

n be locally closed subvari-eties. A regular morphism f : X → Y is a continuous map in Zariski topology, suchthat for any open subset V ⊂ Y and any regular function φ ∈ O(V ), the compositemap φ ◦ f : f−1(V ) → C is a regular function on f−1(V ).

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5.40 Regular morphisms to C Show that a regular morphism f : X → C isthe same as a regular function f , that is, an element of O(X).

5.41 Regular morphisms to Cn Show that a regular morphism f : X → C

n

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 morphism f : X → Y is the sameas a regular morphism f : X → C

n whose image lies in Y .

5.42 Exercise Show that for any closed subvariety X ⊂ Cm, a map X → C

n isa regular morphism if and only if there exist polynomials g1, . . . , gn ∈ C[x1, . . . , xm]such that fi = gi|X . In other words, when X is a closed subvariety, every regularmorphism f : X → C

n admits a prolongation to a regular morphism g : Cm → C

n

(with g|X = f).

5.43 Exercise If X ⊂ Cm and Y ⊂ C

n are Zariski closed subsets and f : X → Ya regular morphism, does there exist a regular morphism g : C

m → Y such thatthe restriction g|X equals f : X → Y ? (Hint: Consider the special case wherem = n = 1, X = Y and f = idX .)

5.44 Exercise: Pull-back of regular functions Let X ⊂ Cm and Y ⊂ C

n belocally closed subvarieties, and let f : X → Y be a regular morphism. Show thatφ 7→ φ ◦ f defines a C-algebra homomorphism f# : O(Y ) → O(X).

5.45 Theorem Let X ⊂ Cm be a locally closed subvariety and Y ⊂ C

n be a closedsubvariety. Then associating the pull-back C-algebra homomorphism f# : O(Y ) →O(X) to a regular morphism f : X → Y defines a bijective correspondence betweenthe set of all regular morphisms X → Y and the set of all C-algebra homomorphismsO(Y ) → O(X).

5.46 Exercise Suppose that in the hypothesis of the above theorem, instead ofY being closed, we take Y to be merely locally closed. Then does the conclusion ofthe theorem remain valid? (Hint: Take Y = C

2 − {P}). Is it true that f 7→ f# isinjective even when Y ⊂ C

n is locally closed?

5.47 Exercise: Composition of regular morphisms Let X ⊂ Cm, Y ⊂ C

n

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 aregular morphism. Moreover, show that the composite f# ◦ g# of the pull-backhomomorphisms f# : O(Y ) → O(X) and g# : O(Z) → O(Y ) equals the pull-back

23

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 asfollows. The objects of QAV are all locally closed subvarieties of C

n, as n variesover all non-negative integers. We call them as quasi-affine varieties. The morphismsin QAV are regular morphisms as defined above.

5.49 Affine variety A quasi-affine variety X ⊂ 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) issaid to be an affine variety if there exists some closed subvariety Z ⊂ C

m such thatX is isomorphic to Z.

5.51 Exercise If a locally closed subvariety X ⊂ Cn can be expressed as an

intersection Z ∩Uf where Z ⊂ Cn is Zariski closed and Uf = C

n −Z(f) is principalopen, then show that X is affine.

5.52 Existence of an affine open cover Deduce from the above exercise thatgiven any locally closed subvariety X ⊂ C

n, there exist an open cover of X bysubsets Vi ⊂ X such that each Vi is an affine variety.

5.53 Equivalence of categories Show that we have a contravariant functor fromthe category of all affine varieties (regarded as a full subcategory of all quasi-affinevarieties) to the category of all finite-type reduced C-algebras, which associates toany affine variety X the C-algebra O(X), and associates to any regular morphismf : X → Y the pull-back homomorphism f# : O(Y ) → O(X). Show that thisfunctor is fully faithful and essentially surjective, and therefore an (anti-)equivalenceof categories.

6 Projective spaces

Let V be a vector space over C. The set of points of the corresponding projectivespace P(V ) is the set of all 1-dimensional vector subspaces L ⊂ V . In particular, ifV = 0 then P(V ) is empty.

For V = Cn+1 where n ≥ 0 is an integer, the corresponding projective space P(Cn+1)

is denoted by Pn, and it is called the n-dimensional complex projective space.The space P1 is called as the complex projective line, and P2 is called as the complexprojective plane. The points of Pn are 1-dimensional vector subspaces L ⊂ C

n+1.

24

Any point L ⊂ Cn+1 of Pn is represented by a vector 0 6= v ∈ L. If v = (a0, . . . , an),

then we call the n-tuple (a0, . . . , an) as the homogeneous coordinates of L. Note thatnot all ai can be zero. Given any (a0, . . . , an) ∈ C

n+1 − 0, we denote by 〈a0, . . . , an〉the corresponding point L of Pn. With this notation, we have

〈a0, . . . , an〉 = 〈λa0, . . . , λan〉 for all λ 6= 0

Conversely, if 〈a0, . . . , an〉 = 〈b0, . . . , bn〉 ∈ Pn, then there exists some λ 6= 0 suchthat (a0, . . . , an) = λ(b0, . . . , bn) ∈ C

n+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 on Pn as the Zariski topology on Pn. If we given Cn+1−{0} the Euclidean

topology, then we call the corresponding quotient topology on Pn as the Euclideantopology on Pn.

6.1 Exercise Show that the Euclidean topology on Pn is finer than the Zariskitopology on Pn. Show that Pn is Hausdorff in the Euclidean topology, but for n ≥ 1Pn is not Hausdorff in the Zariski topology.

6.2 Exercise Show that the map φ : C → P1 : z 7→ 〈z, 1〉 gives an open inclusionof C in P1, with P1 − φ(C) equal to the singleton set {〈1, 0〉}. This extra pointwas regarded by Riemann as the ‘point at infinity’ to be added to C, 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 ⊂C

n+1 to the projective space Pn. Deduce that Pn is compact in the Euclideantopology.

6.4 Projective linear subspaces If W ⊂ V is a vector subspace, then we getan inclusion P(W ) ↪→ P(V ). For various linear subspaces W of C

n+1, show thatthe corresponding subsets P(W ) ⊂ Pn are closed in Zariski topology.

6.5 Open cover by affine spaces Let f : V → C be a non-zero linear functional.Let K = ker(f), and let v ∈ V with f(v) 6= 0. This gives an injective map

φ : K → Pn : w 7→ 〈v + w〉

When V = Cn+1, show that any such map is open in both the Zariski topology and

Euclidean topology (where K becomes isomorphic to Cn on choosing a linear basis,

and so has a well-defined Zariski as well as Euclidean topology). We have

φ(K) = P(V ) − P(K)

25

Taking f to be the linear functional xi for i = 0, . . . , n, show that the correspondingφ(K) define an open cover of Pn by n + 1 copies of C

n. When f = xi, v = ei, andK 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→ 〈z0, . . . , zi−1, 1, zi+1, . . . , zn〉

6.6 Exercise Show that any two projective linear subspaces P(V ) ⊂ Pn andP(W ) ⊂ Pn have intersection P(V ∩W ). In particular, any two distinct projectivelines in P2 intersect in exactly a single point.

7 Algebraic subvarieties of projective spaces

7.1 Closed subsets Z(S) ⊂ Pn Let S ⊂ C[x0, . . . , xn] be a set, all whose elementsare homogeneous polynomials. Then it defines a subset Z(S) ⊂ Pn, which consistsof all points 〈a0, . . . , an〉 such that f(a0, . . . , an) = 0 for all f ∈ S. This condition iswell-defined, as

f(λa0, . . . , λan) = λdf(a0, . . . , an)

for any homogeneous polynomial of degree d. Under the quotient map

q : Cn+1 − {0} → Pn

the inverse image of Z(S) ⊂ Pn is the subset Z(S)∩(Cn+1−{0}), where Z(S) ⊂ Cn+1

denotes the Zariski closed subset defined by S. Hence by definition of quotienttopology, Z(S) is Zariski closed in Pn. Similarly, Z(S) is Euclidean closed in Pn.

7.2 Exercise Let X ⊂ Cn+1 be a closed subvariety, which is conical (that is, if v ∈

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 iffor any f ∈ I, all the homogeneous components of f are in I. Deduce from theabove exercise that a subset X ⊂ Pn is Zariski closed if and only if there exists ahomogeneous ideal I ⊂ C[x0, . . . , xn] such that X = Z(I).

7.4 Projective variety A projective variety is a Zariski closed subset of Pn.

7.5 Homogeneous Nullstellensatz Let I ⊂ C[x0, . . . , xn] be a homogeneousideal. Let f ∈ C[x0, . . . , xn] be a homogeneous polynomial of degree ≥ 1. Show thatZ(I) ⊂ Z(f) if and only if f ∈

√I.

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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 betweenthe set HRI of all homogeneous radical ideals in S = C[x0, . . . , xn] which do notcontain the ideal S+, and the set of all non-empty closed subvarieties of Pn, givenby I 7→ Z(I), with inverse given by

X 7→ IX = { f ∈ C[x0, . . . , xn] | f(a0, . . . , an) = 0 for all 〈a0, . . . , an〉 ∈ X }

Under this correspondence, the set of all maximal elements of HRI is in bijectionwith the set of all points of Pn. The set of all prime ideals in HRI is in bijectionwith the set of all irreducible closed subvarieties in Pn.

7.8 Affine cone X over a projective variety X ⊂ Pn Given any closed subsetX ⊂ Pn defined by a homogeneous radical ideal I ⊂ C[x0, . . . , xn], the closed subset

X = Z(I) ⊂ Cn+1 is called the affine cone over X. The ring C[x0, . . . , xn]/I = O(X)

is called the homogeneous coordinate ring of X ⊂ Pn.

7.9 Exercise: Noetherianness Show that Pn under 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, anylocally closed subvariety X ⊂ Pn is quasi-compact, that is, every Zariski open coverof X has a finite subcover. Moreover, X can be uniquely written as a minimal finiteunion of irreducible closed subsets of X.

7.10 Principal open subsets Let f ∈ C[x0, . . . , xn] be a homogeneous polyno-mial. The Zariski open subset Uf = Pn − Z(f) is called the principal open subsetdefined by f . Show that principal open subsets form a basis of open sets for theZariski topology on Pn.

7.11 Regular functions on an open subvariety A function φ : V → C onan open subvariety V ⊂ Pn is called a regular function if given any principal openUf ⊂ V , the restriction φ|Uf

is of the form g/f r where g ∈ C[x1, . . . , xn] is ahomogeneous polynomial and r is a non-negative integer such that deg(g) = r deg(f)(so that the rational function g/f r 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/fri

i where gi ∈ C[x1, . . . , xn] is a homogeneous polynomialand ri is a non-negative integer such that deg(gi) = ri deg(fi), then φ is regular.

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7.13 Exercise Show that every regular function is continuous in Zariski topology.Is the converse true?

7.14 Exercise Show that every regular function is holomorphic. Is the conversetrue?

7.15 Exercise Show that all regular functions on an open subvariety V ⊂ Pn

form a C-algebra (which we denote by O(V )) under point-wise operations.

7.16 Exercise For any 0 ≤ i ≤ n, let Ui ⊂ Pn be the principal open set Uxi.

Show that its ring of regular functions is the polynomial ring

O(Ui) = C

[x0

xi

, . . . ,xi−1

xi

,xi+1

xi

, . . . ,xn

xi

]

More generally, for any principal open Uf ⊂ Pn, show that

O(Uf ) = C[x0, . . . , xn, 1/f ]0 ⊂ C[x0, . . . , xn, 1/f ]

the homogeneous component of degree zero in C[x0, . . . , xn, 1/f ].

7.17 Regular functions on a locally closed subvariety Let X ⊂ Pn be alocally closed subvariety. A function φ : X → C is called a regular function if foreach P ∈ X, there exists a Zariski open neighbourhood U ⊂ Pn together with aregular function on ψ : U → C such that φ|X∩U = ψ|X∩U .

7.18 Exercise Show that the regular functions on any locally closed subvarietyX ⊂ C

n form a C-algebra under point-wise operations. (We denote this algebra byO(X)) over C.

7.19 Exercise Show that O(Pn) = C, that is, the only regular functions on all ofPn are the constant functions.

7.20 Exercise More generally, for any connected closed subvariety X ⊂ Pn, showthat the only regular functions are constants, that is, O(X) = C.

The category of all quasi-affine or quasi-projective varieties

We now define a category V of varieties. The objects of this category are all quasi-affine or quasi-projective varieties (locally closed subsets of C

n or Pn). Given twovarieties X and Y (objects of V) a morphism f : X → Y in the category V is bydefinition a continuous map f from X to Y in the Zariski topology such that forany open subset V ⊂ Y and any regular function φ ∈ O(V ), the composite mapφ ◦ f : f−1(V ) → C is a regular function on f−1(V ). These will be called regularmorphisms.

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7.21 Pull-back homomorphism f# : O(V ) → O(f−1(V ) If f : X → Y isa morphism in the category V and if V ⊂ Y is any open subset, then we get aC-algebra homomorphism f# : O(V ) → O(f−1(V )) under which φ 7→ φ ◦ f .

7.22 Exercise Show that the map ψi : Cn → Pn defined earlier by putting

ψi : Cn → Pn : (z0, . . . , zi−1, zi+1, . . . , zn) 7→ 〈z0, . . . , zi−1, 1, zi+1, . . . , zn〉

is an isomorphism in the category V of Cn with the open subvariety Uxi

⊂ Pn.(Hint: Show that ψi is a homeomorphism which has the property that for any opensubset V ⊂ Ui and function φ : V → C, the composite map φ ◦ ψi : ψ−1

i (V ) → C isa regular function on ψ−1

i (V ) if and only if φ : V → C is regular.

7.23 Exercise: Composition of regular morphisms Let X, Y and Z beobjects of V and let f : X → Y and g : Y → Z be regular morphisms. Then showthat the composite g ◦f : X → Z is again a regular morphism. Moreover, show thatthe composite f# ◦ g# of the pull-back homomorphisms f# : O(Y ) → O(X) andg# : O(Z) → O(Y ) equals the pull-back homomorphism (g ◦ f)# for the composite,that is, f# ◦ g# = (g ◦ f)# : O(Z) → O(X).

7.24 Exercise If a locally closed subvariety X ⊂ Pn can be expressed as anintersection Z ∩ Uf where Z ⊂ Pn is Zariski closed and Uf = Pn − Z(f) is aprincipal open defined by some homogeneous polynomial f ∈ C[x0, . . . , xn] of degree≥ 1, then show that X is affine, that is, X is isomorphic to a closed subvariety ofsome C

m.

8 Products, Dimension

8.1 Product of affine spaces For any variety X, the set of all regular morphismsf : X → C

m+n is in bijection with the set of all ordered pairs (g, h) where g : X →C

m and h : X → Cn are regular morphisms. The projections p1 : C

m+n → Cm and

p2 : Cm+n → C

n are regular, and the bijection is given in terms of the projections

Hom(X, Cm+n) → Hom(X, Cm) × Hom(X, Cn) : f 7→ (p1 ◦ f, p2 ◦ f)

8.2 Categorical universal property The above shows that the ordered triple(Cm+n, p1, p2) is a product of C

m and Cn in the category V of all quasi-affine and

quasi-projective varieties.

8.3 Tensor product: categorical characterisation In the category of com-mutative C-algebras, the tensor product has the following meaning. Let A and Bbe commutative C-algebras, let A⊗C B their tensor product over C, which is again

29

a commutative C-algebra, and let θ1 : A → A ⊗C B and θ2 : B → A ⊗C B beC-algebra homomorphisms defined respectively by sending a 7→ a⊗1 and b 7→ 1⊗ b.Then the triple (A ⊗C B, θ1, θ2) is a co-product of A and B. That is, given anytriple (R, η1, η2) consisting of a commutative C-algebra R together with C-algebrahomomorphisms η1 : A → R and η2 : B → R, there exists a unique C-algebrahomomorphism α : A ⊗C B → R such that ηi = α ◦ θi for i = 1, 2. Note that α isdefined by the formula α(

∑ak ⊗ bk) =

∑η1(ak)η2(bk).

We denote α by the symbol (η1, η2).

8.4 Tensor product of polynomial algebras Given m + n algebraically in-dependent variables x1, . . . , xm, y1, . . . , yn over C, show that the pair of C-algebrahomomorphisms η1 : C[x1, . . . , xm] → C[x1, . . . , xm, y1, . . . , yn] : xi 7→ xi and η2 :C[y1, . . . , yn] → C[x1, . . . , xm, y1, . . . , yn] : yj 7→ yj induces an isomorphism of C-algebras (η1, η2) : C[x1, . . . , xm] ⊗C C[y1, . . . , yn]

∼→ C[x1, . . . , xm, y1, . . . , yn].

8.5 Product of affine varieties Let X ⊂ Cm and Y ⊂ C

n be closed subvarieties,defined by radical ideals I ⊂ C[x1, . . . , xm] and J ⊂ C[y1, . . . , yn] respectively. LetK ⊂ C[x1, . . . , xm, y1, . . . , yn] be the ideal generated by I ∪ J , where we regard Iand J as subsets of C[x1, . . . , xm, y1, . . . , yn] under the inclusions C[x1, . . . , xm] ↪→C[x1, . . . , xm, y1, . . . , yn] : xi 7→ xi and C[y1, . . . , yn] ↪→ C[x1, . . . , xm, y1, . . . , yn] :yj 7→ yj. Then K is a radical ideal, and Z(K) ⊂ C

m+n is the subset X × Y ⊂C

m ×Cn = C

m+n (where we have made the standard identification of Cm ×C

n withC

m+n). This shows that X × Y is again an affine variety.

8.6 Universal property in V Let p1 : X × Y → X and p2 : X × Y → Y againdenote the projections. These are regular, being restrictions of p1 : C

m+n → Cm

and p2 : Cm+n → C

n. Show that the triple (X × Y, p1, p2) is a product of X and Yin the category V of all quasi-affine and quasi-projective varieties.

8.7 Correspondence with tensor product With notation as above, let p#1 :

O(X) → O(X ×Y ) and p#2 : O(Y ) → O(X ×Y ) be the pull-back homomorphisms.

Then show that the induced homomorphism

(p#1 , p#

2 ) : O(X) ⊗C O(Y ) → O(X × Y )

is an isomorphism. In terms of the defining ideals, this is the isomorphism

C[x1, . . . , xm]

I⊗C

C[y1, . . . , yn]

J→ C[x1, . . . , xm, y1, . . . , yn]

(I ∪ J)

8.8 Note By the anti-equivalence of categories between affine varieties and finite-type reduced C-algebras, it follows that a product (X ×Y, p1, p2) of X and Y in thecategory of affine varieties will exist and will correspond to O(X)⊗CO(Y ). However,

30

what we have shown is the stronger statement that the product (X × Y, p1, p2) inthe category of affine varieties in fact has the universal property of a product in thelarger category V of all quasi-affine and quasi-projective varieties.

8.9 Product of quasi-affine varieties Let X ⊂ Cm and Y ⊂ C

n be locallyclosed subvarieties. Then show that the subset X × Y ⊂ C

m × Cn = C

m+n is alocally closed subvariety of C

m+n. Let p1 : X × Y → X and p2 : X × Y → Y againdenote the projections (these are regular, being restrictions of p1 : C

m+n → Cm and

p2 : Cm+n → C

n). Show that the triple (X × Y, p1, p2) is a product of X and Y inthe category V of all quasi-affine and quasi-projective varieties.

Product of projective spaces

Let V = Cm with linear coordinates (x0, . . . , xm), and let W = C

n with linearcoordinates (y0, . . . , yn). Let T = V ⊗C W be the tensor product, with inducedlinear coordinates (zi,j) where 0 ≤ i ≤ m and 0 ≤ j ≤ n. If v = (x0, . . . , xm) ∈ Vand w = (y0, . . . , yn) ∈ W are any two vectors, then the vector v⊗w ∈ T has linearcoordinates zi,j = xiyj in terms of the basis (vi ⊗ wj) of T where (vi) and (wj) arethe standard bases of V and W . We thus get a morphism of varieties

t : V × W → T : (v, w) 7→ v ⊗ w

8.10 Exercise: Image of t Show that the image of the above map is the closedsubvariety X ⊂ T defined by the ideal I ⊂ C[zi,j] generated by all the monomialsof the form zi,jzk,` − zi,`zk,j where 0 ≤ i, k ≤ m and 0 ≤ j, ` ≤ n. In terms of tensorproduct, this in particular shows that the set of so called ‘decomposable tensors’ inV ⊗ W (means those which can be expressed as v ⊗ w) is Zariski closed.

8.11 The Segre map s : Pm × Pn → Pmn+m+n If one of v and w is multipliedby a scalar, v ⊗ w gets multiplied by the same scalar. This shows that we get awell-defined set-map

s : Pm × Pn → Pmn+m+n

given in terms of homogeneous coordinates by

(〈x0, . . . , xm〉, 〈y0, . . . , yn〉) 7→ 〈x0y0, . . . , x0yn, x1y0, . . . , x1yn, . . . , xmyn〉

Note that in the above, Pm ×Pn just denote the product set (we have not yet givenit a topology or the structure of a variety).

Let Z ⊂ Pmn+m+n denote the closed subvariety defined by the homogeneous idealI ⊂ C[zi,j] defined above.

8.12 Proposition With notation as above, the map s is injective with image Z,so it induces a bijection ψ : Pm × Pn → Z. The maps π1 = p1 ◦ ψ−1 : Z → Pm and

31

π2 = p2 ◦ ψ−1 : Z → Pn are regular morphisms, where p1 and p2 are the projections(regarded as set-maps). The resulting triple (Z, π1, π2) is a product of Pm and Pn

in the category V .

In view of the above proposition, we will define the variety Pm × Pn to be theprojective variety Z ⊂ Pmn+m+n constructed above, together with projections π1

and π2. The bijection ψ and the equations pi = πi ◦ ψ show that if we forget thevariety structure (that is, if we apply the forgetful functor from varieties to sets) theproduct of Pm and Pn as varieties reduces to the product as sets.

8.13 Exercise Show that if m,n ≥ 1 then the Zariski topology on Pm ×Pn = Zis not the product of the Zariski topologies on the factors.

8.14 Product of quasi-projective varieties Let X ⊂ Pm and Y ⊂ Pn belocally closed subvarieties. Then show that the subset X × Y ⊂ Pm ×Pn is locallyclosed, and the triple (X × Y, p1, p2), where the πi are restrictions to X × Y of theprojections Pm × Pn → Pm and Pm × Pn → Pn, has the universal property ofproduct of X and Y in the category V .

8.15 Product of varieties in V Let X and Y be objects of V . As shown earlier,any locally closed subvariety X ⊂ C

m is isomorphic to the locally closed subvarietyψ0(X) ⊂ Pm. As a product is defined by its universal property, this allows us toreplace any variety by an isomorphic variety and thereby assume that both X and Yare quasi-projective varieties. As we have constructed a product of quasi-projectivevarieties in V , we get a product of any two objects of V .

8.16 Exercise: Separatedness If X is any object of V , show that the image ofthe morphism

(idX , idX) : X → X × X : P 7→ (P, P )

is a closed subset ∆X ⊂ X ×X. This property is expressed by saying that varietiesin V are separated. Note however that the topology on X × X is generally not theproduct topology, so ∆X can be closed without X being Hausdorff. The propertyof separatedness is partly a substitute for the property of being Hausdorff in thegeometry of varieties.

Dimension of varieties

8.17 Definition Let X be a variety. The dimension of X is the supremum ofall integers n such that there exists a chain of distinct irreducible closed subsetsX0 ⊂ X1 ⊂ . . . ⊂ Xn = X.

The dimension of the empty variety is −∞.

32

8.18 Caution The above definition only used the topology on X and not anythingabout regular functions, so the definition makes logical sense for arbitrary topologicalspaces. But it does not give the ‘correct’ notion of dimension for the usual topologicalspaces that are important in algebraic topology or differential geometry or analysis.For example, instead of giving the answer n for the dimension of the Euclidean spaceR

n, it will give the answer 0.)

8.19 Dimension of an affine variety as Krull dimension of O(X) Recallthat if X ⊂ C

n is an affine variety, then irreducible closed subsets of X correspondto prime ideals in O(X). From this conclude that dim(X) is equal to the Krulldimension of the ring O(X).

8.20 Exercise Determine the dimension of the affine variety Cn.

8.21 Exercise Show that Pn is irreducible.

8.22 Exercise Determine the dimension of the projective variety Pn. Is it equalto the Krull dimension of O(Pn)?

8.23 Exercise Show that for any quasi-affine variety X ⊂ Cn, dim(X) = dim(X)

where X ⊂ Cn is the closure of X. Show that for any quasi-projective variety

X ⊂ Pn, dim(X) = dim(X) where X ⊂ Pn is the closure of X. Conclude inparticular that varieties are finite dimensional.

8.24 Local nature of dimension We define the dimension of a variety X ata point P to be the infimum of the dimensions of open neighbourhoods of P inX. Show that this is the same as the maximum of the dimensions of irreduciblecomponents of X which contain P . Show that the dimension of X is the maximumof the local dimensions of X at all points.

8.25 Krull dimension of local ring Show that the local dimension of X at Pequals the Krull dimension of the local ring of germs OX,P .

Function field of an irreducible variety

8.26 Rational functions Let X be an irreducible variety. Consider all orderedpairs (V, f) where V ⊂ X is a non-empty open subset and f ∈ O(V ). We say thattwo such pairs (V, f) and (W, g) are equivalent if f |V ∩W = g|V ∩W (notice that V ∩Wis again non-empty by irreducibility of X). Let C(X) denote the set of equivalenceclasses of such pairs. Each element of C(X) is called a rational function on X. We

33

have an inclusion C ⊂ C(X) where any c ∈ C is represented by the pair (X, c).Given any (V, f) and (W, g) representing elements of C(X), we define their sum

(V, f) + (W, g) = (V ∩ W, f |V ∩W + g|V ∩W )

and their product

(V, f)(W, g) = (V ∩ W, (f |V ∩W )(g|V ∩W ))

which can be shown to be well-defined operations on C(X).

8.27 Exercise Prove that the above sum and product are well-defined and makeC(X) and extension field of C.

8.28 Exercise Prove that if X is an irreducible variety and V ⊂ X a non-emptyopen subset, then V is irreducible and restriction of functions gives an isomorphismC(X)

∼→ C(V ) over C of the function fields.

8.29 Dimension of an irreducible variety as transcendence degree Showthat the dimension of an irreducible variety X is the same as the dimension ofany non-empty open affine subvariety of X, which in turn equals the transcendencedegree over C of its function field. Hence dimension of an irreducible variety X isthe same as the transcendence degree of C(X) over C.

8.30 Application of Krull’s principal ideal theorem Let X be an irreduciblevariety, and let f ∈ O(X) − {0}. Let Y ⊂ X denote the closed subset consistingof all P ∈ X such that f(P ) = 0. Let Z be any irreducible component of Y . Thenshow that dim(Z) = dim(X) − 1.

Is the above conclusion true without the hypothesis that X is irreducible?

9 Tangent Space to a Variety

Real directional derivatives and differentials

9.1 Let V be a finite dimensional real vector space, U ⊂ V be an open subset andf : U → W a C∞ function. Let P ∈ U be a point. Then vP (f) ∈ R denotes thelimit

vP (f) = limt→0

f(P + tv) − f(P )

tShow that the above limit indeed exists, and it depends only on the germ of f at P .

9.2 With the above notation, show that the following holds: vP (λ) = 0 for aconstant function λ, vP (f + g) = vP (f) + vP (f), vP (fg) = f(P )vP (g) + g(P )vP (f).

34

9.3 Let C∞V,P denote the local ring of germs of C∞ functions at a point P ∈ V where

V is a finite dimensional real vector space, and let aP ⊂ C∞V,P denote its maximal

ideal. Let a2P denote the square of the ideal. Show that if f ∈ a

2P then vP (f) = 0

for each v.

9.4 Exercise If V = Rn with linear coordinates xi, and if v = (v1, . . . vn), then

show (using chain rule) that

vP (f) =∑

i

∂f

∂xi

(P )vi

9.5 Differential dfP defined From the above exercise, conclude that the mapdfP : V → R defined by v 7→ vP (f) is a real linear functional on the vector spaceV . This linear functional is denoted dfP , and is called the differential of f at P . Itdepends only on the germ of f at P .

9.6 Exercise With notation as above, show that the map

U → HomR(V, R) : P 7→ dfP

is a C∞ map from U to the dual vector space HomR(V, R) of V .

9.7 C∞ total derivative Let V and W be finite dimensional real vector spaces,and let v ∈ V . Let U ⊂ V be an open subset and let f : U → W be a C∞ map. LetP ∈ U be a point. Then vP (f) ∈ W denotes the limit

vP (f) = limt→0

f(P + tv) − f(P )

t

Show that the above limit indeed exists. If V = Rn with linear coordinates xi, if

v = (v1, . . . vn), and if W = Rm with linear coordinates yj so that f can be written

as (f1, . . . , fm) where each fi is in C∞(U), then show that the column vector vP (f)is given by applying the m × n Jacobian matrix

(∂fk

∂xi

(P )

)

to the column vector v. Conclude that we thus get an R-linear homomorphism

DfP : V → W : v 7→ vP (f)

and as P varies we get a C∞ map

Df : U → HomR(V,W ) : P 7→ DfP

The linear map DfP ∈ HomR(V,W ) is called the total derivative of f at P .

35

9.8 If P = (a1, . . . , an) ∈ Rn is any point, show that any germ f ∈ C∞

Rn,P of aC∞-function at P can be expressed as

f = f(P ) +∑

1≤i≤n

∂f

∂xi

(P ) (xi − ai) +∑

1≤i≤j≤n

gi,j (xi − ai)(xj − aj)

where gi,j are germs of C∞-functions at P . In particular, if f(P ) = 0 and dfP = 0then f ∈ a

2P where aP is ideal formed by all germ of all C∞ functions which take the

value 0 at P .

Holomorphic directional derivatives and differentials

9.9 Cauchy-Riemann equations Let V = Cn, U ⊂ V an open subset, and

f : U → C a C∞ map (where U is regarded as an open subset of Cn = R

2n andC = R

2 where the identifications are made as usual by considering real and imaginaryparts of coordinates. Then f is holomorphic if and only if for each P ∈ U we have

(√−1v)P (f) =

√−1 · vP (f)

for all vectors v ∈ Cn. This is equivalent to the condition that for each P ∈ U the

R-linear map dfP : V → C should actually be C-linear, that is,

dfP ∈ HomC(V, C) ⊂ HomR(V, C)

If xi = si+√−1ti are the complex coordinates on C

n with real parts si and imaginaryparts ti, then show that the above condition is equivalent to the set of n equations

∂f

∂si

= −√−1

∂f

∂ti

called the Cauchy-Riemann equations.

9.10 Holomorphic directional derivative Let V be a finite dimensional com-plex vector space, and let v ∈ V . Let U ⊂ V be an open subset and let f : U → C

be a holomorphic map. Let P ∈ U be a point. Then vP (f) ∈ C denotes the limit

vP (f) = limt→0

f(P + tv) − f(P )

t

Show that the above limit indeed exists, and it depends only on the germ of f at P .

9.11 With the above notation, show that the following holds: vP (λ) = 0 for aconstant function λ, vP (f + g) = vP (f) + vP (f), vP (fg) = f(P )vP (g) + g(P )vP (f).

36

9.12 Let HV,P denote the local ring of germs of holomorphic functions at a pointP ∈ V where V is a finite dimensional complex vector space, and let aP ⊂ HV,P

denote its maximal ideal. Let a2P denote the square of the ideal. Show that if f ∈ a

2P

then vP (f) = 0 for each v.

9.13 Holomorphic differentials With notation as above, we denote by dfP themap V → C defined by v 7→ vP (f). Show that this is a C-linear functional on V .The element dfP ∈ HomC(V, C) is called the holomorphic differential of f at P . Itdepends only on the germ of f at P .

9.14 Holomorphic total derivative Let V and W be finite dimensional complexvector spaces, and let v ∈ V . Let U ⊂ V be an open subset and let f : U → W bea holomorphic map. Let P ∈ U be a point. Then vP (f) ∈ W denotes the limit

vP (f) = limt→0

f(P + tv) − f(P )

t

Show that the above limit indeed exists. If V = Cn with complex linear coordinates

xi, if v = (v1, . . . vn), and if W = Cm with complex linear coordinates yj so that f

can be written as (f1, . . . , fm) where each fi is in H(U), then show that the columnvector vP (f) is given by applying the m × n Jacobian matrix

(∂fk

∂xi

(P )

)

to the column vector v. Conclude that we thus get a C-linear homomorphism

DfP : V → W : v 7→ vP (f)

and as P varies we get a holomorphic map

Df : U → HomC(V,W ) : P 7→ DfP

The linear map DfP ∈ HomC(V,W ) is called the holomorphic total derivative of fat P .

9.15 If P = (a1, . . . , an) ∈ Cn is any point, show that any germ f ∈ HRn,P of a

holomorphic function at P can be expressed as

f = f(P ) +∑

1≤i≤n

∂f

∂xi

(P ) (xi − ai) +∑

1≤i≤j≤n

gi,j (xi − ai)(xj − aj)

where gi,j are germs of holomorphic functions at P . In particular, if f(P ) = 0and dfP = 0 then f ∈ a

2P where aP is ideal formed by all germ of all holomorphic

functions which take the value 0 at P .

37

Tangent spaces to a manifold

9.16 C∞ (respectively, holomorphic) Zariski tangent space Let V be a finitedimensional vector space over R (respectively, over C), and let X ⊂ V be an arbi-trary subset. For any point P ∈ X, let TP (X) ⊂ V (respectively, let TP (X) ⊂ V )denote the set of all vectors v ∈ V such that for each germ (U, f) of C∞-function(respectively, of holomorphic function) on V at P such that f |X∩U = 0, we havevP (f) = 0. Then show that TP X is a real (respectively, TP X is a complex) vec-tor subspace of V . This vector space TP X is called the C∞ Zariski tangent space(respectively, TP X is called the holomorphic Zariski tangent space) to X at P .

9.17 Let V be a finite dimensional vector space complex vector space of complexdimension d, and let X ⊂ V and let P ∈ V . Let the holomorphic Zariski tangentspace TP X ⊂ V be the complex vector subspace defined above. We can regardV as a real vector space of dimension 2d, so we have the C∞ Zariski tangent spaceTP X ⊂ V as defined above, which is a real vector subspace of V . Then from the factthat the real and imaginary components of a holomorphic function are C∞ functions,deduce that TP X ⊂ TP X.

9.18 With the above notation, if V is real (respectively, complex) and X ⊂ Vis a locally closed C∞-submanifold (respectively, holomorphic submanifold ) of Vwith P ∈ X, then the Zariski tangent space TP X (respectively, TP X) is calledthe tangent space to the manifold X at P ∈ X. Show that if P is the centre of aC∞ (respectively, holomorphic) cubical coordinate chart (W ; u1, . . . , un) in V (wheren = dim(V ) such that X ∩W is given by ud+1 = . . . = un = 0 (where d = dim(X)),then

TP (X) = {v ∈ V | vP (ui) = 0 for all d + 1 ≤ i ≤ n}(respectively, TP X = {v ∈ V | vP (ui) = 0 for all d + 1 ≤ i ≤ n.) With respect tolinear coordinates (x1, . . . , xn) on V , deduce that the tangent space TP (X) (respec-tively, TP X) is described as follows. We have v = (v1, . . . , vn) ∈ TP (X) (respectively,TP X) if and only if

∑

1≤j≤n

∂ui

∂xj

(P )vj = 0 for all d + 1 ≤ i ≤ n

In particular, as the rank of the matrix (∂ui/∂xj)d+1≤i≤n,1≤j≤n is n−d, it follows thatdimR TP (X) = d = dimR(X) for any C∞ (respectively, dimC TP (X) = d = dimC(X)for any holomorphic) manifold X and point P ∈ X.

9.19 With the above notation, if V is a complex vector space and X ⊂ V isa locally closed holomorphic submanifold of V with P ∈ X, then as real vectorsubspaces of V we have an equality TP X = TP X ⊂ V . This equality follows fromthe description of these tangent spaces given above in terms of local coordinates.

38

9.20 Let V be a finite dimensional vector space over R, and let X ⊂ V be anarbitrary subset. For any point P ∈ X, let IX,P ⊂ aP be the ideal which consistsof all germs (U, f) which vanish on X ∩ U . Consider the linear subspace N∗

X,P ofV ∗ spanned by all differentials dfP where f ∈ IX,P . Then show that TP X ⊂ V isthe subspace annihilated by N∗

X,P ⊂ V ∗ under the dual pairing of V with V ∗. Inparticular, dim(TP X) = n− r where r = dim(N∗

X,P ). State and prove an analogousstatement for the holomorphic tangent space TP X.

9.21 Let V be a finite dimensional vector space over the field k = R (respectively,over k = C), and let X ⊂ V be an arbitrary subset. For any point P ∈ X, let C∞

V,P

(respectively, HV,P ) denote the local ring of germs of C∞-functions (respectively,holomorphic functions) on V at P , and let aP denote its maximal ideal. Let IX,P ⊂aP be the ideal which consists of all germs (U, f) which vanish on X ∩U . We definea map

TP X × aP

IX,P + a2P

→ k : (v, f) 7→ vP (f)

(where recall k denotes the field R or C as stated above). Note that vP (IX,P ) = 0by definition of TP X and vP (a2) = 0 by 9.3 (respectively, by 9.12) so this map iswell-defined. Show that that above map defines a non-degenerate bilinear pairing,making TP X and aP /(IX,P + a

2P ) dual vector spaces of each other.

Proof If v 6= 0 and xi are linear coordinates on V with P = (a1, . . . , an), thenv(xi − ai) 6= 0 for some i, showing the ‘left-kernel’ to be zero in the above pairing.To see the ‘right-kernel’ to be zero, suppose f ∈ aP is such that vP (f) = 0 for allv ∈ TP X (or TP X). Let f1, . . . , fr ∈ IX,P such that (df1)P , . . . , d(fr)P is a basisfor N∗

X,P . As TP X ⊂ V is exactly the subspace annihilated by N∗X,P , and as dfP

annihilates TP X, dfP is a linear combination∑

cid(fi)P for constants ci. Henced(f − ∑

cifi)P = 0. This shows using 9.8 or 9.15 that f − ∑cifi ∈ a

2P , so f = 0 in

aP /(IX,P + a2P ), as was to be shown.

Tangent spaces to a variety

9.22 Algebraic Zariski tangent space Let V be a finite dimensional vectorspace over C, and let X ⊂ V be an arbitrary subset. For any point P ∈ V , letTP (X) ⊂ V denote the set of all vectors v ∈ V such that for each germ (U, f) ofregular function on V at P such that f |X∩U = 0, we have vP (f) = 0. Then showthat TP X is a complex vector subspace of V . This vector space TP X is called thealgebraic Zariski tangent space to X at P .

9.23 If TP X (respectively, TP X) denotes the tangent space in the C∞ (respectively,in the holomorphic) category defined in 9.16, and if TP X denotes the tangent spacein the algebraic category defined in 9.22, then as every regular germ is a holomorphic

39

germ and as the real and imaginary parts of every holomorphic germ are C∞ germs,we have the inclusions

TP X ⊂ TP X ⊂ TP X

If X is a locally closed algebraic subvariety with P ∈ X, then we get the equalityTP X = TP X as an immediate consequence of Proposition 10.12.

9.24 Let V is a finite dimensional complex vector space, and let X ⊂ V be anarbitrary subset. For any point P ∈ X, let OV,P denote the local ring of germs ofregular functions on V at P , and let aP denote its maximal ideal. Let IX,P ⊂ aP

be the ideal which consists of all germs (U, f) which vanish on X ∩ U . We define amap

TP X × aP

IX,P + a2P

→ C : (v, f) 7→ vP (f)

Note that vP (IX,P ) = 0 by definition of TP X and vP (a2) = 0 by 9.12, so this mapis well-defined. Let N

∗X,P ⊂ V ∗ be the vector space spanned by all differentials dfP

where f ∈ IX,P . Show following the proof of 9.21 that above map defines a non-degenerate bilinear pairing, making TP X and aP /(IX,P + a

2P ) dual vector spaces of

each other.

9.25 Let V is a finite dimensional complex vector space, and let X ⊂ V be a locallyclosed subvariety. Let OX,P be the local ring at P ∈ X and let mP ⊂ OX,P be itsmaximal ideal. As above, let OV,P denote the local ring of germs of regular functionson V at P with maximal ideal aP , and let IX,P ⊂ aP be the ideal which consistsof all germs (U, f) which vanish on X ∩ U . Then show that OX,P = OV,P /IX,P ,mP = aP /IX,P , and consequently

aP

IX,P + a2P

=mP

m2P

Hence we have a non-degenerate pairing

TP X × mP

m2P

→ C : (v, f) 7→ vP (f)

9.26 Let V be a finite dimensional complex vector space, and let X ⊂ V be anarbitrary subset and let P ∈ X. From the proof of 9.21, we have the following. Letf1, . . . , fm ∈ IX,P be any finite sequence of elements such that (df1)P , . . . , d(fm)P

generates N∗X,P ⊂ V ∗ which is the vector space of all differentials dfP where f ∈ IX,P

(for example, we may take f1, . . . , fm to be any set of generators of the ideal IX,P ). AsTP X ⊂ V is exactly the subspace annihilated by N

∗X,P , in terms of linear coordinates

x1, . . . , xn on V , the tangent space TP X is the kernel of the linear map Cn → C

m

defined by the m × n Jacobian matrix (∂fi/∂xj)(P ). In particular, we have

rank(∂fi/∂xj)(P ) + dimC(TP X) = dimC V

40

10 Singular and Non-singular Varieties

Regular local rings: definition

10.1 Proposition If A is a noetherian local ring with maximal ideal m and residuefield k = A/m, then m/m2 as a k-vector space has dimension ≥ the Krull dimensionof A.

dimk(m/m2) ≥ dim(A)

10.2 Definition: Regular local ring A noetherian local ring with maximalideal m and residue field k = A/m is called regular if equality holds: dimk(m/m2) =dim(A).

Non-singularity: intrinsic definition

10.3 Non-singular point on a subvariety A point P on a variety X is saidto be non-singular point of X (or a variety X is said to be non-singular at a pointP ∈ X) if the local ring OX,P is a regular local ring. Otherwise P is said to be asingular point of X (or X is said to be singular at P ).

10.4 Non-singular variety A variety X is said to be non-singular if each pointP ∈ X is a non-singular point of X.

10.5 Lemma Any regular local ring is an integral domain.

10.6 Exercise Let X be a variety, let X1 ⊂ X and X2 ⊂ X be two differentirreducible components of X, and let P ∈ X1 ∩ X2. Then show using the abovelemma that X is singular at P . In particular, if X is a non-singular variety, then Xis the disjoint union of all its irreducible components.

Embedded non-singularity: Jacobian criterion

10.7 Lemma Let X ⊂ Cn be a locally closed subvariety, and let P ∈ X. Let mP ⊂

OX,P denote the maximal ideal of the local ring of X at P , and let dimC(mP /m2P )

denote the dimension of the C-vector space mP /m2P . Then the following holds.

Given any principal open subset Uh ⊂ Cn such that P ∈ Uh and X ∩Uh is closed in

Uh, together with any finite set of generators f1, . . . , fm of the radical ideal IX∩Uh⊂

O(Uh) = C[x1, . . . , xn, 1/h], we have

rank(∂fi/∂xj)(P ) + dimC(mP/m2P ) = n

where rank(∂fi/∂xj)(P ) is the rank of the m × n Jacobian matrix (∂fi/∂xj)(P ).

41

Proof By 9.26, we have rank(∂fi/∂xj)(P ) + dimC(TP X) = n. By 9.25, the vectorspaces TP X and m/m2 are dual of each other so have the same dimension.

10.8 Theorem For any point P on a locally closed subvariety X ⊂ Cn, the

following conditions are equivalent.

(1) X is non-singular at P , that is, the local ring OX,P is regular.

(2) The algebraic Zariski tangent vector space TP X has dimension equal to the localdimension dim(OX,P ) of X at P .

(3) Given any principal open subset Uh ⊂ Cn such that P ∈ Uh and X ′ = X ∩ Uh

is closed in Uh, together with any finite set of generators f1, . . . , fm of the radicalideal IX′ ⊂ O(Uh) = C[x1, . . . , xn, 1/h], the following holds: the rank of the m × nJacobian matrix (∂fi/∂xj)(P ) equals n − d(P ) where d(P ) = dim(OX,P ) denotesthe local dimension of X at P .

Proof As principal open sets form a basis of open sets in Cn, and as X is locally

closed in Cn, there exists a principal open subset Uh ⊂ Cn such that P ∈ Uh and

X ∩Uh is closed in Uh. For any finite set of generators f1, . . . , fm of the radical idealIX′ ⊂ O(Uh) = C[x1, . . . , xn, 1/h], by the above lemma we have

rank(∂fi/∂xj)(P ) + dimC(mP/m2P ) = n

Therefore, X is non-singular at X (that is, dimC(mP /m2P ) = dim(OX,P ) = d(P ) the

local dimension of X at P ) if and only if rank(∂fi/∂xj)(P ) = n−d(P ). This provesthe theorem.

Non-singular locus is open and non-empty.

10.9 Theorem For any non-empty variety X, there exists a nonempty open sub-variety U ⊂ X such that a point P lies in U if and only if X is non-singular atP .

Proof (1) Openness of the set U of nonsingular points: Let P ∈ U . As by assump-tion X is non-singular at P , by an above exercise (which uses the fact that regularlocal rings are domains), it follows that P lies on exactly one irreducible componentof X. Replacing X by the complement in X of all other irreducible components ofX, we can assume without loss of generality that X is irreducible. As the questionis local on X, we may assume that X is a closed subvariety of C

n. For any finiteset of generators f1, . . . , fm of the radical ideal I = IX , by the above theorem at thepoint P , we have equality (∂fi/∂xj)(P ) equals n − d(P ) where d(P ) = dim(OX,P )denotes the local dimension of X at P . As X is irreducible, dim(OX,P ) = dim(X),hence we get an equality

rank(∂fi/∂xj)(P ) = n − dim(X)

However, the rank of a matrix of continuous functions is lower-semicontinuous (asif a certain minor has determinant non-zero at a point, it remains non-zero in a

42

neighbourhood). Hence for all Q ∈ V where V is a Zariski-open neighbourhood ofP in X we have the inequality

rank(∂fi/∂xj)(Q) ≥ n − dim(X)

On the other hand, we know that

rank(∂fi/∂xj)(Q) = n − dim(mQ/m2Q) ≤ n − dim(OX,Q) = n − dim(X)

Therefore we have the equality rank(∂fi/∂xj)(Q) = n − dim(OX,Q) at all points ofV . Hence by the above theorem X is non-singular at all points of V , showing thatV ⊂ U . Therefore U is open as claimed.

(2) U is non-empty when X is non-empty: Without loss of generality, we can assumethat X is a closed irreducible subvariety of C

n. Let I ⊂ C[x1, . . . , xn] be the corre-sponding radical ideal, which is therefore a prime ideal, and O(X) is the C-algebraC[x1, . . . , xn]/I which is a domain of finite-type over C. Let y1, . . . , yd ∈ O(X) bea maximal set of algebraically independent elements over C (where d is the dimen-sion of X). The field of fractions K(X) of O(X) is therefore a finite extension therational function field C(y1, . . . , yd). As C is characteristic zero, this is a separableextension. (More generally, when working over an algebraically closed base field kof finite characteristic instead of C, we use the theorem on the existence of a sep-arating transcendence basis y1, . . . , yd). Hence by the primitive element theorem,there exists an element α ∈ K(X) such that

K(X) = C(y1, . . . , yd)[α]

The element α satisfies a monic irreducible polynomial g(t) ∈ C(y1, . . . , yd)[t] so thatK(X) = C(y1, . . . , yd)[t]/(g) with α the class of t.

The coefficients of g(t) are rational functions in y1, . . . , yd. If b ∈ C[y1, . . . , yd] isthe L.C.M. of their denominators (determined up to a non-zero constant by uniquefactorisation on polynomials), then we obtain an irreducible polynomial

f(t) = bg(t) ∈ C[y1, . . . , yd, t]

of degree ≥ 1 in t, such that K(X) is isomorphic to C(y1, . . . , yd)[t]/(f).

Let Cd+1 have linear coordinates denoted by y1, . . . , yd, t. Consider the closed sub-

varietyY = Z(f(t)) ⊂ C

d+1

defined by the vanishing of f . We claim that X has a non-empty open subvarietyV which is isomorphic to a non-empty open subvariety W of Y . Let the chosenprimitive element α ∈ K(X) be written as a quotient α = u/v where u, v ∈ O(X),v 6= 0. Then the vanishing of v defines a proper closed subvariety Z(v) ⊂ X, whichhas a non-empty open complement V = X − Z(v). Then we have y1, . . . , yd, α ∈O(V ). Hence we have a morphism

θ = (y1, . . . , yd, α) : V → Cd+1

43

As the image of θ lies in Y ⊂ Cd+1, this defines a morphism V → Y which we again

denote by θ.

Note that x1, . . . , xn ∈ O(X) ⊂ K(X) = C(y1, . . . , yd)[α]. Hence each xi can beexpressed as a ratio of polynomials ai(y1, . . . , yd, α)/bi(y1, . . . , yd), where the de-nominators are not identically equal to zero. As t does not occur in bi, note thatf(t) does not divide bi. Hence the vanishing each bi defines a proper closed subva-riety Z(bi) ⊂ Y . As Y is irreducible, Y is not a union of any collection of finitelymany proper closed subvarieties, hence the open complement W = Y − ∪iZ(bi) isnon-empty. Then x1, . . . , xn ∈ O(W ), hence we get a morphism

η = (x1, . . . , xn) : W → Cn

As the image of η lies in X ⊂ Cn, this defines a morphism W → X which we again

denote by η.

It follows from their definitions that η ◦ θ = idV and θ ◦ η = idW .

We will now show that for the subvariety Y ⊂ Cd+1, the set of non-singular points is

nonempty. As Y is closed of dimension d, with its radical ideal equal to the principalideal (f), by the above theorem it is non-singular at all points P ∈ Y where the1 × (d + 1)-matrix (∂f/∂y1, . . . , ∂f/∂yd, ∂f/∂t)(P ) is not zero. As t occurs in fwith degree ≥ 1, the derivative ∂f/∂t is not the zero polynomial. (Here we usedthat C has characteristic zero. However, it is possible to appropriately modify theargument so as to apply to any perfect field of finite characteristic). The t-degree of∂f/∂t is less that that of f , so f cannot divide it, so ∂f/∂t is not identically zeroon Y . Hence Y has a non-empty open subset N of non-singular points.

By irreducibility of Y , it follows that N ∩ W is non-empty. Using the isomorphismθ : V → W , it follows that V (and hence X) has at least one non-singular point.

This completes the proof of the theorem.

10.10 Note Inside the part (2) of the above proof is hidden the basic concept ofbirationality. It will be made explicit later in some other series of lectures in thisInstructional School.

Some holomorphic facts

10.11 Exercise If V is a non-empty connected open subset of Cn in the Euclidean

topology, then the ring H(V ) of all holomorphic functions on V is a domain.

10.12 Proposition Let X ⊂ Cn be Zariski closed, and let f1, . . . , fm generate

IX ⊂ C[x1, . . . , xn], the ideal consisting of all polynomial functions vanishing onX. Let V ⊂ C

n be a Euclidean-open subset and let g : V → C be a holomorphicfunction such that g vanishes on X ∩ V . Then there exist holomorphic functions

44

h1, . . . , hm : V → C such that g =∑

1≤i≤m hifi. In other words, the ideal JX∩V ⊂H(V ) of all holomorphic functions on V vanishing on X ∩ V is given by

JX∩V = H(V )IX

Proof (Sketch) Take any P = (a1, . . . , an) ∈ X ∩ V . The ideal I is radical, hence

its extension in the completion OCn,P = C[[x1 − a1, . . . , xn − an]] is again a radical

ideal. Note that OCn,P ⊂ HCn,P ⊂ OCn,P , so IX generates a radical ideal in HCn,P .By analytic nullstellensatz, if g is a germ of a holomorphic function around P suchthat g vanishes on X, then g lies in the radical of the analytic ideal generated byIX . From this the result follows.

10.13 GAGA The above proposition also follows by GAGA, but the above proofis more elementary.

10.14 Exercise Let X ⊂ Cm and Y ⊂ C

n be irreducible closed subvarieties, andlet f : X → Y be a regular morphism. Suppose that the induced C-algebra homo-morphism f# : O(Y ) → O(X) makes O(X) a finite algebra over O(Y ). Then showthat (i) the map f is closed in Zariski topology, and (ii) the map f is proper henceclosed in Euclidean topology (hint: show that the inverse image of any bounded setis bounded).

10.15 Proposition If X ⊂ Cn is an irreducible closed subvariety, then any proper

Zariski-closed subset of X is no-where dense in the Euclidean topology. Conse-quently, any non-empty Zariski-open subset is Euclidean-dense, and any non-emptyEuclidean-open subset is Zariski-dense in X.

Proof (Sketch) When X = Cn, we leave this as an exercise. In the general case, use

Noether normalisation to find a finite projection X → Cm, and then use the inverse

function theorem to show that outside a Zariski-closed subset it is a finite coveringprojection, to prove the result for X.

Comparison theorem: Nonsingularity and submanifolds

10.16 Theorem Let X ⊂ Cn be a locally closed subvariety, and let P ∈ X. Then

the following statements are equivalent.

(1) The variety X is non-singular at the point P .

(2) There exists a Euclidean open neighbourhood W of P in Cn such that X ∩ W

is a closed connected holomorphic submanifold of W .

(3) There exists a Euclidean open neighbourhood W of P in Cn such that X ∩ W

is a closed connected real C∞-submanifold of W .

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(4) There exists a Euclidean open neighbourhood W of P in Cn such that X ∩ W

is a closed connected real differential submanifold of W of class C1.

Moreover, when the above equivalent conditions are fulfilled, the dimension of X∩Was a topological manifold is equal to twice the the local dimension of X at P as analgebraic variety.

Proof (1) ⇒ (2) Suppose X is non-singular at P . Then there exists a principalopen neighbourhood Uh of P in C

n such that X ∩Uh is closed in Uh, irreducible andnon-singular. If X ∩ Uh defined by the radical ideal I ⊂ O(Uh), then for any set ofgenerators f1, . . . , fm of the radical ideal and any point Q ∈ X ∩Uh, the rank of them × n Jacobian matrix (∂fi/∂xj)(Q) equals n − d where d = dim(X ∩ Uh).

Let (after possible re-indexing) the first r = n − d rows and columns of the matrix(∂fi/∂xj)(P ) be linearly independent. The non-vanishing of the determinant of ther×r-minor (∂fi/∂xj)1≤i≤r,1≤j≤r defines a Zariski-open neighbourhood N of P in Uh.

Let φ : N → Cr be the map (f1, . . . , fr). Hence applying the implicit function

theorem to the map φ, we see that the vanishing of φ defines a closed holomorphicsubmanifold Y of N of dimension n−r = d. As dfr+1, . . . , dfm are linear combinationsof df1, . . . , dfr on Y , it follows by integration that fr+1, . . . , fm are locally constanton Y . Hence a Euclidean open neighbourhood of P in Y lies in X. This shows thatX is a holomorphic manifold of dimension d.

(2) ⇒ (3) ⇒ (4) : Obvious. (Note: Actually, we have not introduced the notion of aC1-submanifold in these lectures so far: the reader may either just ignore statement(4) and go from (3) back to (1) using the below argument for (4) implies (1), orotherwise the reader can learn the definition of C1-submanifolds elsewhere.)

(4) ⇒ (2) Suppose that there exists a Euclidean open neighbourhood W of P in Cn

such that X ∩ W is a closed connected C1-submanifold of W of dimension r.

As non-singular points are dense in the Euclidean topology, the real tangent spaceTP X to X at P is the limit of tangent spaces at a sequence of regular points tendingto P . Hence TP X is a complex-linear subspace, say of complex dimension d (so rmust be even, equal to 2d). Then there exists a coordinate plane E of dimension d inC

n such that the projection π of TP X on E is an isomorphism. Hence in a Euclideanneighbourhood of π(P ) ∈ E, there exist n − d complex C1 functions h1, . . . , hn−d

whose graph is the intersection of X with a cubical neighbourhood of P inside W .Again, as regular points are dense, it follows that the functions h1, . . . , hn−d satisfythe Cauchy-Riemann equations. Hence X ∩W is in fact a holomorphic submanifoldof W .

(2) ⇒ (1) Let W be a Euclidean open neighbourhood of P in Cn such that X∩W is a

closed connected holomorphic submanifold of W of complex dimension d. Let Uh bea principal open neighbourhood of X in C

n such that X ′ = X∩Uh is closed. We firstshow that P lies on a unique irreducible component of X ′. Let P ∈ D ⊂ X ′ where Dis Euclidean open in X ′ and is holomorphically isomorphic to a polydisk. If X1 and

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X2 are two irreducible components of X ′ passing through P , then by a Propositionabove (which says that any non-empty Euclidean open subset is Zariski-dense in anirreducible variety) it follows that D ∩ Xi is Zariski-dense in Xi for i = 1, 2. Hencefrom the fact that H(D) is a domain, we see that X1 = X2.

Hence we can now assume that X ′ = X ∩ Uh is closed in Uh and irreducible.Let d′ denote the dimension of X ′ as an algebraic variety, which by irreducibil-ity equals dim(OX,Q) for all Q ∈ X ′. There exists a holomorphic coordinate chart(U ; z1, . . . , zn) in Uh such that X ′ ∩ U is defined by vanishing of zd+1, . . . , zn. The(n− d)× n Jacobian matrix (∂zi/∂xj) where d + 1 ≤ i ≤ n and 1 ≤ j ≤ n has rankn− d. If f1, . . . , fm generate the radical ideal of X ′ in O(Uh), then by a propositionabove, the functions zd+1, . . . , zn are holomorphic linear combinations of f1, . . . , fm

around P . Clearly, f1, . . . , fm are holomorphic linear combinations of zd+1, . . . , zn.Hence we get

rank(∂fi/∂xj)(P ) = n − d

As any non-empty Zariski-open is Euclidean-dense, there exists some Q ∈ B suchthat X ′ is non-singular at Q. Then rank(∂fi/∂xj)(Q) = n− d′ where d′ = dim(X ′).This shows d′ = d. Hence the equality rank(∂fi/∂xj)(P ) = n − d now shows thatX ′ is non-singular at P .

This completes the proof of the theorem. ¤

The above proofs of Proposition 10.12 and Theorem 10.16 are taken from page 13 ofJohn Milnor: Singular points of complex hypersurfaces, Princeton University Press,1968. I thank Gurjar for showing me this reference.

11 Abstract Manifolds and Varieties

Abstract C∞-manifolds

An abstract C∞-manifold of dimension n consists of a topological space X togetherwith an R-algebra C∞(V ) of continuous functions f : V → R for each open subsetV ⊂ X, such that the following conditions are satisfied.

(1) If U ⊂ V ⊂ X are open and f ∈ C∞(V ), then the restriction f |U is in C∞(U).

(2) If (Ui) is an open cover of an open subset V ⊂ X, and if fi ∈ C∞(Ui) suchthat fi|Ui∩Uj

= fj|Ui∩Ujfor all i, j, then there exists a unique f ∈ C∞(V ) such that

fi = f |Uifor all i.

(3) Each point P ∈ X has an open neighbourhood V together with a homeomor-phism φ : V → R

n, such that for any open subset U ⊂ V , a function f : U → R isin C∞(U) if and only if the function f ◦ φ−1 : φ(U) → R is a C∞ function on φ(U).

The C∞-functions on an open subset V ⊂ X are by definition the elements ofC∞(V ). A C∞-morphism φ : X → Y of C∞-manifolds is the same as a continuousmap φ : X → Y such that for any open W ⊂ Y and any g ∈ C∞(W ), the compositeg ◦ φ : φ−1(W ) → R is in C∞(φ−1W ).

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11.1 Exercise Relate this definition of C∞-manifolds, C∞-functions and C∞-mor-phisms to the definition in terms of overlapping patches that you may have encoun-tered previously.

Abstract holomorphic manifolds

An abstract holomorphic manifold of dimension n consists of a topological space Xtogether with an C-algebra H(V ) of continuous functions f : V → C for each opensubset V ⊂ X, such that the following conditions are satisfied.

(1) If U ⊂ V ⊂ X are open and f ∈ H(V ), then the restriction f |U is in H(U).

(2) If (Ui) is an open cover of an open subset V ⊂ X, and if fi ∈ H(Ui) suchthat fi|Ui∩Uj

= fj|Ui∩Ujfor all i, j, then there exists a unique f ∈ H(V ) such that

fi = f |Uifor all i.

(3) Each point P ∈ X has an open neighbourhood V together with a homeo-morphism φ : V → B where B is an open subset of C

n, such that for any opensubset U ⊂ V , a function f : U → C is in H(U) if and only if the functionf ◦ φ−1 : φ(U) → C is a holomorphic function on φ(U).

The holomorphic functions on an open subset V ⊂ X are by definition the elementsof H(V ). A holomorphic morphism φ : X → Y of holomorphic manifolds is thesame as a continuous map φ : X → Y such that for any open W ⊂ Y and anyg ∈ H(W ), the composite g ◦ φ : φ−1(W ) → C is in H(φ−1W ).

11.2 Exercise Relate this definition of holomorphic manifolds, holomorphic func-tions and holomorphic morphisms to the definition in terms of overlapping patchesthat you may have encountered previously.

Abstract varieties of Serre

It is an important property of quasi-affine and quasi-projective varieties that theyadmit open covers by affine subvarieties. This property was made the key point ofthe definition of an abstract variety by Serre.

An abstract variety in the sense of Serre is a topological space X together with aC-algebra O(V ) of continuous functions f : V → C for each open V ⊂ X such thatthe following conditions are satisfied.

(1) If U ⊂ V ⊂ X are open and f ∈ O(V ), then the restriction f |U is in O(U).

(2) If (Ui) is an open cover of an open subset V ⊂ X, and if fi ∈ O(Ui) suchthat fi|Ui∩Uj

= fj|Ui∩Ujfor all i, j, then there exists a unique f ∈ O(V ) such that

fi = f |Uifor all i.

(3) Each point P ∈ X has an open neighbourhood V together with a homeomor-phism φ : V → Y where Y is an affine variety, such that for any open subset U ⊂ V ,a function f : U → C is in O(U) if and only if the function f ◦ φ−1 : φ(U) → C is aregular function on the quasi-affine variety φ(U).

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The regular functions on an open subset V ⊂ X are by definition the elements ofO(V ). A regular morphism φ : X → Y of varieties is the same as a continuousmap φ : X → Y such that for any open W ⊂ Y and any g ∈ O(W ), the compositeg ◦ φ : φ−1(W ) → C is in O(φ−1W ).

11.3 Exercise Give another equivalent definition of an abstract variety, regularfunctions and regular morphisms, in terms of overlapping affine open patches, in thespirit of the definition of a manifold in terms of overlapping coordinate patches.

11.4 Note In the above definition, one can replace the field C by any algebraicallyclosed field k of arbitrary characteristic, to get the definition of the category ofabstract varieties over k.

11.5 An abstract variety X is said to be non-singular at a point P if the localring OX,P is regular. This generalises the earlier definition of non-singularity forquasi-projective varieties.

11.6 If X and Y are abstract varieties, then a product variety X × Y togetherwith regular morphisms (projections) p1 : X × Y → X and p2 : X × Y → Yexists in the category of abstract varieties. It is a simple exercise to manufacture(X × Y, p1, p2) by taking affine open covers (Ui) and (Vj) respectively of X and Y ,taking the products Ui × Vj of the affine open sets, and then gluing them togetheras in Exercise 11.3.

11.7 If the diagonal subvariety ∆X ⊂ X ×X is closed then X is called separated.Note that all quasi-projective varieties are automatically separated by Exercise 8.16.

12 Some suggestions for further reading

There is a very large number of good text-books available on Algebraic Geometry.The student can begin with Mumford: The Red Book of Varieties and Schemesand then go on to Hartshorne: Algebraic Geometry for laying the foundations. Forstudying the subject from an analytical (rather than algebraic) point of view, a goodsource is the text-book Griffiths and Harris: Algebraic Geometry. A reader of thesenotes would also be equipped with the necessary background to study basic text-books on the subject of Algebraic Groups such as Borel: Linear Algebraic Groupsor Humphreys: Linear Algebraic Groups.

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