Introduction

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Just as the starting point of linear algebra is the study of the solutions of systems

of linear equations,

_n

j=1

aijXj = di, i= 1, . . . ,m, (*)

the starting point for algebraic geometry is the study of the solutions of systems of

polynomial equations,

fi(X1, . . . ,Xn) = 0, i= 1, . . . ,m, fi ∈ k[X1, . . . ,Xn].

Note immediately one difference between linear equations and polynomial equations:

theorems for linear equations don’t depend on which field k you are working over,1

but those for polynomial equations depend on whether or not k is algebraically closed

and (to a lesser extent) whether k has characteristic zero. Since I intend to emphasize

the geometry in this course, we will work over algebraically closed fields for the major

part of the course.

A better description of algebraic geometry is that it is the study of polynomial functions

and the spaces on which they are defined (algebraic varieties), just as topology

is the study of continuous functions and the spaces on which they are defined (topological

spaces), differential geometry (=advanced calculus) the study of differentiable

functions and the spaces on which they are defined (differentiable manifolds), and

complex analysis the study of holomorphic functions and the spaces on which they

are defined (Riemann surfaces and complex manifolds). The approach adopted in

this course makes plain the similarities between these different fields. Of course, the

polynomial functions form a much less rich class than the others, but by restricting

our study to polynomials we are able to do calculus over any field:w e simply define

d

dX

_

aiXi =

_

iaiXi−1.

Moreover, calculations (on a computer) with polynomials are easier than with more

general functions.

Consider a differentiable function f(x, y, z). In calculus, we learn that the equation

f(x, y, z) = C (**)

defines a surface S in R3, and that the tangent space to S at a point P = (a, b, c) has

equation2

_

∂f

∂x

_

P

(x − a) +

_

∂f

∂y

_

P

(y − b) +

_

∂f

∂z

_

P

(z − c) = 0. (***).

The inverse function theorem says that a differentiable map α : S → S_ of surfaces is

a local isomorphism at a point P ∈ S if it maps the tangent space at P isomorphically

onto the tangent space at P_ = α(P).

1For example, suppose that the system (*) has coefficients aij ∈ k and that K is a field containing

k. Then (*) has a solution in kn if and only if it has a solution in Kn, and the dimension of the

space of solutions is the same for both fields. (Exercise!)

2Think of S as a level surface for the function f, and note that the equation is that of a plane

through (a, b, c) perpendicular to the gradient vector (_f)P at P.)

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Consider a polynomial f(x, y, z) with coefficients in a field k. In this course, we

shall learn that the equation (**) defines a surface in k3, and we shall use the equation

(***) to define the tangent space at a point P on the surface. However, and this is

one of the essential differences between algebraic geometry and the other fields, the

inverse function theorem doesn’t hold in algebraic geometry. One other essential

difference:1 /X is not the derivative of any rational function of X; nor is Xnp−1 in

characteristic p _= 0. Neither can be integrated in the ring of polynomial functions.

Some notations. Recall that a field k is said to be algebraically closed if every

polynomial f(X) with coefficients in k factors completely in k. Examples: C, or the

subfield Q al of C consisting of all complex numbers algebraic over Q. Every field k

is contained in an algebraically closed field.

A field of characteristic zero contains a copy of Q, the field of rational numbers. A

field of characteristic p contains a copy of Fp, the field Z/pZ. The symbol N denotes

the natural numbers, N = {0, 1, 2, . . . }. Given an equivalence relation, [∗] sometimes

denotes the equivalence class containing ∗.

“Ring” will mean “commutative ring with 1”, and a homomorphism of rings will

always carry 1 to 1. For a ring A, A× is the group of units in A:

A× = {a ∈ A | ∃b ∈ A such that ab = 1}.

A subset R of a ring A is a subring if it is closed under addition, multiplication, the

formation of negatives, and contains the identity element.3 We use Gothic (fraktur)

letters for ideals:

a b c m n p q A B C M N P Q

a b c m n p q A B C M N P Q

We use the following notations:

X ≈Y Xand Y are isomorphic;

X

∼=

Y Xand Y are canonically isomorphic (or there is a given or unique isomorphism);

X df =Y Xis defined to be Y , or equals Y by definition;

X ⊂Y Xis a subset of Y (not necessarily proper).

3The definition on page 2 of Atiyah and MacDonald 1969 is incorrect, since it omits the condition

that x ∈ R ⇒−x ∈ R — the subset N of Z satisfies their conditions, but it is not a subring of Z.

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