The purpose of this section is to convince you that commutative algebras are really spaces seen from the other side of your brain.

For a compact hausdorff space X, we have the algebra C(X) of continuous, complex-valued functions a : X → ℂ. The addition and multiplication are obtained pointwise from ℂ.

A continuous function ƒ : X → Y gives rise to an algebra morphism C(ƒ) : C(Y) → C(X) (note the reversal of direction!) via C(ƒ)(b) = a, where a (x) = b(ƒ(x)). In particular, the unique X → 1 gives the algebra morphism η : ℂ = C(1) → C(X), while each point x : 1 → X of the space gives an algebra morphism C(X) → ℂ.

Actually C(X) is more than just a ℂ-algebra; it is what is called a commutative C*-algebra (there is a norm and an involution (_)* coming from conjugation). With this extra structure the duality becomes precise:

This result is commonly referred to as Gelfand duality.

Algebraic geometry is the study of spaces called varieties: the solutions to polynomial equations in several variables. In studying the variety given by x2 + 2y3 = z4 over the field k, we pass to the k-algebra

By k[x, y, z] we mean the k-algebra of polynomials in three commuting indeterminates x, y, z; the elements are expressions

where αijk ∈ k and (i, j,k) runs over a finite subset of ℕ3.