Morphisms are ring maps, reversed
A morphism φ: X → Y of affine varieties is a map given in coordinates by polynomials, φ = (φ_1, …, φ_m). The decisive fact: composing a regular function on Y with φ gives a regular function on X, so φ induces a ring homomorphism φ*: k[Y] → k[X] going the *other way*. This is a functor from varieties to k-algebras — contravariant, and in fact an anti-equivalence onto reduced finitely generated k-algebras.
Example: the parametrization of the parabola.
phi : A^1 -> X = V(y - x^2) in A^2, t |-> (t, t^2).
Pull back coordinate functions:
phi*(x) = t, phi*(y) = t^2.
So phi* : k[X] = k[x,y]/(y - x^2) -> k[t],
x |-> t, y |-> t^2.
We checked in guide 2 that this is an ISOMORPHISM, so phi
is an isomorphism of varieties: A^1 ~= parabola.
Non-example: the cusp parametrization for V(y^2 - x^3),
psi : A^1 -> Y, t |-> (t^2, t^3).
psi* : k[Y] -> k[t], x |-> t^2, y |-> t^3.
The image is the subring k[t^2, t^3], which MISSES t, so
psi* is injective but NOT surjective. psi is a bijective
morphism that is NOT an isomorphism -- the cusp obstructs it.
Moral: morphisms of varieties = ring maps of coordinate
rings, and isomorphism of varieties = isomorphism of rings.Projective space: adding points at infinity
Affine varieties leak: two parallel lines never meet, a parabola runs off to nowhere. Projective space P^n fixes this by adding a 'horizon'. Points of P^n are lines through the origin in A^{n+1}, written in homogeneous coordinates [a_0 : ⋯ : a_n] where scaling all coordinates by the same λ ≠ 0 gives the same point. Equations must be homogeneous so that vanishing is well-defined on these classes; their zero sets are projective varieties.
The payoff is that geometry becomes uniform and complete. Bézout's theorem becomes clean — two projective plane curves of degrees d and e meet in exactly de points (counted with multiplicity), once the missing intersections at infinity are included. P^n is covered by n+1 affine charts (set one coordinate to 1), so locally it is just ordinary affine geometry glued along overlaps.
Spec R: when points become primes
The Nullstellensatz tied points of X to maximal ideals of k[X] — but only over an algebraically closed field. Grothendieck's leap is to take *any* commutative ring R and *define* its space of points to be its prime spectrum Spec R = { prime ideals of R }, with the Zariski topology whose closed sets are V(I) = { primes containing I }. Maximal ideals are the 'classical' closed points; the other primes are new.
Two telling spectra.
(1) R = Z. Spec Z = { (0) } union { (p) : p prime }.
The (p) are CLOSED points -- the usual primes 2,3,5,...
The point (0) is NOT closed; its closure is ALL of
Spec Z. It is the GENERIC POINT: a single 'fat' point
whose presence remembers the whole space. So number
theory becomes the geometry of a curve-like object.
(2) R = k[x], k alg. closed. Spec k[x] =
{ (x - a) : a in k } (the classical points of A^1)
plus the generic point (0). Adding (0) is exactly the
irreducible variety A^1 itself reappearing AS a point.
The STRUCTURE SHEAF O puts a ring of functions on each open
set (on a basic open D(f) = { p : f not in p } it is the
localization R[1/f]). The pair (Spec R, O) is an AFFINE
SCHEME; gluing these is a SCHEME. Varieties over an alg.
closed field embed as the closed-point locus of such schemes.