態射就是反向的環映射
仿射簇之間的 態射 φ: X → Y 在座標下由多項式給出，φ = (φ_1, …, φ_m)。決定性的事實：把 Y 上的正則函數與 φ 複合，得到 X 上的正則函數，於是 φ 誘導一個 反向 的環同態 φ*: k[Y] → k[X]。這是從簇到 k-代數的一個 函子——反變的，實際上是到既約有限生成 k-代數的一個反等價。
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.射影空間：添上無窮遠點
仿射簇會『漏』：兩條平行線永不相交，拋物線奔向虛無。射影空間 P^n 添上一道『地平線』來修補它。P^n 的點是 A^{n+1} 中過原點的直線，用齊次座標 [a_0 : ⋯ : a_n] 表示，將所有座標同乘以同一個 λ ≠ 0 給出同一點。方程必須是 齊次 的，使得在這些類上為零有定義；它們的零點集是 射影簇。
回報是幾何變得一致而完備。Bézout 定理變得乾淨——兩條次數為 d 和 e 的射影平面曲線恰好相交於 de 個點（計重數），一旦把無窮遠處缺失的交點納入。P^n 由 n+1 個仿射卡覆蓋（令某個座標為 1），所以局部上它不過是普通的仿射幾何，沿重疊黏合。
Spec R：當點變成素理想
零點定理把 X 的點繫於 k[X] 的極大理想——但僅在代數封閉體上。Grothendieck 的飛躍是取 任意 交換環 R，並 定義 它的點空間為它的 素譜 Spec R = { R 的素理想 }，配以 Zariski 拓樸，其閉集為 V(I) = { 包含 I 的素理想 }。極大理想是『經典的』閉點；其餘素理想是新東西。
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.