态射就是反向的环映射
仿射簇之间的 态射 φ: X → Y 在坐标下由多项式给出，φ = (φ_1, …, φ_m)。决定性的事实：把 Y 上的正则函数与 φ 复合，得到 X 上的正则函数，于是 φ 诱导一个 反向 的环同态 φ*: k[Y] → k[X]。这是从簇到 k-代数的一个 函子——反变的，实际上是到既约有限生成 k-代数的一个反等价。
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.射影空间：添上无穷远点
仿射簇会‘漏’：两条平行线永不相交，抛物线奔向虚无。射影空间 P^n 添上一道‘地平线’来修补它。P^n 的点是 A^{n+1} 中过原点的直线，用齐次坐标 [a_0 : ⋯ : a_n] 表示，将所有坐标同乘以同一个 λ ≠ 0 给出同一点。方程必须是 齐次 的，使得在这些类上为零有定义；它们的零点集是 射影簇。
回报是几何变得一致而完备。Bézout 定理变得干净——两条次数为 d 和 e 的射影平面曲线恰好相交于 de 个点（计重数），一旦把无穷远处缺失的交点纳入。P^n 由 n+1 个仿射卡覆盖（令某个坐标为 1），所以局部上它不过是普通的仿射几何，沿重叠粘合。
Spec R：当点变成素理想
零点定理把 X 的点系于 k[X] 的极大理想——但仅在代数闭域上。Grothendieck 的飞跃是取 任意 交换环 R，并 定义 它的点空间为它的 素谱 Spec R = { R 的素理想 }，配以 Zariski 拓扑，其闭集为 V(I) = { 包含 I 的素理想 }。极大理想是‘经典的’闭点；其余素理想是新东西。
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.