Spec(ZZ).base_scheme() is Spec(QQ).base_scheme(); 整数环素谱的基本概型等于有理数域素谱的的基本概型
A.base_morphism();AQ.base_morphism();AAQ.base_morphism() 上面几个基本概型的基本态射
Scheme morphism: From: Affine Space of dimension 3 over Integer Ring To: Spectrum of Integer Ring Defn: Structure map
Scheme morphism: From: Affine Space of dimension 3 over Rational Field To: Spectrum of Rational Field Defn: Structure map
Scheme morphism: From: Affine Space of dimension 3 over Finite Field of size 5 To: Spectrum of Finite Field of size 5 Defn: Structure map

AAQ.identity_morphism(),自态射
A.dimension_relative(),维度关系
AQ.dimension_relative(),
AAQ.dimension_relative(),
from sage.schemes.generic.point import SchemePoint加载函数库
P1 = SchemePoint(A);P1;概型的点，很抽象
P2 = SchemePoint(AQ);P2;
P3 = SchemePoint(AAQ);P3;
Point on Affine Space of dimension 3 over Integer Ring
Point on Affine Space of dimension 3 over Rational Field
Point on Affine Space of dimension 3 over Finite Field of size 5
A.gens(),生成元
AQ.gens(),
AAQ.gens(),
P.ngens（）生成元个数
A.is_projective() 判断射影还是仿射空间
AAQ.rational_points(); 有理点
[(0, 0, 0), (1, 0, 0), (2, 0, 0), (3, 0, 0), (4, 0, 0), (0, 1, 0), (1, 1, 0), (2, 1, 0), (3, 1, 0), (4, 1, 0), (0, 2, 0), (1, 2, 0), (2, 2, 0), (3, 2, 0), (4, 2, 0), (0, 3, 0), (1, 3, 0), (2, 3, 0), (3, 3, 0), (4, 3, 0), (0, 4, 0), (1, 4, 0), (2, 4, 0), (3, 4, 0), (4, 4, 0), (0, 0, 1), (1, 0, 1), 。。。。。。
定义子概型，射影的要齐次
X = A.subscheme([x+1, 1-y^2, x*y^2-5]); X;
XX = PQ.subscheme([x^3-y^3, y^3+7*x^3, x*y^2]); XX
Closed subscheme of Affine Space of dimension 3 over Integer Ring defined by: x + 1, -y^2 + 1, x*y^2 - 5
Closed subscheme of Projective Space of dimension 7 over Finite Field of size 3 defined by: x0^3 - x1^3, x0^3 + x1^3, x0*x1^2

X.defining_polynomials ();定义概型的多项式
I = X.defining_ideal(); I;定义概型理想
XX.defining_polynomials ();
II = XX.defining_ideal(); II;
(x + 1, -y^2 + 1, x*y^2 - 5)
Ideal (x + 1, -y^2 + 1, x*y^2 - 5) of Multivariate Polynomial Ring in x, y, z over Integer Ring
(x0^3 - x1^3, x0^3 + x1^3, x0*x1^2) Ideal (x0^3 - x1^3, x0^3 + x1^3, x0*x1^2) of Multivariate Polynomial Ring in x0, x1, x2, x3, x4, x5, x6, x7 over Finite Field of size 3

一楼不具体定义生成元代表字母，系统默人x1,x2,x3.......
P = ProjectiveSpace(7, ZZ);
P;PQ = P.base_extend(GF(3)); PQ;
P.dimension_absolute();
PQ.dimension_absolute();
P.dimension();PQ.dimension();P.dimension_relative();
PQ.dimension_relative();
P.gen(1);PQ.gen(3);P.gens()
;PQ.gens();P.ngens();
PQ.is_projective()
Projective Space of dimension 7 over Integer Ring
Projective Space of dimension 7 over Finite Field of size 3
8
7
8
7
7
7
x1
x3
(x0, x1, x2, x3, x4, x5, x6, x7)
(x0, x1, x2, x3, x4, x5, x6, x7)
8
True

用在线的不好吗，文档自动保存，下载的在LINUX上才好用，WINDOW的1G多,还要其它的虚拟软件。。

以后挪窝了，上GOOGLE网盘和文档，百毒快了。。。。