Module: sage.schemes.generic.spec
Spec of a ring
Module-level Functions
X) |
sage: is_Spec(QQ^3) False sage: X = Spec(QQ); X Spectrum of Rational Field sage: is_Spec(X) True
Class: Spec
Note:
Calling Spec(R)
twice produces two distinct
(but equal) schemes, which is important for gluing to
construct more general schemes.
sage: Spec(QQ) Spectrum of Rational Field sage: Spec(PolynomialRing(QQ, 'x')) Spectrum of Univariate Polynomial Ring in x over Rational Field sage: Spec(PolynomialRing(QQ, 'x', 3)) Spectrum of Multivariate Polynomial Ring in x0, x1, x2 over Rational Field sage: X = Spec(PolynomialRing(GF(49,'a'), 3, 'x')); X Spectrum of Multivariate Polynomial Ring in x0, x1, x2 over Finite Field in a of size 7^2 sage: loads(X.dumps()) == X True sage: A = Spec(ZZ); B = Spec(ZZ) sage: A is B False sage: A == B True
A TypeError is raised if the input is not a CommutativeRing.
sage: Spec(5) Traceback (most recent call last): ... TypeError: R (=5) must be a commutative ring sage: Spec(FreeAlgebra(QQ,2, 'x')) Traceback (most recent call last): ... TypeError: R (=Free Algebra on 2 generators (x0, x1) over Rational Field) must be a commutative ring
sage: X = Spec(ZZ) sage: X Spectrum of Integer Ring sage: X.base_scheme() Spectrum of Integer Ring sage: X.base_ring() Integer Ring sage: X.dimension() 1
self, R, [S=None], [check=True]) |
Functions: coordinate_ring,
dimension
self) |
Return the underlying ring of this scheme.
sage: Spec(QQ).coordinate_ring() Rational Field sage: Spec(PolynomialRing(QQ,3, 'x')).coordinate_ring() Multivariate Polynomial Ring in x0, x1, x2 over Rational Field
self) |
Return the relative dimension of this scheme over its base.
Special Functions: __call__,
__init__,
_cmp_,
_repr_
self, x) |
Create a point of this scheme.
self, X) |
Anything that is not a Spec is less than X. Spec's are compared with self using comparison of the underlying rings.
sage: Spec(QQ) == Spec(QQ) True sage: Spec(QQ) == Spec(ZZ) False sage: Spec(QQ) == 5 False sage: Spec(GF(5)) < Spec(GF(7)) True sage: Spec(GF(7)) < Spec(GF(5)) False