The Spec functor#
AUTHORS:
William Stein (2006): initial implementation
Peter Bruin (2014): rewrite Spec as a functor
- sage.schemes.generic.spec.Spec(R, S=None)[source]#
Apply the Spec functor to \(R\).
INPUT:
R
– either a commutative ring or a ring homomorphismS
– a commutative ring (optional), the base ring
OUTPUT:
AffineScheme
– the affine scheme \(\mathrm{Spec}(R)\)
EXAMPLES:
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 # needs sage.rings.finite_rings Spectrum of Multivariate Polynomial Ring in x0, x1, x2 over Finite Field in a of size 7^2 sage: TestSuite(X).run() # needs sage.rings.finite_rings
>>> from sage.all import * >>> Spec(QQ) Spectrum of Rational Field >>> Spec(PolynomialRing(QQ, 'x')) Spectrum of Univariate Polynomial Ring in x over Rational Field >>> Spec(PolynomialRing(QQ, 'x', Integer(3))) Spectrum of Multivariate Polynomial Ring in x0, x1, x2 over Rational Field >>> X = Spec(PolynomialRing(GF(Integer(49),'a'), Integer(3), 'x')); X # needs sage.rings.finite_rings Spectrum of Multivariate Polynomial Ring in x0, x1, x2 over Finite Field in a of size 7^2 >>> TestSuite(X).run() # needs sage.rings.finite_rings
Applying
Spec
twice to the same ring gives identical output (see Issue #17008):sage: A = Spec(ZZ); B = Spec(ZZ) sage: A is B True
>>> from sage.all import * >>> A = Spec(ZZ); B = Spec(ZZ) >>> A is B True
A
TypeError
is raised if the input is not a commutative ring:sage: Spec(5) Traceback (most recent call last): ... TypeError: x (=5) is not in Category of commutative rings sage: Spec(FreeAlgebra(QQ, 2, 'x')) # needs sage.combinat sage.modules Traceback (most recent call last): ... TypeError: x (=Free Algebra on 2 generators (x0, x1) over Rational Field) is not in Category of commutative rings
>>> from sage.all import * >>> Spec(Integer(5)) Traceback (most recent call last): ... TypeError: x (=5) is not in Category of commutative rings >>> Spec(FreeAlgebra(QQ, Integer(2), 'x')) # needs sage.combinat sage.modules Traceback (most recent call last): ... TypeError: x (=Free Algebra on 2 generators (x0, x1) over Rational Field) is not in Category of commutative rings
- class sage.schemes.generic.spec.SpecFunctor(base_ring=None)[source]#
Bases:
Functor
,UniqueRepresentation
The Spec functor.