Coverage for local/lib/python2.7/site-packages/sage/schemes/generic/spec.py : 51%

""" 

The Spec functor 

AUTHORS: 

- William Stein (2006): initial implementation 

- Peter Bruin (2014): rewrite Spec as a functor 

""" 

#******************************************************************************* 

# Copyright (C) 2006 William Stein 

# Distributed under the terms of the GNU General Public License (GPL) 

# http://www.gnu.org/licenses/ 

#******************************************************************************* 

from sage.categories.functor import Functor 

from sage.rings.integer_ring import ZZ 

from sage.schemes.generic.scheme import AffineScheme, is_AffineScheme 

from sage.structure.unique_representation import UniqueRepresentation 

def Spec(R, S=None): 

r""" 

Apply the Spec functor to `R`. 

INPUT: 

- ``R`` -- either a commutative ring or a ring homomorphism 

- ``S`` -- 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 

Spectrum of Multivariate Polynomial Ring in x0, x1, x2 over Finite Field in a of size 7^2 

sage: TestSuite(X).run() 

Applying ``Spec`` twice to the same ring gives identical output 

(see :trac:`17008`):: 

sage: A = Spec(ZZ); B = Spec(ZZ) 

sage: 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')) 

Traceback (most recent call last): 

... 

TypeError: x (=Free Algebra on 2 generators (x0, x1) over Rational Field) is not in Category of commutative rings 

TESTS:: 

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 

sage: Spec(QQ,QQ).base_scheme() 

Spectrum of Rational Field 

sage: Spec(RDF,QQ).base_scheme() 

Spectrum of Rational Field 

""" 

return SpecFunctor(S)(R) 

class SpecFunctor(Functor, UniqueRepresentation): 

""" 

The Spec functor. 

""" 

def __init__(self, base_ring=None): 

""" 

EXAMPLES:: 

sage: from sage.schemes.generic.spec import SpecFunctor 

sage: SpecFunctor() 

Spec functor from Category of commutative rings to Category of schemes 

sage: SpecFunctor(QQ) 

Spec functor from Category of commutative rings to 

Category of schemes over Rational Field 

""" 

from sage.categories.all import CommutativeAlgebras, CommutativeRings, Schemes 

if base_ring is None: 

domain = CommutativeRings() 

codomain = Schemes() 

elif base_ring in CommutativeRings(): 

# We would like to use CommutativeAlgebras(base_ring) as 

# the domain; we use CommutativeRings() instead because 

# currently many algebras are not yet considered to be in 

# CommutativeAlgebras(base_ring) by the category framework. 

domain = CommutativeRings() 

codomain = Schemes(AffineScheme(base_ring)) 

else: 

raise TypeError('base (= {}) must be a commutative ring'.format(base_ring)) 

self._base_ring = base_ring 

super(SpecFunctor, self).__init__(domain, codomain) 

def _repr_(self): 

""" 

Return a string representation of ``self``. 

EXAMPLES:: 

sage: from sage.schemes.generic.spec import SpecFunctor 

sage: SpecFunctor(QQ) 

Spec functor from Category of commutative rings to 

Category of schemes over Rational Field 

""" 

return 'Spec functor from {} to {}'.format(self.domain(), self.codomain()) 

def _latex_(self): 

r""" 

Return a LaTeX representation of ``self``. 

EXAMPLES:: 

sage: from sage.schemes.generic.spec import SpecFunctor 

sage: latex(SpecFunctor()) 

\mathrm{Spec}\colon \mathbf{CommutativeRings} \longrightarrow \mathbf{Schemes} 

""" 

return r'\mathrm{{Spec}}\colon {} \longrightarrow {}'.format( 

self.domain()._latex_(), self.codomain()._latex_()) 

def _apply_functor(self, A): 

""" 

Apply the Spec functor to the commutative ring ``A``. 

EXAMPLES:: 

sage: from sage.schemes.generic.spec import SpecFunctor 

sage: F = SpecFunctor() 

sage: F(RR) # indirect doctest 

Spectrum of Real Field with 53 bits of precision 

""" 

# The second argument of AffineScheme defaults to None. 

# However, AffineScheme has unique representation, so there is 

# a difference between calling it with or without explicitly 

# giving this argument. 

if self._base_ring is None: 

return AffineScheme(A) 

return AffineScheme(A, self._base_ring) 

def _apply_functor_to_morphism(self, f): 

""" 

Apply the Spec functor to the ring homomorphism ``f``. 

EXAMPLES:: 

sage: from sage.schemes.generic.spec import SpecFunctor 

sage: F = SpecFunctor(GF(7)) 

sage: A.<x, y> = GF(7)[] 

sage: B.<t> = GF(7)[] 

sage: f = A.hom((t^2, t^3)) 

sage: Spec(f) # indirect doctest 

Affine Scheme morphism: 

From: Spectrum of Univariate Polynomial Ring in t over Finite Field of size 7 

To: Spectrum of Multivariate Polynomial Ring in x, y over Finite Field of size 7 

Defn: Ring morphism: 

From: Multivariate Polynomial Ring in x, y over Finite Field of size 7 

To: Univariate Polynomial Ring in t over Finite Field of size 7 

Defn: x |--> t^2 

y |--> t^3 

""" 

A = f.domain() 

B = f.codomain() 

return self(B).hom(f, self(A)) 

SpecZ = Spec(ZZ) 

# Compatibility with older versions of this module 

from sage.misc.superseded import deprecated_function_alias 

is_Spec = deprecated_function_alias(16158, is_AffineScheme) 

from sage.structure.sage_object import register_unpickle_override 

register_unpickle_override('sage.schemes.generic.spec', 'Spec', AffineScheme)