Schemes¶
- class sage.categories.schemes.AbelianVarieties(base)[source]¶
- Bases: - Schemes_over_base- The category of abelian varieties over a given field. - EXAMPLES: - sage: AbelianVarieties(QQ) Category of abelian varieties over Rational Field sage: AbelianVarieties(ZZ) Traceback (most recent call last): ... ValueError: category of abelian varieties is only defined over fields - >>> from sage.all import * >>> AbelianVarieties(QQ) Category of abelian varieties over Rational Field >>> AbelianVarieties(ZZ) Traceback (most recent call last): ... ValueError: category of abelian varieties is only defined over fields - class Homsets(category, *args)[source]¶
- Bases: - HomsetsCategory- Overloaded - Homsetsclass to register the homset as an additive abelian group.- EXAMPLES: - sage: AbelianVarieties(QQ).Homsets().is_subcategory(CommutativeAdditiveGroups()) True - >>> from sage.all import * >>> AbelianVarieties(QQ).Homsets().is_subcategory(CommutativeAdditiveGroups()) True - class Endset(base_category)[source]¶
- Bases: - CategoryWithAxiom- Overloaded - Endsetclass to register the endset as a ring.- sage: AbelianVarieties(QQ).Endsets().is_subcategory(Rings()) True 
 - extra_super_categories()[source]¶
- Register the homset as an additive abelian group. - EXAMPLES: - sage: Hom(EllipticCurve(j=1), EllipticCurve(j=2)) in CommutativeAdditiveGroups() True - >>> from sage.all import * >>> Hom(EllipticCurve(j=Integer(1)), EllipticCurve(j=Integer(2))) in CommutativeAdditiveGroups() True 
 
 - base_scheme()[source]¶
- EXAMPLES: - sage: Schemes(Spec(ZZ)).base_scheme() Spectrum of Integer Ring - >>> from sage.all import * >>> Schemes(Spec(ZZ)).base_scheme() Spectrum of Integer Ring 
 - super_categories()[source]¶
- EXAMPLES: - sage: AbelianVarieties(QQ).super_categories() [Category of schemes over Rational Field, Category of commutative additive groups] - >>> from sage.all import * >>> AbelianVarieties(QQ).super_categories() [Category of schemes over Rational Field, Category of commutative additive groups] 
 
- class sage.categories.schemes.Jacobians(base)[source]¶
- Bases: - Schemes_over_base- The category of Jacobians attached to curves or function fields. - EXAMPLES: - sage: Jacobians(QQ) Category of Jacobians over Rational Field - >>> from sage.all import * >>> Jacobians(QQ) Category of Jacobians over Rational Field - class ParentMethods[source]¶
- Bases: - object- base_curve()[source]¶
- Return the curve to which this Jacobian is attached. - EXAMPLES: - sage: # needs sage.rings.function_field sage: K.<x> = FunctionField(GF(2)) sage: J = K.jacobian() sage: J.base_curve() Rational function field in x over Finite Field of size 2 - >>> from sage.all import * >>> # needs sage.rings.function_field >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1) >>> J = K.jacobian() >>> J.base_curve() Rational function field in x over Finite Field of size 2 
 
 - base_scheme()[source]¶
- Return the base scheme of this Jacobians category. - EXAMPLES: - sage: Jacobians(QQ).base_scheme() Spectrum of Rational Field - >>> from sage.all import * >>> Jacobians(QQ).base_scheme() Spectrum of Rational Field 
 - super_categories()[source]¶
- Return the super categories of this Jacobians category. - EXAMPLES: - sage: Jacobians(QQ).super_categories() [Category of abelian varieties over Rational Field] - >>> from sage.all import * >>> Jacobians(QQ).super_categories() [Category of abelian varieties over Rational Field] 
 
- class sage.categories.schemes.Schemes[source]¶
- Bases: - Category- The category of all schemes. - EXAMPLES: - sage: Schemes() Category of schemes - >>> from sage.all import * >>> Schemes() Category of schemes - Schemescan also be used to construct the category of schemes over a given base:- sage: Schemes(Spec(ZZ)) Category of schemes over Integer Ring sage: Schemes(ZZ) Category of schemes over Integer Ring - >>> from sage.all import * >>> Schemes(Spec(ZZ)) Category of schemes over Integer Ring >>> Schemes(ZZ) Category of schemes over Integer Ring - Todo - Make - Schemes()a singleton category (and remove- Schemesfrom the workaround in- category_types.Category_over_base._test_category_over_bases()).- This is currently incompatible with the dispatching below. 
- class sage.categories.schemes.Schemes_over_base(base, name=None)[source]¶
- Bases: - Category_over_base- The category of schemes over a given base scheme. - EXAMPLES: - sage: Schemes(Spec(ZZ)) Category of schemes over Integer Ring - >>> from sage.all import * >>> Schemes(Spec(ZZ)) Category of schemes over Integer Ring