Localization

Localization is an important ring construction tool. Whenever you have to extend a given integral domain such that it contains the inverses of a finite set of elements but should allow non injective homomorphic images this construction will be needed. See the example on Ariki-Koike algebras below for such an application.

EXAMPLES:

sage: # needs sage.modules
sage: LZ = Localization(ZZ, (5,11))
sage: m = matrix(LZ, [[5, 7], [0,11]])
sage: m.parent()
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring localized at (5, 11)
sage: ~m      # parent of inverse is different: see documentation of m.__invert__
[  1/5 -7/55]
[    0  1/11]
sage: _.parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: mi = matrix(LZ, ~m)
sage: mi.parent()
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring localized at (5, 11)
sage: mi == ~m
True

The next example defines the most general ring containing the coefficients of the irreducible representations of the Ariki-Koike algebra corresponding to the three colored permutations on three elements:

sage: R.<u0, u1, u2, q> = ZZ[]
sage: u = [u0, u1, u2]
sage: S = Set(u)
sage: I = S.cartesian_product(S)
sage: add_units = u + [q, q + 1] + [ui - uj for ui, uj in I if ui != uj]
sage: add_units += [q*ui - uj for ui, uj in I if ui != uj]
sage: L = R.localization(tuple(add_units)); L                                       # needs sage.libs.pari
Multivariate Polynomial Ring in u0, u1, u2, q over Integer Ring localized at
 (q, q + 1, u2, u1, u1 - u2, u0, u0 - u2, u0 - u1, u2*q - u1, u2*q - u0,
  u1*q - u2, u1*q - u0, u0*q - u2, u0*q - u1)

Define the representation matrices (of one of the three dimensional irreducible representations):

sage: # needs sage.libs.pari sage.modules
sage: m1 = matrix(L, [[u1, 0, 0], [0, u0, 0], [0, 0, u0]])
sage: m2 = matrix(L, [[(u0*q - u0)/(u0 - u1), (u0*q - u1)/(u0 - u1), 0],
....:                 [(-u1*q + u0)/(u0 - u1), (-u1*q + u1)/(u0 - u1), 0],
....:                 [0, 0, -1]])
sage: m3 = matrix(L, [[-1, 0, 0],
....:                 [0, u0*(1 - q)/(u1*q - u0), q*(u1 - u0)/(u1*q - u0)],
....:                 [0, (u1*q^2 - u0)/(u1*q - u0), (u1*q^ 2 - u1*q)/(u1*q - u0)]])
sage: m1.base_ring() == L
True

Check relations of the Ariki-Koike algebra:

sage: # needs sage.libs.pari sage.modules
sage: m1*m2*m1*m2 == m2*m1*m2*m1
True
sage: m2*m3*m2 == m3*m2*m3
True
sage: m1*m3 == m3*m1
True
sage: m1**3 - (u0+u1+u2)*m1**2 + (u0*u1+u0*u2+u1*u2)*m1 - u0*u1*u2 == 0
True
sage: m2**2 - (q-1)*m2 - q == 0
True
sage: m3**2 - (q-1)*m3 - q == 0
True
sage: ~m1 in m1.parent()
True
sage: ~m2 in m2.parent()
True
sage: ~m3 in m3.parent()
True

Obtain specializations in positive characteristic:

sage: # needs sage.libs.pari sage.modules
sage: Fp = GF(17)
sage: f = L.hom((3,5,7,11), codomain=Fp); f
Ring morphism:
  From: Multivariate Polynomial Ring in u0, u1, u2, q over Integer Ring localized at
        (q, q + 1, u2, u1, u1 - u2, u0, u0 - u2, u0 - u1, u2*q - u1, u2*q - u0,
         u1*q - u2, u1*q - u0, u0*q - u2, u0*q - u1)
  To:   Finite Field of size 17
  Defn: u0 |--> 3
        u1 |--> 5
        u2 |--> 7
        q |--> 11
sage: mFp1 = matrix({k: f(v) for k, v in m1.dict().items()}); mFp1
[5 0 0]
[0 3 0]
[0 0 3]
sage: mFp1.base_ring()
Finite Field of size 17
sage: mFp2 = matrix({k: f(v) for k, v in m2.dict().items()}); mFp2
[ 2  3  0]
[ 9  8  0]
[ 0  0 16]
sage: mFp3 = matrix({k: f(v) for k, v in m3.dict().items()}); mFp3
[16  0  0]
[ 0  4  5]
[ 0  7  6]

Obtain specializations in characteristic 0:

sage: # needs sage.libs.pari
sage: fQ = L.hom((3,5,7,11), codomain=QQ); fQ
Ring morphism:
  From: Multivariate Polynomial Ring in u0, u1, u2, q over Integer Ring
        localized at (q, q + 1, u2, u1, u1 - u2, u0, u0 - u2, u0 - u1,
        u2*q - u1, u2*q - u0, u1*q - u2, u1*q - u0, u0*q - u2, u0*q - u1)
  To:   Rational Field
  Defn: u0 |--> 3
        u1 |--> 5
        u2 |--> 7
        q |--> 11

sage: # needs sage.libs.pari sage.modules sage.rings.finite_rings
sage: mQ1 = matrix({k: fQ(v) for k, v in m1.dict().items()}); mQ1
[5 0 0]
[0 3 0]
[0 0 3]
sage: mQ1.base_ring()
Rational Field
sage: mQ2 = matrix({k: fQ(v) for k, v in m2.dict().items()}); mQ2
[-15 -14   0]
[ 26  25   0]
[  0   0  -1]
sage: mQ3 = matrix({k: fQ(v) for k, v in m3.dict().items()}); mQ3
[    -1      0      0]
[     0 -15/26  11/26]
[     0 301/26 275/26]

sage: # needs sage.libs.pari sage.libs.singular
sage: S.<x, y, z, t> = QQ[]
sage: T = S.quo(x + y + z)
sage: F = T.fraction_field()
sage: fF = L.hom((x, y, z, t), codomain=F); fF
Ring morphism:
  From: Multivariate Polynomial Ring in u0, u1, u2, q over Integer Ring
        localized at (q, q + 1, u2, u1, u1 - u2, u0, u0 - u2, u0 - u1,
        u2*q - u1, u2*q - u0, u1*q - u2, u1*q - u0, u0*q - u2, u0*q - u1)
  To:   Fraction Field of Quotient of Multivariate Polynomial Ring in x, y, z, t
        over Rational Field by the ideal (x + y + z)
  Defn: u0 |--> -ybar - zbar
        u1 |--> ybar
        u2 |--> zbar
        q |--> tbar
sage: mF1 = matrix({k: fF(v) for k, v in m1.dict().items()}); mF1                   # needs sage.modules
[        ybar            0            0]
[           0 -ybar - zbar            0]
[           0            0 -ybar - zbar]
sage: mF1.base_ring() == F                                                          # needs sage.modules
True

AUTHORS:

• Sebastian Oehms 2019-12-09: initial version.

• Sebastian Oehms 2022-03-05: fix some corner cases and add factor() (github issue #33463)

class sage.rings.localization.Localization(base_ring, extra_units, names=None, normalize=True, category=None, warning=True)

Bases: IntegralDomain, UniqueRepresentation

The localization generalizes the construction of the field of fractions of an integral domain to an arbitrary ring. Given a (not necessarily commutative) ring \(R\) and a subset \(S\) of \(R\), there exists a ring \(R[S^{-1}]\) together with the ring homomorphism \(R \longrightarrow R[S^{-1}]\) that “inverts” \(S\); that is, the homomorphism maps elements in \(S\) to unit elements in \(R[S^{-1}]\) and, moreover, any ring homomorphism from \(R\) that “inverts” \(S\) uniquely factors through \(R[S^{-1}]\).

The ring \(R[S^{-1}]\) is called the localization of \(R\) with respect to \(S\). For example, if \(R\) is a commutative ring and \(f\) an element in \(R\), then the localization consists of elements of the form \(r/f, r\in R, n \geq 0\) (to be precise, \(R[f^{-1}] = R[t]/(ft-1)\).

The above text is taken from \(Wikipedia\). The construction here used for this class relies on the construction of the field of fraction and is therefore restricted to integral domains.

Accordingly, this class is inherited from IntegralDomain and can only be used in that context. Furthermore, the base ring should support sage.structure.element.CommutativeRingElement.divides() and the exact division operator \(//\) (sage.structure.element.Element.__floordiv__()) in order to guarantee a successful application.

INPUT:

• base_ring – an instance of Ring allowing the construction of fraction_field() (that is an integral domain)

• extra_units – tuple of elements of base_ring which should be turned into units

• names – passed to IntegralDomain

• normalize – (optional, default: True) passed to IntegralDomain

• category – (optional, default: None) passed to IntegralDomain

• warning – (optional, default: True) to suppress a warning which is thrown if self cannot be represented uniquely

REFERENCES:

• Wikipedia article Ring_(mathematics)#Localization

EXAMPLES:

sage: L = Localization(ZZ, (3,5))
sage: 1/45 in L
True
sage: 1/43 in L
False

sage: Localization(L, (7,11))
Integer Ring localized at (3, 5, 7, 11)
sage: _.is_subring(QQ)
True

sage: L(~7)
Traceback (most recent call last):
...
ValueError: factor 7 of denominator is not a unit

sage: Localization(Zp(7), (3, 5))                                               # needs sage.rings.padics
Traceback (most recent call last):
...
ValueError: all given elements are invertible in
7-adic Ring with capped relative precision 20

sage: # needs sage.libs.pari
sage: R.<x> = ZZ[]
sage: L = R.localization(x**2 + 1)
sage: s = (x+5)/(x**2+1)
sage: s in L
True
sage: t = (x+5)/(x**2+2)
sage: t in L
False
sage: L(t)
Traceback (most recent call last):
...
TypeError: fraction must have unit denominator
sage: L(s) in R
False
sage: y = L(x)
sage: g = L(s)
sage: g.parent()
Univariate Polynomial Ring in x over Integer Ring localized at (x^2 + 1,)
sage: f = (y+5)/(y**2+1); f
(x + 5)/(x^2 + 1)
sage: f == g
True
sage: (y+5)/(y**2+2)
Traceback (most recent call last):
...
ValueError: factor x^2 + 2 of denominator is not a unit

sage: Lau.<u, v> = LaurentPolynomialRing(ZZ)                                    # needs sage.modules
sage: LauL = Lau.localization(u + 1)                                            # needs sage.modules
sage: LauL(~u).parent()                                                         # needs sage.modules
Multivariate Polynomial Ring in u, v over Integer Ring localized at (v, u, u + 1)

More examples will be shown typing sage.rings.localization?

Element

alias of LocalizationElement

characteristic()

Return the characteristic of self.

EXAMPLES:

sage: # needs sage.libs.pari
sage: R.<a> = GF(5)[]
sage: L = R.localization((a**2 - 3, a))
sage: L.characteristic()
5

fraction_field()

Return the fraction field of self.

EXAMPLES:

sage: # needs sage.libs.pari
sage: R.<a> = GF(5)[]
sage: L = Localization(R, (a**2 - 3, a))
sage: L.fraction_field()
Fraction Field of Univariate Polynomial Ring in a over Finite Field of size 5
sage: L.is_subring(_)
True

gen(i)

Return the i-th generator of self which is the i-th generator of the base ring.

EXAMPLES:

sage: R.<x, y> = ZZ[]
sage: R.localization((x**2 + 1, y - 1)).gen(0)                              # needs sage.libs.pari
x

sage: ZZ.localization(2).gen(0)
1

gens()

Return a tuple whose entries are the generators for this object, in order.

EXAMPLES:

sage: R.<x, y> = ZZ[]
sage: Localization(R, (x**2 + 1, y - 1)).gens()                             # needs sage.libs.pari
(x, y)

sage: Localization(ZZ, 2).gens()
(1,)

is_field(proof=True)

Return True if this ring is a field.

INPUT:

• proof – (default: True) Determines what to do in unknown cases

ALGORITHM:

If the parameter proof is set to True, the returned value is correct but the method might throw an error. Otherwise, if it is set to False, the method returns True if it can establish that self is a field and False otherwise.

EXAMPLES:

sage: R = ZZ.localization((2, 3))
sage: R.is_field()
False

krull_dimension()

Return the Krull dimension of this localization.

Since the current implementation just allows integral domains as base ring and localization at a finite set of elements the spectrum of self is open in the irreducible spectrum of its base ring. Therefore, by density we may take the dimension from there.

EXAMPLES:

sage: R = ZZ.localization((2, 3))
sage: R.krull_dimension()
1

ngens()

Return the number of generators of self according to the same method for the base ring.

EXAMPLES:

sage: R.<x, y> = ZZ[]
sage: Localization(R, (x**2 + 1, y - 1)).ngens()                            # needs sage.libs.pari
2

sage: Localization(ZZ, 2).ngens()
1

class sage.rings.localization.LocalizationElement(parent, x)

Bases: IntegralDomainElement

Element class for localizations of integral domains

INPUT:

• parent – instance of Localization

• x – instance of FractionFieldElement whose parent is the fraction

  field of the parent’s base ring

EXAMPLES:

sage: # needs sage.libs.pari
sage: from sage.rings.localization import LocalizationElement
sage: P.<x,y,z> = GF(5)[]
sage: L = P.localization((x, y*z - x))
sage: LocalizationElement(L, 4/(y*z-x)**2)
(-1)/(y^2*z^2 - 2*x*y*z + x^2)
sage: _.parent()
Multivariate Polynomial Ring in x, y, z over Finite Field of size 5
 localized at (x, y*z - x)

denominator()

Return the denominator of self.

EXAMPLES:

sage: L = Localization(ZZ, (3,5))
sage: L(7/15).denominator()
15

factor(proof=None)

Return the factorization of this polynomial.

INPUT:

• proof – (optional) if given it is passed to the corresponding method of the numerator of self

EXAMPLES:

sage: P.<X, Y> = QQ['x, y']
sage: L = P.localization(X - Y)
sage: x, y = L.gens()
sage: p = (x^2 - y^2)/(x-y)^2                                               # needs sage.libs.singular
sage: p.factor()                                                            # needs sage.libs.singular
(1/(x - y)) * (x + y)

inverse_of_unit()

Return the inverse of self.

EXAMPLES:

sage: P.<x,y,z> = ZZ[]
sage: L = Localization(P, x*y*z)
sage: L(x*y*z).inverse_of_unit()                                            # needs sage.libs.singular
1/(x*y*z)
sage: L(z).inverse_of_unit()                                                # needs sage.libs.singular
1/z

is_unit()

Return True if self is a unit.

EXAMPLES:

sage: # needs sage.libs.pari sage.singular
sage: P.<x,y,z> = QQ[]
sage: L = P.localization((x, y*z))
sage: L(y*z).is_unit()
True
sage: L(z).is_unit()
True
sage: L(x*y*z).is_unit()
True

numerator()

Return the numerator of self.

EXAMPLES:

sage: L = ZZ.localization((3,5))
sage: L(7/15).numerator()
7

sage.rings.localization.normalize_extra_units(base_ring, add_units, warning=True)

Function to normalize input data.

The given list will be replaced by a list of the involved prime factors (if possible).

INPUT:

• base_ring – an instance of IntegralDomain

• add_units – list of elements from base ring

• warning – (optional, default: True) to suppress a warning which is thrown if no normalization was possible

OUTPUT:

List of all prime factors of the elements of the given list.

EXAMPLES:

sage: from sage.rings.localization import normalize_extra_units
sage: normalize_extra_units(ZZ, [3, -15, 45, 9, 2, 50])
[2, 3, 5]
sage: P.<x,y,z> = ZZ[]
sage: normalize_extra_units(P,                                                  # needs sage.libs.pari
....:                       [3*x, z*y**2, 2*z, 18*(x*y*z)**2, x*z, 6*x*z, 5])
[2, 3, 5, z, y, x]
sage: P.<x,y,z> = QQ[]
sage: normalize_extra_units(P,                                                  # needs sage.libs.pari
....:                       [3*x, z*y**2, 2*z, 18*(x*y*z)**2, x*z, 6*x*z, 5])
[z, y, x]

sage: # needs sage.libs.singular
sage: R.<x, y> = ZZ[]
sage: Q.<a, b> = R.quo(x**2 - 5)
sage: p = b**2 - 5
sage: p == (b-a)*(b+a)
True
sage: normalize_extra_units(Q, [p])                                             # needs sage.libs.pari
doctest:...: UserWarning: Localization may not be represented uniquely
[b^2 - 5]
sage: normalize_extra_units(Q, [p], warning=False)                              # needs sage.libs.pari
[b^2 - 5]