quotient(...,Strategy=>...)

Description

Suppose that I is the image of a free module FI in a quotient module G, and J is the image of the free module FJ in G.

Available strategies for the computation can be listed using the function hooks:

i1 : hooks methods(quotient, Ideal, Ideal)

o1 = {0 => (quotient, Ideal, Ideal, Strategy => Quotient)}
     {1 => (quotient, Ideal, Ideal, Strategy => Iterate) }
     {2 => (quotient, Ideal, Ideal, Strategy => Monomial)}

o1 : NumberedVerticalList

The strategy Quotient computes the first components of the syzygies of the map $R\oplus(FJ^\vee\otimes FI) \to FJ^\vee \otimes G$. The Macaulay2 code for each strategy can be viewed using the function code:

i2 : code(quotient, Ideal, Ideal, Strategy => Quotient)

o2 = -- code for method: quotient(Ideal,Ideal)
     /usr/share/Macaulay2/Saturation.m2:212:30-223:23: --source code:
         Quotient => (opts, I, J) -> (
             R := ring I;
             mR := transpose generators J ** (R / I);
             -- if J is a single element, this is the same as
             -- computing syz gb(matrix{{f}} | generators I, ...)
             g := syz gb(mR, opts,
                 Strategy   => LongPolynomial,
                 Syzygies   => true,
                 SyzygyRows => 1);
             -- The degrees of g are not correct, so we fix that here:
             -- g = map(R^1, null, g);
             lift(ideal g, R)),

If Strategy => Iterate then quotient first computes the quotient I1 by the first generator of J. It then checks whether this quotient already annihilates the second generator of J mod I. If so, it goes on to the third generator; else it intersects I1 with the quotient of I by the second generator to produce a new I1. It then iterates this process, working through the generators one at a time.

To use Strategy=>Linear the argument J must be a principal ideal, generated by a linear form. A change of variables is made so that this linear form becomes the last variable. Then a reverse lex Gröbner basis is used, and the quotient of the initial ideal by the last variable is computed combinatorially. This set of monomial is then lifted back to a set of generators for the quotient.

The following examples show timings for the different strategies. Strategy => Iterate is sometimes faster for ideals with a small number of generators:

i3 : n = 6

o3 = 6

i4 : S = ZZ/101[vars(0..n-1)];

i5 : I = monomialCurveIdeal(S, 1..n-1);

o5 : Ideal of S

i6 : J = ideal(map(S^1, S^n, (p, q) -> S_q^5));

o6 : Ideal of S

i7 : time quotient(I^3, J^2, Strategy => Iterate);
     -- used 0.607105 seconds

o7 : Ideal of S

i8 : time quotient(I^3, J^2, Strategy => Quotient);
     -- used 1.01358 seconds

o8 : Ideal of S

Strategy => Quotient is faster in other cases:

i9 : S = ZZ/101[vars(0..4)];

i10 : I = ideal vars S;

o10 : Ideal of S

i11 : time quotient(I^5, I^3, Strategy => Iterate);
     -- used 0.034072 seconds

o11 : Ideal of S

i12 : time quotient(I^5, I^3, Strategy => Quotient);
     -- used 0.0137244 seconds

o12 : Ideal of S

Further information

Default value: null
Function: quotient -- quotient or division
Option key: Strategy -- an optional argument

References

For further information see for example Exercise 15.41 in Eisenbud's Commutative Algebra with a View Towards Algebraic Geometry.

Functions with optional argument named `Strategy` :

"addHook(...,Strategy=>...)" -- see addHook -- add a hook function to an object for later processing
"annihilator(...,Strategy=>...)" -- see annihilator -- the annihilator ideal
"associatedPrimes(...,Strategy=>...)" -- see associatedPrimes -- find associated primes
"basis(...,Strategy=>...)" -- see basis -- basis or generating set of all or part of a ring, ideal or module
"mingens(...,Strategy=>...)" -- see Complement -- a Strategy option value
"trim(...,Strategy=>...)" -- see Complement -- a Strategy option value
determinant(...,Strategy=>...) -- choose between Bareiss and Cofactor algorithms
"dual(MonomialIdeal,List,Strategy=>...)" -- see dual(MonomialIdeal,Strategy=>...)
"dual(MonomialIdeal,RingElement,Strategy=>...)" -- see dual(MonomialIdeal,Strategy=>...)
dual(MonomialIdeal,Strategy=>...)
exteriorPower(...,Strategy=>...) -- choose between Bareiss and Cofactor algorithms
"gb(...,Strategy=>...)" -- see gb -- compute a Gröbner basis
gcdLLL(...,Strategy=>...) (missing documentation)
"GF(...,Strategy=>...)" -- see GF -- make a finite field
"groebnerBasis(...,Strategy=>...)" -- see groebnerBasis -- Gröbner basis, as a matrix
hermite(...,Strategy=>...) (missing documentation)
"hooks(...,Strategy=>...)" -- see hooks -- list hooks attached to a key
"idealizer(...,Strategy=>...)" -- see idealizer -- compute Hom(I,I) as a quotient ring
integralClosure(...,Strategy=>...) -- control the algorithm used
"intersect(Ideal,Ideal,Strategy=>...)" -- see intersect(Ideal,Ideal) -- compute an intersection of a sequence of ideals or modules
"intersect(Module,Module,Strategy=>...)" -- see intersect(Ideal,Ideal) -- compute an intersection of a sequence of ideals or modules
"intersectInP(...,Strategy=>...)" -- see intersectInP(...,BasisElementLimit=>...) -- Option for intersectInP
"isPrimary(...,Strategy=>...)" -- see isPrimary -- determine whether a submodule is primary
"isPrime(Ideal,Strategy=>...)" -- see isPrime(Ideal) -- whether an ideal is prime
LLL(...,Strategy=>...) -- choose among different algorithms
"localize(...,Strategy=>...)" -- see localize -- localize an ideal at a prime ideal
"match(...,Strategy=>...)" -- see match -- regular expression matching
"decompose(Ideal,Strategy=>...)" -- see minimalPrimes -- minimal primes of an ideal
"minimalPrimes(...,Strategy=>...)" -- see minimalPrimes -- minimal primes of an ideal
minors(...,Strategy=>...) -- choose between Bareiss and Cofactor algorithms
"primaryComponent(...,Strategy=>...)" -- see primaryComponent -- find a primary component corresponding to an associated prime
pushForward(...,Strategy=>...) (missing documentation)
quotient(...,Strategy=>...)
"radical(...,Strategy=>...)" -- see radical -- the radical of an ideal
"radicalContainment(...,Strategy=>...)" -- see radicalContainment -- whether an element is contained in the radical of an ideal
"analyticSpread(...,Strategy=>...)" -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
"distinguished(...,Strategy=>...)" -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
"isLinearType(...,Strategy=>...)" -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
"isReduction(...,Strategy=>...)" -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
"minimalReduction(...,Strategy=>...)" -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
"multiplicity(...,Strategy=>...)" -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
"normalCone(...,Strategy=>...)" -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
"reesAlgebra(...,Strategy=>...)" -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
"specialFiber(...,Strategy=>...)" -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
"specialFiberIdeal(...,Strategy=>...)" -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
"regSeqInIdeal(...,Strategy=>...)" -- see regSeqInIdeal -- a regular sequence contained in an ideal
resolution(...,Strategy=>...)
saturate(...,Strategy=>...)
"primaryDecomposition(...,Strategy=>...)" -- see strategies for computing primary decomposition
"syz(...,Strategy=>...)" -- see syz(Matrix) -- compute the syzygy matrix
"tangentCone(...,Strategy=>...)" -- see tangentCone(Ideal)

quotient(...,Strategy=>...)

Description

Further information

References

Functions with optional argument named Strategy :

Functions with optional argument named `Strategy` :