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

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.m2:224:30-235:23: --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 0.278934 seconds

o7 : Ideal of S
i8 : time quotient(I^3, J^2, Strategy => Quotient);
     -- used 0.502808 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 0.014117 seconds

o11 : Ideal of S
i12 : time quotient(I^5, I^3, Strategy => Quotient);
     -- used 0.00581886 seconds

o12 : Ideal of S

Further information

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 :