# 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.m2:227:30-239:23: --source code: Quotient => (opts, I, J) -> ( R := ring I; -- FIXME: this line computes a gb for 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 0.764164 seconds o7 : Ideal of S i8 : time quotient(I^3, J^2, Strategy => Quotient); -- used 1.22538 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 0.0451759 seconds o11 : Ideal of S i12 : time quotient(I^5, I^3, Strategy => Quotient); -- used 0.0184546 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.