If Strategy => IdealStrategy, then dualize assume the module is isomorphic to an ideal, embeds it as an ideal, and computes the dual by forming a colon. ModuleStrategy simply computes the Hom. The default Strategy for modules is ModuleStrategy, and the default Strategy for ideals is IdealStrategy. This is because there is overhead using the opposite strategy (involving embedding modules as ideals). Frequently IdealStrategy is faster, but not always. Consider first a D4 singularity in characteristic 2.
i1 : R = ZZ/2[x,y,z]/ideal(z^2-x*y*z-x^2*y-x*y^2); |
i2 : m = ideal(x,y,z); o2 : Ideal of R |
i3 : J = m^9; o3 : Ideal of R |
i4 : M = J*R^1; |
i5 : time dualize(J, Strategy=>IdealStrategy); -- used 0.0793123 seconds o5 : Ideal of R |
i6 : time dualize(J, Strategy=>ModuleStrategy); -- used 0.94745 seconds o6 : Ideal of R |
i7 : time dualize(M, Strategy=>IdealStrategy); -- used 1.19006 seconds |
i8 : time dualize(M, Strategy=>ModuleStrategy); -- used 0.00274742 seconds |
i9 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy); -- used 0.00193363 seconds o9 : Ideal of R |
For monomial ideals in toric rings, frequently ModuleStrategy appears faster.
i10 : R = ZZ/7[x,y,u,v]/ideal(x*y-u*v); |
i11 : I = ideal(x,u); o11 : Ideal of R |
i12 : J = I^15; o12 : Ideal of R |
i13 : time dualize(J, Strategy=>IdealStrategy); -- used 0.0607565 seconds o13 : Ideal of R |
i14 : time dualize(J, Strategy=>ModuleStrategy); -- used 0.00528944 seconds o14 : Ideal of R |