There are two strategies for computing a reflexification (at least if the module embeds as an ideal).
IdealStrategy. In the case that R is a domain, and our module is isomorphic to an ideal I, then one can compute the reflexification by computing colons.
ModuleStrategy. This computes the reflexification simply by computing Hom twice.
ModuleStrategy is the default strategy for modules, IdealStrategy is the default Strategy for Ideals. In our experience, IdealStrategy is faster on average. Note that calling ModuleStrategy for Ideals or IdealStrategy for modules creates overhead which can slow things down substantially (since we must embed various modules as ideals).
i1 : R = ZZ/13[x,y,z]/ideal(x^3 + y^3-z^11*x*y); |
i2 : I = ideal(x-4*y, z); o2 : Ideal of R |
i3 : J = I^20; o3 : Ideal of R |
i4 : M = J*R^1; |
i5 : J1 = time reflexify( J, Strategy=>IdealStrategy ) -- used 0.131455 seconds 2 2 9 9 11 o5 = ideal (x + 5x*y + 3y , x*z - 4y*z , z + x - 4y) o5 : Ideal of R |
i6 : J2 = time reflexify( J, Strategy=>ModuleStrategy ) -- used 11.9387 seconds 2 2 9 9 11 o6 = ideal (x + 5x*y + 3y , x*z - 4y*z , - 4z - 4x + 3y) o6 : Ideal of R |
i7 : J1 == J2 o7 = true |
i8 : time reflexify( M, Strategy=>IdealStrategy ); -- used 11.0221 seconds |
i9 : time reflexify( M, Strategy=>ModuleStrategy ); -- used 0.910936 seconds |
However, sometimes ModuleStrategy is faster, especially for Monomial ideals.
i10 : R = QQ[x,y,u,v]/ideal(x*y-u*v); |
i11 : I = ideal(x,u); o11 : Ideal of R |
i12 : J = I^20; o12 : Ideal of R |
i13 : M = I^20*R^1; |
i14 : time reflexify( J, Strategy=>IdealStrategy ) -- used 0.113502 seconds 20 19 2 18 3 17 4 16 5 15 6 14 7 13 8 12 o14 = ideal (u , x*u , x u , x u , x u , x u , x u , x u , x u , ----------------------------------------------------------------------- 9 11 10 10 11 9 12 8 13 7 14 6 15 5 16 4 17 3 18 2 x u , x u , x u , x u , x u , x u , x u , x u , x u , x u , ----------------------------------------------------------------------- 19 20 x u, x ) o14 : Ideal of R |
i15 : time reflexify( J, Strategy=>ModuleStrategy ) -- used 0.0301788 seconds 20 19 2 18 3 17 4 16 5 15 6 14 7 13 8 12 o15 = ideal (u , x*u , x u , x u , x u , x u , x u , x u , x u , ----------------------------------------------------------------------- 9 11 10 10 11 9 12 8 13 7 14 6 15 5 16 4 17 3 18 2 x u , x u , x u , x u , x u , x u , x u , x u , x u , x u , ----------------------------------------------------------------------- 19 20 x u, x ) o15 : Ideal of R |
i16 : time reflexify( M, Strategy=>IdealStrategy ); -- used 0.29097 seconds |
i17 : time reflexify( M, Strategy=>ModuleStrategy ); -- used 0.0126362 seconds |