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 0.0598019 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 7.76844 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 9.69243 seconds |
i9 : time reflexify( M, Strategy=>ModuleStrategy ); -- used 0.435874 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 0.0724363 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 0.0161962 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 0.189106 seconds |
i17 : time reflexify( M, Strategy=>ModuleStrategy ); -- used 0.00642223 seconds |