Given a singly-graded module, this function computes a non-minimal free resolution. If the input is an ideal I ⊂S, it computes a non-minimal resolution of S1/I.
The key benefit of this function is that it allows a much faster method for computing the betti numbers of the minimal free resolution. If that is your only interest (i.e. you don’t need the complex itself), instead use minimalBetti.
i1 : I = Grassmannian(1,6, CoefficientRing => ZZ/101); ZZ o1 : Ideal of ---[p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p ] 101 0,1 0,2 1,2 0,3 1,3 2,3 0,4 1,4 2,4 3,4 0,5 1,5 2,5 3,5 4,5 0,6 1,6 2,6 3,6 4,6 5,6 |
i2 : S = ring I o2 = S o2 : PolynomialRing |
i3 : elapsedTime C = res(I, FastNonminimal => true) -- 1.05492 seconds elapsed 1 35 241 841 1781 2464 2294 1432 576 135 14 o3 = S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- 0 0 1 2 3 4 5 6 7 8 9 10 11 o3 : ChainComplex |
i4 : elapsedTime C1 = res ideal(I_*) -- 1.71321 seconds elapsed 1 35 140 385 819 1080 819 385 140 35 1 o4 = S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- 0 0 1 2 3 4 5 6 7 8 9 10 11 o4 : ChainComplex |
i5 : betti(C, Minimize => true) == betti C1 o5 = true |
For a non-minimal resolution, betti gives the actual Betti numbers, and using the betti(..., Minimize => ...) option gives the ranks in a minimal resolution (which is itself not computed).
i6 : betti C 0 1 2 3 4 5 6 7 8 9 10 o6 = total: 1 35 241 841 1781 2464 2294 1432 576 135 14 0: 1 . . . . . . . . . . 1: . 35 140 290 402 402 293 152 53 11 1 2: . . 101 514 1174 1577 1365 780 287 62 6 3: . . . 37 204 479 621 480 221 56 6 4: . . . . 1 6 15 20 15 6 1 o6 : BettiTally |
i7 : betti(C, Minimize => true) 0 1 2 3 4 5 6 7 8 9 10 o7 = total: 1 35 140 385 819 1080 819 385 140 35 1 0: 1 . . . . . . . . . . 1: . 35 140 189 84 . . . . . . 2: . . . 196 735 1080 735 196 . . . 3: . . . . . . 84 189 140 35 . 4: . . . . . . . . . . 1 o7 : BettiTally |
As mentioned above, if you are just interested in the minimal betti numbers of the ideal or module, then use minimalBetti, as it avoids construction of the non-minimal free resolution.
i8 : minimalBetti I 0 1 2 3 4 5 6 7 8 9 10 o8 = total: 1 35 140 385 819 1080 819 385 140 35 1 0: 1 . . . . . . . . . . 1: . 35 140 189 84 . . . . . . 2: . . . 196 735 1080 735 196 . . . 3: . . . . . . 84 189 140 35 . 4: . . . . . . . . . . 1 o8 : BettiTally |
If the resolution is not large, this function can be slower than the usual function resolution. But for larger examples, if one is only interested in the betti numbers, this function can be hundreds or thousands of times faster.
Released in M2 1.9, still experimental. Only works over finite prime fields. Uses quite alot of memory.
The object FastNonminimal is a symbol.