The minimal resolution of a module over a complete intersection has betti numbers that grow as a polynomial of degree at most equal to the codimension-1. The complexity is one more than the degree of this polynomial.
i1 : S = ZZ/101[a,b,c,d]; |
i2 : ff1 = matrix"a3,b3,c3,d3"; 1 4 o2 : Matrix S <--- S |
i3 : ff =ff1*random(source ff1, source ff1); 1 4 o3 : Matrix S <--- S |
i4 : R = S/ideal ff; |
i5 : M = highSyzygy (R^1/ideal"a2b2"); |
i6 : complexity M o6 = 2 |
i7 : mf = matrixFactorization (ff, M) o7 = {{7} | a -36b 0 0 |, {8} | -35a2 -48b -43a2 0 0 |} {6} | -b2 a2 7b2 0 | {8} | -35b2 -35a 0 0 0 | {7} | 0 0 a -b | {8} | 0 0 -35a2 -48b 0 | {8} | 0 0 -48b2 -48a 33b2 | o7 : List |
i8 : complexity mf o8 = 2 |
i9 : betti res (R^1/ideal"a2b2", LengthLimit=>10) 0 1 2 3 4 5 6 7 8 9 10 o9 = total: 1 1 2 3 4 5 6 7 8 9 10 0: 1 . . . . . . . . . . 1: . . . . . . . . . . . 2: . . . . . . . . . . . 3: . 1 2 1 . . . . . . . 4: . . . 2 4 2 . . . . . 5: . . . . . 3 6 3 . . . 6: . . . . . . . 4 8 4 . 7: . . . . . . . . . 5 10 o9 : BettiTally |