We compute the equation and nonminimal resolution F of the carpet of type (a,b) where a ≥b over a larger finite prime field, lift the complex to the integers, which is possible since the coefficients are small. Finally we study the nonminimal strands over ZZ by computing the Smith normal form. The resulting data allow us to compute the Betti tables for arbitrary primes.
i1 : a=5,b=5 o1 = (5, 5) o1 : Sequence |
i2 : h=carpetBettiTables(a,b) -- 0.00151242 seconds elapsed -- 0.00547527 seconds elapsed -- 0.0236456 seconds elapsed -- 0.0103217 seconds elapsed -- 0.00301096 seconds elapsed 0 1 2 3 4 5 6 7 8 9 o2 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1} 0: 1 . . . . . . . . . 1: . 36 160 315 288 . . . . . 2: . . . . . 288 315 160 36 . 3: . . . . . . . . . 1 0 1 2 3 4 5 6 7 8 9 2 => total: 1 36 167 370 476 476 370 167 36 1 0: 1 . . . . . . . . . 1: . 36 160 322 336 140 48 7 . . 2: . . 7 48 140 336 322 160 36 . 3: . . . . . . . . . 1 0 1 2 3 4 5 6 7 8 9 3 => total: 1 36 160 315 302 302 315 160 36 1 0: 1 . . . . . . . . . 1: . 36 160 315 288 14 . . . . 2: . . . . 14 288 315 160 36 . 3: . . . . . . . . . 1 o2 : HashTable |
i3 : T= carpetBettiTable(h,3) 0 1 2 3 4 5 6 7 8 9 o3 = total: 1 36 160 315 302 302 315 160 36 1 0: 1 . . . . . . . . . 1: . 36 160 315 288 14 . . . . 2: . . . . 14 288 315 160 36 . 3: . . . . . . . . . 1 o3 : BettiTally |
i4 : J=canonicalCarpet(a+b+1,b,Characteristic=>3); ZZ o4 : Ideal of --[x , x , x , x , x , x , y , y , y , y , y , y ] 3 0 1 2 3 4 5 0 1 2 3 4 5 |
i5 : elapsedTime T'=minimalBetti J -- 0.250437 seconds elapsed 0 1 2 3 4 5 6 7 8 9 o5 = total: 1 36 160 315 302 302 315 160 36 1 0: 1 . . . . . . . . . 1: . 36 160 315 288 14 . . . . 2: . . . . 14 288 315 160 36 . 3: . . . . . . . . . 1 o5 : BettiTally |
i6 : T-T' 0 1 2 3 4 5 6 7 8 9 o6 = total: . . . . . . . . . . 1: . . . . . . . . . . 2: . . . . . . . . . . 3: . . . . . . . . . . o6 : BettiTally |
i7 : elapsedTime h=carpetBettiTables(6,6); -- 0.00387262 seconds elapsed -- 0.0191576 seconds elapsed -- 0.112225 seconds elapsed -- 1.18526 seconds elapsed -- 0.477561 seconds elapsed -- 0.0415347 seconds elapsed -- 0.00536286 seconds elapsed -- 8.52788 seconds elapsed |
i8 : keys h o8 = {0, 2, 3, 5} o8 : List |
i9 : carpetBettiTable(h,7) 0 1 2 3 4 5 6 7 8 9 10 11 o9 = total: 1 55 320 891 1408 1155 1155 1408 891 320 55 1 0: 1 . . . . . . . . . . . 1: . 55 320 891 1408 1155 . . . . . . 2: . . . . . . 1155 1408 891 320 55 . 3: . . . . . . . . . . . 1 o9 : BettiTally |
i10 : carpetBettiTable(h,5) 0 1 2 3 4 5 6 7 8 9 10 11 o10 = total: 1 55 320 891 1408 1275 1275 1408 891 320 55 1 0: 1 . . . . . . . . . . . 1: . 55 320 891 1408 1155 120 . . . . . 2: . . . . . 120 1155 1408 891 320 55 . 3: . . . . . . . . . . . 1 o10 : BettiTally |