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finiteBettiNumbers -- betti numbers of finite resolution computed from a matrix factorization

Synopsis

Description

Uses the ranks of the B matrices in a matrix factorization for a module M over S/(f_1,..,f_c) to compute the betti numbers of the minimal resolution of M over S, which is the sum of the Koszul complexes K(f_1..f_{j-1}) tensored with B(j)

i1 : setRandomSeed 0

o1 = 0
 
i2 : kk = ZZ/101

o2 = kk

o2 : QuotientRing
 
i3 : S = kk[a,b,u,v]

o3 = S

o3 : PolynomialRing
 
i4 : ff = matrix"au,bv"

o4 = | au bv |

             1      2
o4 : Matrix S  <-- S
 
i5 : R = S/ideal ff

o5 = R

o5 : QuotientRing
 
i6 : M0 = R^1/ideal"a,b"

o6 = cokernel | a b |

                            1
o6 : R-module, quotient of R
 
i7 : F = res(M0, LengthLimit =>3)

      1      2      3      4
o7 = R  <-- R  <-- R  <-- R
                           
     0      1      2      3

o7 : ChainComplex
 
i8 : M = coker F.dd_3;
 
i9 : MF = matrixFactorization(ff,M);
 
i10 : betti res pushForward(map(R,S),M)

             0 1 2
o10 = total: 3 5 2
          2: 3 4 .
          3: . 1 2

o10 : BettiTally
 
i11 : finiteBettiNumbers MF

o11 = {3, 5, 2}

o11 : List
 
i12 : infiniteBettiNumbers(MF,5)

o12 = {3, 4, 5, 6, 7, 8}

o12 : List
 
i13 : betti res (M, LengthLimit => 5)

             0 1 2 3 4 5
o13 = total: 3 4 5 6 7 8
          2: 3 4 5 6 7 8

o13 : BettiTally

See also

Ways to use finiteBettiNumbers :

For the programmer

The object finiteBettiNumbers is a method function.