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# BRanks -- ranks of the modules B_i(d) in a matrixFactorization

## Synopsis

• Usage:
br = BRanks MF
• Inputs:
• MF, a list, output of a matrixFactorization computation
• Outputs:
• br, a list, list of pairs {rank B_1(d), rank B_0(d)}

## Description

 i1 : c = 2 o1 = 2 i2 : S = ZZ/32003[x_0..x_(c-1),a_(0,0)..a_(c-1,c-1)]; i3 : A = genericMatrix(S,a_(0,0),c,c); 2 2 o3 : Matrix S <-- S i4 : f = matrix{{x_0..x_(c-1)}}*map(S^{c:-1},S^{c:-2},A) o4 = | x_0a_(0,0)+x_1a_(0,1) x_0a_(1,0)+x_1a_(1,1) | 1 2 o4 : Matrix S <-- S i5 : R = S/ideal f; i6 : kR = R^1/ideal(x_0..x_(c-1)) o6 = cokernel | x_0 x_1 | 1 o6 : R-module, quotient of R i7 : MF = matrixFactorization(f,highSyzygy kR) o7 = {{2} | a_(0,0) -a_(0,1) a_(1,0) -a_(1,1) |, {3} | x_0 a_(0,1) 0 {2} | x_1 x_0 0 0 | {3} | -x_1 a_(0,0) 0 {2} | 0 0 x_1 x_0 | {3} | 0 0 x_0 {3} | 0 0 -x_1 ------------------------------------------------------------------------ a_(1,1) 0 |, {2} | 0 0 1 |} a_(1,0) 0 | {2} | 0 1 0 | a_(0,1) a_(1,1) | {2} | 1 0 0 | a_(0,0) a_(1,0) | o7 : List i8 : netList BRanks MF +-+-+ o8 = |2|2| +-+-+ |1|2| +-+-+ i9 : netList dMaps MF +-----------------------------------------+ o9 = |{2} | a_(0,0) -a_(0,1) | | |{2} | x_1 x_0 | | +-----------------------------------------+ |{2} | a_(0,0) -a_(0,1) a_(1,0) -a_(1,1) || |{2} | x_1 x_0 0 0 || |{2} | 0 0 x_1 x_0 || +-----------------------------------------+ i10 : netList bMaps MF +------------------------+ o10 = |{2} | a_(0,0) -a_(0,1) || |{2} | x_1 x_0 || +------------------------+ |{2} | x_1 x_0 | | +------------------------+ i11 : netList psiMaps MF +------------------------+ o11 = |{2} | a_(1,0) -a_(1,1) || |{2} | 0 0 || +------------------------+

• matrixFactorization -- Maps in a higher codimension matrix factorization
• bMaps -- list the maps d_p:B_1(p)-->B_0(p) in a matrixFactorization
• dMaps -- list the maps d(p):A_1(p)--> A_0(p) in a matrixFactorization
• psiMaps -- list the maps psi(p): B_1(p) --> A_0(p-1) in a matrixFactorization
• hMaps -- list the maps h(p): A_0(p)--> A_1(p) in a matrixFactorization

• BRanks(List)

## For the programmer

The object BRanks is .