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InvolutiveBases :: janetResolution

janetResolution -- construct a free resolution for a given ideal or module using Janet bases

Synopsis

Description

The computed Janet basis for each homological degree can be extracted with janetBasis.

The sets of multiplicative variables can also be extracted from the Janet basis in each homological degree with multVar.

Note that janetResolution can be combined with resolution: when providing the option 'Strategy => Involutive' to resolution, janetResolution constructs the resolution.

i1 : R = QQ[x,y,z];
i2 : M = matrix {{x,y,z}};

             1       3
o2 : Matrix R  <--- R
i3 : C = janetResolution M

      1      3      3      1
o3 = R  <-- R  <-- R  <-- R  <-- 0
                                  
     0      1      2      3      4

o3 : ChainComplex
i4 : janetBasis(C, 2)

     +------+---------+
o4 = || -y ||{z, y, x}|
     ||  x ||         |
     ||  0 ||         |
     +------+---------+
     || -z ||{z, y, x}|
     ||  0 ||         |
     ||  x ||         |
     +------+---------+
     ||  0 ||{z, y}   |
     || -z ||         |
     ||  y ||         |
     +------+---------+

o4 : InvolutiveBasis
i5 : multVar(C, 2)

o5 = {set {x, y, z}, set {x, y, z}, set {y, z}}

o5 : List
i6 : R = QQ[x,y,z];
i7 : I = ideal(x,y,z);

o7 : Ideal of R
i8 : res(I, Strategy => Involutive)

      1      3      3      1
o8 = R  <-- R  <-- R  <-- R  <-- 0
                                  
     0      1      2      3      4

o8 : ChainComplex

See also

Ways to use janetResolution :