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CompleteIntersectionResolutions :: makeFiniteResolutionCodim2

makeFiniteResolutionCodim2 -- Maps associated to the finite resolution of a high syzygy module in codim 2

Synopsis

Description

Given a codim 2 matrix factorization, makes all the components of the differential and of the homotopies that are relevant to the finite resolution, as in 4.2.3 of Eisenbud-Peeva "Minimal Free Resolutions and Higher Matrix Factorizations"

i1 : kk=ZZ/101

o1 = kk

o1 : QuotientRing
i2 : S = kk[a,b]

o2 = S

o2 : PolynomialRing
i3 : ff = matrix"a4,b4"

o3 = | a4 b4 |

             1       2
o3 : Matrix S  <--- S
i4 : R = S/ideal ff

o4 = R

o4 : QuotientRing
i5 : N = R^1/ideal"a2, ab, b3"

o5 = cokernel | a2 ab b3 |

                            1
o5 : R-module, quotient of R
i6 : N = coker vars R

o6 = cokernel | a b |

                            1
o6 : R-module, quotient of R
i7 : M = highSyzygy N

o7 = cokernel {2} | 0 -b3 a3 0 |
              {4} | b a   0  0 |
              {4} | 0 0   b  a |

                            3
o7 : R-module, quotient of R
i8 : MS = pushForward(map(R,S),M)

o8 = cokernel {2} | 0 b3 a3 0 0  |
              {4} | b -a 0  0 0  |
              {4} | 0 0  b  a b4 |

                            3
o8 : S-module, quotient of S
i9 : mf = matrixFactorization(ff, M)

o9 = {{4} | a -b 0 0  |, {5} | a3 b 0   0  0  |, {2} | 0 -1 0 |}
      {2} | 0 a3 0 b3 |  {5} | 0  a -b3 0  0  |  {4} | 0 0  1 |
      {4} | 0 0  b a  |  {5} | 0  0 0   -a b3 |  {4} | 1 0  0 |
                         {5} | 0  0 a3  b  0  |

o9 : List
i10 : G = makeFiniteResolutionCodim2(ff,mf)

o10 = HashTable{alpha => {5} | 0   0 |              }
                         {5} | -b3 0 |
                b => {4} | b a |
                h1' => {5} | 0   0  0  |
                       {5} | -b3 0  0  |
                       {5} | 0   -a b3 |
                       {5} | a3  b  0  |
                h1 => {5} | 0   0  0  |
                      {5} | -b3 0  0  |
                      {5} | 0   -a b3 |
                      {5} | a3  b  0  |
                mu => {5} | a3 b |
                      {5} | 0  a |
                partial => {4} | a -b |
                           {2} | 0 a3 |
                psi => {4} | 0 0  |
                       {2} | 0 b3 |
                               3      5      2
                resolution => S  <-- S  <-- S  <-- 0
                                                    
                              0      1      2      3
                sigma => {5} | b3 |
                         {5} | 0  |
                tau => 0
                u => {8} | 1 0 |
                v => {9} | 0 -1 |
                     {9} | 1 0  |
                X => 0
                Y => 0

o10 : HashTable
i11 : F = G#"resolution"

       3      5      2
o11 = S  <-- S  <-- S  <-- 0
                            
      0      1      2      3

o11 : ChainComplex

See also

Ways to use makeFiniteResolutionCodim2 :

For the programmer

The object makeFiniteResolutionCodim2 is a method function with options.