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
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i2 : S = kk[a,b]
o2 = S
o2 : PolynomialRing
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i3 : ff = matrix"a4,b4"
o3 = | a4 b4 |
1 2
o3 : Matrix S <--- S
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i4 : R = S/ideal ff
o4 = R
o4 : QuotientRing
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i5 : N = R^1/ideal"a2, ab, b3"
o5 = cokernel | a2 ab b3 |
1
o5 : R-module, quotient of R
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i6 : N = coker vars R
o6 = cokernel | a b |
1
o6 : R-module, quotient of R
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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
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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
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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
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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
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i11 : F = G#"resolution"
3 5 2
o11 = S <-- S <-- S <-- 0
0 1 2 3
o11 : ChainComplex
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