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

## Synopsis

• Usage:
maps = makeFiniteResolutionCodim2(ff,mf)
• Inputs:
• mf, a list, matrix factorization
• ff, , regular sequence
• Optional inputs:
• Check => ..., default value false
• Outputs:
• maps, , many maps

## 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