# localResolution -- find a resolution over a local ring

## Description

This method has option inputs that it inherits from

resolution.

This function iterates

localsyz to obtain a resolution over the local ring.
 i1 : R = ZZ/32003[x,y,z,w,SkewCommutative=>true] o1 = R o1 : PolynomialRing, 4 skew commutative variables i2 : m = matrix{{x,y*z},{z*w,x}} o2 = | x yz | | zw x | 2 2 o2 : Matrix R <--- R i3 : setMaxIdeal(ideal(x,y,z,w)) o3 = ideal (x, y, z, w) o3 : Ideal of R i4 : C = localResolution(coker m, LengthLimit=>10) 2 2 2 2 2 2 2 2 2 2 2 o4 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R 0 1 2 3 4 5 6 7 8 9 10 o4 : ChainComplex i5 : C = localResolution(coker m) 2 2 2 2 2 2 2 2 2 2 2 o5 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R 0 1 2 3 4 5 6 7 8 9 10 o5 : ChainComplex i6 : C^2 o6 = 0 o6 : R-module i7 : C.dd_4 o7 = {6} | -zw x | {6} | -x yz | 2 2 o7 : Matrix R <--- R
 i8 : R = QQ[x,y,z] o8 = R o8 : PolynomialRing i9 : setMaxIdeal ideal vars R o9 = ideal (x, y, z) o9 : Ideal of R i10 : m = matrix {{x-1, y, z-1}} o10 = | x-1 y z-1 | 1 3 o10 : Matrix R <--- R i11 : C = resolution coker m 1 3 3 1 o11 = R <-- R <-- R <-- R <-- 0 0 1 2 3 4 o11 : ChainComplex i12 : C.dd 1 3 o12 = 0 : R <----------------- R : 1 | x-1 y z-1 | 3 3 1 : R <------------------------- R : 2 {1} | -y -z+1 0 | {1} | x-1 0 -z+1 | {1} | 0 x-1 y | 3 1 2 : R <--------------- R : 3 {2} | z-1 | {2} | -y | {2} | x-1 | 1 3 : R <----- 0 : 4 0 o12 : ChainComplexMap i13 : LC = localResolution coker m 1 3 2 o13 = R <-- R <-- R <-- 0 0 1 2 3 o13 : ChainComplex i14 : LC.dd 1 3 o14 = 0 : R <----------------- R : 1 | z-1 y x-1 | 3 2 1 : R <--------------------- R : 2 {1} | -x+1 y | {1} | 0 -z+1 | {1} | z-1 0 | 2 2 : R <----- 0 : 3 0 o14 : ChainComplexMap