# replacements for functions from version 1.0

This page describes the replacements for functions implemented in version 1.0 of this package by Mike Stillman and David Eisenbud. That version implemented functionality for finding minimal generators, syzygies and resolutions for polynomial rings localized at a maximal ideal.

Defining a local ring using setMaxIdeal and localRing:

 i1 : S = ZZ/32003[x,y,z,w] o1 = S o1 : PolynomialRing i2 : P = ideal(x,y,z,w) o2 = ideal (x, y, z, w) o2 : Ideal of S i3 : setMaxIdeal P -- version 1.0 o3 = ideal (x, y, z, w) o3 : Ideal of S i4 : R = localRing(S, P) -- version 2.0 and above o4 = R o4 : LocalRing, maximal ideal (x, y, z, w)

Computing syzygies using localsyz and syz:

 i5 : use S o5 = S o5 : PolynomialRing i6 : m = matrix{{x,y*z},{z*w,x}} o6 = | x yz | | zw x | 2 2 o6 : Matrix S <--- S i7 : m * localsyz m o7 = 0 2 o7 : Matrix S <--- 0 i8 : use R o8 = R o8 : LocalRing, maximal ideal (x, y, z, w) i9 : m = matrix{{x,y*z},{z*w,x}} o9 = | x yz | | zw x | 2 2 o9 : Matrix R <--- R i10 : m * syz m o10 = 0 2 o10 : Matrix R <--- 0

Computing syzygies using localMingens and mingens:

 i11 : use S o11 = S o11 : PolynomialRing i12 : localMingens matrix{{x-1,x,y},{x-1,x,y}} o12 = | x-1 | | x-1 | 2 1 o12 : Matrix S <--- S i13 : use R o13 = R o13 : LocalRing, maximal ideal (x, y, z, w) i14 : mingens image matrix{{x-1,x,y},{x-1,x,y}} o14 = | x-1 | | x-1 | 2 1 o14 : Matrix R <--- R

Computing syzygies using localModulo and modulo:

 i15 : use S o15 = S o15 : PolynomialRing i16 : localModulo(matrix {{x-1,y}}, matrix {{y,z}}) o16 = {1} | z y 0 | {1} | 0 -x+1 -1 | 2 3 o16 : Matrix S <--- S i17 : use R o17 = R o17 : LocalRing, maximal ideal (x, y, z, w) i18 : modulo(matrix {{x-1,y}}, matrix {{y,z}}) o18 = {1} | 0 y z | {1} | -1 -x+1 0 | 2 3 o18 : Matrix R <--- R

Computing syzygies using localPrune and prune:

 i19 : use S o19 = S o19 : PolynomialRing i20 : localPrune image matrix{{x-1,x,y},{x-1,x,y}} 1 o20 = S o20 : S-module, free, degrees {1} i21 : use R o21 = R o21 : LocalRing, maximal ideal (x, y, z, w) i22 : prune image matrix{{x-1,x,y},{x-1,x,y}} 1 o22 = R o22 : R-module, free, degrees {1}

Computing syzygies using localResolution and resolution:

 i23 : use S o23 = S o23 : PolynomialRing i24 : localResolution coker matrix{{x,y*z},{z*w,x}} 2 2 o24 = S <-- S <-- 0 0 1 2 o24 : ChainComplex i25 : oo.dd 2 2 o25 = 0 : S <------------- S : 1 | yz x | | x zw | 2 1 : S <----- 0 : 2 0 o25 : ChainComplexMap i26 : use R o26 = R o26 : LocalRing, maximal ideal (x, y, z, w) i27 : res coker matrix{{x,y*z},{z*w,x}} 2 2 o27 = R <-- R 0 1 o27 : ChainComplex i28 : oo.dd 2 2 o28 = 0 : R <------------- R : 1 | yz x | | x zw | o28 : ChainComplexMap