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) |
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 |