i1 : R = QQ[x,y,z,Degrees=>{2,3,1}]/(y^2-x^3) o1 = R o1 : QuotientRing |
i2 : H = Hom(ideal(x,y), R^1) -- ker (39) called with OptionTable: OptionTable{SubringLimit => infinity} -- ker (39) returned CacheFunction: -*a cache function*- -- ker (39) called with Matrix: {-5} | y -x | -- {-6} | x2 -y | -- ker (39) returned Module: image {-2} | x y | -- {-3} | y x2 | assert( ker(map(R^{{5}, {6}},R^{{2}, {3}},{{y,-x}, {x^2,-y}})) === (image(map(R^{{2}, {3}},R^{{0}, {-1}},{{x,y}, {y,x^2}})))) o2 = image {-2} | x y | {-3} | y x2 | 2 o2 : R-module, submodule of R |
i3 : f = H_{1} o3 = {0} | 0 | {1} | 1 | o3 : Matrix |
i4 : g = homomorphism f o4 = | y x2 | o4 : Matrix |
i5 : source g === module ideal(x,y) o5 = true |
i6 : target g === R^1 o6 = true |
Except for a possible redistribution of degrees between the map and modules, we can undo the process with homomorphism'.
i7 : f' = homomorphism' g -- ker (40) called with OptionTable: OptionTable{SubringLimit => infinity} -- ker (40) returned CacheFunction: -*a cache function*- -- ker (40) called with Matrix: 0 -- 2 -- ker (40) returned Module: R assert( ker(map(R^0,R^{{2}, {3}},0)) === (R^{{2}, {3}})) o7 = {0} | 0 | {1} | 1 | o7 : Matrix |
i8 : f === f' o8 = false |
i9 : f - f' o9 = 0 o9 : Matrix |
i10 : degree f, degree f' o10 = ({0}, {1}) o10 : Sequence |
i11 : degrees f, degrees f' o11 = ({{{0}, {1}}, {{1}}}, {{{0}, {1}}, {{0}}}) o11 : Sequence |
After pruning a Hom module, one cannot use homomorphism directly. Instead, first apply the pruning map:
i12 : H1 = prune H o12 = cokernel {0} | y x2 | {1} | -x -y | 2 o12 : R-module, quotient of R |
i13 : homomorphism(H1.cache.pruningMap * H1_{1}) o13 = | y x2 | o13 : Matrix |
The object homomorphism is a method function.