basic construction, source and target of a ring map

basic construction, source and target of a ring map

Sections:

constructing a ring map

source and target

obtaining the matrix defining a map

constructing a ring map

Use the function map construct a map between two rings. The input, in order, is the target, the source, and the images of the variables of the source ring. The images can be given as a matrix or a list.

i1 : S = QQ[x,y,z]/ideal(x^3+y^3+z^3);

i2 : T = QQ[u,v,w]/ideal(u^3+v^3+w^3);

i3 : G = map(T,S,matrix{{u,v,w^2}})



                     2

o3 = map(T,S,{u, v, w })



o3 : RingMap T <--- S

i4 : G(x^3+y^3+z)



        6    2

o4 = - w  + w



o4 : T

If the third argument is not given there are two possibilities. If a variable in the source ring also appears in the target ring then that variable is mapped to itself and if a variable does not appear in the target ring then it is mapped to zero.

i5 : R = QQ[x,y,w];

i6 : F = map(S,R)



o6 = map(S,R,{x, y, 0})



o6 : RingMap S <--- R

i7 : F(x^3)



        3    3

o7 = - y  - z



o7 : S

source and target

Once a ring map is defined the functions source and target can be used to find out what the source and target of a map are. These functions are particularly useful when working with matrices (see the next example).

i8 : U = QQ[s,t,u, Degrees => {{1,2},{1,1},{1,3}}];

i9 : H = map(U,R,matrix{{s^2,t^3,u^4}})



               2   3   4

o9 = map(U,R,{s , t , u })



o9 : RingMap U <--- R

i10 : use R; H(x^2+y^2+w^2)



       8    6    4

o11 = u  + t  + s



o11 : U

i12 : source H



o12 = R



o12 : PolynomialRing

i13 : target H



o13 = U



o13 : PolynomialRing

obtaining the matrix defining a map

Use F.matrix to obtain the matrix defining the map F.

i14 : H.matrix



o14 = | s2 t3 u4 |



              1       3

o14 : Matrix U  <--- U

i15 : source H.matrix



       3

o15 = U



o15 : U-module, free, degrees {{2, 4}, {3, 3}, {4, 12}}

i16 : target H.matrix



       1

o16 = U



o16 : U-module, free

For more on matrices from maps see input a matrix.