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);
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i2 : T = QQ[u,v,w]/ideal(u^3+v^3+w^3);
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i3 : G = map(T,S,matrix{{u,v,w^2}})
2
o3 = map(T,S,{u, v, w })
o3 : RingMap T <--- S
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i4 : G(x^3+y^3+z)
6 2
o4 = - w + w
o4 : T
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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];
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i6 : F = map(S,R)
o6 = map(S,R,{x, y, 0})
o6 : RingMap S <--- R
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i7 : F(x^3)
3 3
o7 = - y - z
o7 : S
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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}}];
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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
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i10 : use R; H(x^2+y^2+w^2)
8 6 4
o11 = u + t + s
o11 : U
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i12 : source H
o12 = R
o12 : PolynomialRing
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i13 : target H
o13 = U
o13 : PolynomialRing
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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
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i15 : source H.matrix
3
o15 = U
o15 : U-module, free, degrees {{2, 4}, {3, 3}, {4, 12}}
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i16 : target H.matrix
1
o16 = U
o16 : U-module, free
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For more on matrices from maps see
inputting a matrix.