Description
For an overview of ring maps, substitution of variables, and finding implicit equations of a set of polynomial or rational functions, see
substitution and maps between rings.
A ring map $F : R \rightarrow{} S$, where $R$ is a polynomial ring, is specified by giving the images in $S$ of the variables of $R$. For a simple example, consider the following map. Notice that, as is usual in Macaulay2, the target ring is given before the source.
i1 : R = QQ[a,b,c]; S = QQ[s,t];
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i3 : F = map(S,R,{s^3-t^2, s^3-t, s-t})
3 2 3
o3 = map (S, R, {s - t , s - t, s - t})
o3 : RingMap S <-- R
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i4 : target F
o4 = S
o4 : PolynomialRing
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i5 : source F
o5 = R
o5 : PolynomialRing
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i6 : F.matrix
o6 = | s3-t2 s3-t s-t |
1 3
o6 : Matrix S <-- S
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There are other ways to define ring maps. See below.
Apply matrices to ring elements, vectors, matrices, and ideals using usual function notation.
i7 : F (a+b)
3 2
o7 = 2s - t - t
o7 : S
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The kernel of such ring maps are constructed with the aid of Gröbner bases. Preimages of ideals are constructed using the same method.
i8 : I = kernel F
6 4 4 5 2 2 2 2 2 3 3
o8 = ideal(c + 3a*c - 3b*c + 3c + 3a c - 6a*b*c + 3b c + 6a*c - 8b*c
------------------------------------------------------------------------
4 3 2 2 3 2 2 2 2
+ 3c + a - 3a b + 3a*b - b + 3a c - 3b c - 3a*c + 2a - a*b - 3a*c)
o8 : Ideal of R
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i9 : F I
o9 = ideal 0
o9 : Ideal of S
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i10 : J = preimage(F, ideal(s-3))
2
o10 = ideal (b - c - 24, c + a - 6c - 18)
o10 : Ideal of R
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i11 : isSubset(F J, ideal(s-3))
o11 = true
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Geometrically, the inverse image of this line is a conic.
Consider the Cremona transform, and its square:
i12 : G = map(R,R,{a=>b*c,b=>a*c,c=>a*b})
o12 = map (R, R, {b*c, a*c, a*b})
o12 : RingMap R <-- R
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i13 : G*G
2 2 2
o13 = map (R, R, {a b*c, a*b c, a*b*c })
o13 : RingMap R <-- R
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These are injective ring maps
i14 : ker G == 0
o14 = true
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i15 : isInjective G
o15 = true
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i16 : coimage G
o16 = R
o16 : PolynomialRing
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Common ways to make a ring map:
Common ways to get information about ring maps:
Common operations on ring maps:
Applying ring maps, and composing ring maps:
Operations involving modules