next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
map > map(Ring,Ring)

map(Ring,Ring) -- make a ring map, using the names of the variables

Synopsis

Description

For example, consider the following rings.
i1 : A = QQ[a..e];
i2 : B = A[x,y,Join=>false];
i3 : C = QQ[a..e,x,y];
The natural inclusion and projection maps between A and B are
i4 : map(B,A)

o4 = map(B,A,{a, b, c, d, e})

o4 : RingMap B <--- A
i5 : map(A,B)

o5 = map(A,B,{0, 0, a, b, c, d, e})

o5 : RingMap A <--- B
The isomorphisms between B and C:
i6 : F = map(B,C)

o6 = map(B,C,{a, b, c, d, e, x, y})

o6 : RingMap B <--- C
i7 : G = map(C,B)

o7 = map(C,B,{x, y, a, b, c, d, e})

o7 : RingMap C <--- B
i8 : F*G

o8 = map(B,B,{x, y, a, b, c, d, e})

o8 : RingMap B <--- B
i9 : oo === id_B

o9 = true
i10 : G*F

o10 = map(C,C,{a, b, c, d, e, x, y})

o10 : RingMap C <--- C
i11 : oo === id_C

o11 = true

The ring maps that are created are not always mathematically well-defined. For example, the map F below is the natural quotient map, but the map G is not mathematically well-defined, although we can use it in Macaulay2 to lift elements of E to D.

i12 : D = QQ[x,y,z];
i13 : E = D/(x^2-z-1,y);
i14 : F = map(E,D)

o14 = map(E,D,{x, 0, z})

o14 : RingMap E <--- D
i15 : G = map(D,E)

o15 = map(D,E,{x, y, z})

o15 : RingMap D <--- E
i16 : x^3

o16 = x*z + x

o16 : E
i17 : G x^3

o17 = x*z + x

o17 : D

Caveat

The map is not always a mathematically well-defined ring map

See also