Currently,
R and
S must both be polynomial rings over the same base field.
This function first checks to see whether M will be a finitely generated R-module via F. If not, an error message describing the codimension of M/(vars of S)M is given (this is equal to the dimension of R if and only if M is a finitely generated R-module.
Assuming that it is, the push forward
F_*(M) is computed. This is done by first finding a presentation for
M in terms of a set of elements that generates
M as an
S-module, and then applying the routine
coimage to a map whose target is
M and whose source is a free module over
R.
Example: The Auslander-Buchsbaum formula
Let's illustrate the Auslander-Buchsbaum formula. First construct some rings and make a module of projective dimension 2.
i1 : R4 = ZZ/32003[a..d];
|
i2 : R5 = ZZ/32003[a..e];
|
i3 : R6 = ZZ/32003[a..f];
|
i4 : M = coker genericMatrix(R6,a,2,3)
o4 = cokernel | a c e |
| b d f |
2
o4 : R6-module, quotient of R6
|
i5 : pdim M
o5 = 2
|
Create ring maps.
i6 : G = map(R6,R5,{a+b+c+d+e+f,b,c,d,e})
o6 = map(R6,R5,{a + b + c + d + e + f, b, c, d, e})
o6 : RingMap R6 <--- R5
|
i7 : F = map(R5,R4,random(R5^1, R5^{4:-1}))
o7 = map(R5,R4,{- 14824a - 14266b + 13791c - 6843d + 1485e, 617a - 2833b - 5274c - 15242d - 9939e, - 12326a + 10673b + 3536c - 15016d + 913e, - 7408a - 8495b - 6017c + 6268d - 4062e})
o7 : RingMap R5 <--- R4
|
The module M, when thought of as an R5 or R4 module, has the same depth, but since depth M + pdim M = dim ring, the projective dimension will drop to 1, respectively 0, for these two rings.
i8 : P = pushForward(G,M)
o8 = cokernel | c -de |
| d bc-ad+bd+cd+d2+de |
2
o8 : R5-module, quotient of R5
|
i9 : pdim P
o9 = 1
|
i10 : Q = pushForward(F,P)
3
o10 = R4
o10 : R4-module, free, degrees {0, 1, 0}
|
i11 : pdim Q
o11 = 0
|
Example: generic projection of a homogeneous coordinate ring
We compute the pushforward N of the homogeneous coordinate ring M of the twisted cubic curve in P^3.
i12 : P3 = QQ[a..d];
|
i13 : M = comodule monomialCurveIdeal(P3,{1,2,3})
o13 = cokernel | c2-bd bc-ad b2-ac |
1
o13 : P3-module, quotient of P3
|
The result is a module with the same codimension, degree and genus as the twisted cubic, but the support is a cubic in the plane, necessarily having one node.
i14 : P2 = QQ[a,b,c];
|
i15 : F = map(P3,P2,random(P3^1, P3^{-1,-1,-1}))
3 7 5 5 9 7
o15 = map(P3,P2,{--a + 2b + c + 4d, -a + -b + c + d, 2a + -b + --c + -d})
10 9 2 6 10 2
o15 : RingMap P3 <--- P2
|
i16 : N = pushForward(F,M)
o16 = cokernel {0} | 16387580956940ab-14724653927274b2-1687572937560ac+4956973677186bc-403641593240c2 163875809569400a2-108724290935454b2-149778220350820ac+171263908414242bc-31380232552680c2 1104778585290145194092834607301200b3-1291884858328181701654565533161000b2c+23110910711273451342626346437500ac2+483642477404520665871508862290650bc2-61148006370496986650484756103400c3 0 |
{1} | 4767182387215a-66573184077431b+2257144971704c -526812206177555a-590541027284211b+590394412586128c -2233496552789285636725869427800695a2+4501740677319060950084782565664927ab+357748070445773529690004522271997b2+3994500220861366455736412888954445ac-5496074274065385115560152579327715bc-1659920395835751659317670711807200c2 33267148550a3-96943220235a2b+59706121683ab2-62060425857b3-52512092475a2c+116138883810abc-1121662530b2c+12605662800ac2-29352091800bc2+7131760400c3 |
2
o16 : P2-module, quotient of P2
|
i17 : hilbertPolynomial M
o17 = - 2*P + 3*P
0 1
o17 : ProjectiveHilbertPolynomial
|
i18 : hilbertPolynomial N
o18 = - 2*P + 3*P
0 1
o18 : ProjectiveHilbertPolynomial
|
i19 : ann N
3 2 2 3
o19 = ideal(33267148550a - 96943220235a b + 59706121683a*b - 62060425857b
-----------------------------------------------------------------------
2 2 2
- 52512092475a c + 116138883810a*b*c - 1121662530b c + 12605662800a*c
-----------------------------------------------------------------------
2 3
- 29352091800b*c + 7131760400c )
o19 : Ideal of P2
|
Note: these examples are from the original Macaulay script by David Eisenbud.