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,{12444a - 12794b - 11327c - 2683d - 9855e, - 8459a - 13348b - 9606c + 1030d + 6402e, 2936a + 8256b - 1877c - 5358d + 8648e, - 4037a - 2783b + 73c + 11428d - 2220e})
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 2 3 9 4 3 1 8
o15 = map(P3,P2,{-a + -b + 2c + -d, -a + -b + 2c + d, -a + -b + -c + 10d})
2 5 2 8 5 2 5 9
o15 : RingMap P3 <--- P2
|
i16 : N = pushForward(F,M)
o16 = cokernel {0} | 24039272124ab-20709412992b2-12679204266ac+5378616648bc+2264832495c2 48078544248a2-43261507872b2-80600040522ac+60105477816bc+11776986090c2 18319156244495722910952000b3-33286956705520576455762000b2c-3187449736019776062119700ac2+19997612660623494136679100bc2-815240653239153387081375c3 0 |
{1} | 103808746568a-54361427466b-20678253957c 615337112924a-498703257552b-105897040785c 501770584879130057189002528a2-1317643725164220037380016864ab+978902904683378552928616608b2-41027747732074736605310136ac-85978074747649712521144176bc+9519230889330074731353348c2 60766095552a3-211919990592a2b+246435903680ab2-97110538176b3-18983382864a2c+43025600928abc-22932949968b2c+1861546212ac2-2240616276bc2-44907615c3 |
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
o19 = ideal(60766095552a - 211919990592a b + 246435903680a*b -
-----------------------------------------------------------------------
3 2 2
97110538176b - 18983382864a c + 43025600928a*b*c - 22932949968b c +
-----------------------------------------------------------------------
2 2 3
1861546212a*c - 2240616276b*c - 44907615c )
o19 : Ideal of P2
|
Note: these examples are from the original Macaulay script by David Eisenbud.