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,{- 12134a + 6800b - 680c + 5281d - 15936e, - 2927a - 272b + 3423c + 5735d + 12761e, 7372a - 6981b - 12544c + 4786d - 14067e, 8717a - 10346b - 4547c + 4702d - 15957e})
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}))
1 1 1 7 7 5 2 5 9 1
o15 = map(P3,P2,{-a + --b + -c + -d, 7a + b + -c + -d, -a + -b + --c + -d})
4 10 2 8 5 9 3 3 10 4
o15 : RingMap P3 <--- P2
|
i16 : N = pushForward(F,M)
o16 = cokernel {0} | 8164698609600ab-405771109920b2+56677606837200ac+928301349420bc-18413502639480c2 4082349304800a2-25007441670b2+17301059892000ac-122751693540bc-3016014604200c2 833512670690234965761689280b3-24984330352527336776916991920b2c-1240156970384548809232780470000ac2+288446161539059955865166570460bc2-773998516307568105706733507160c3 0 |
{1} | 2034936349348a-7208084570447b-51984415669638c -2153519432704a-905235739489b-17238596739216c 56824617083982695009830098240a2-78186220361822779379120523472ab+40553169255453420681854176568b2-401978498313259480532672111244ac+480856906499315137498000351653bc+1197051998585364381814097252274c2 530276573120a3-755972822160a2b+412756675668ab2-11940672267b3-3507953885280a2c+4410589444848abc-253401457050b2c+2034064503504ac2+1752514250076bc2-10253863858824c3 |
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(530276573120a - 755972822160a b + 412756675668a*b -
-----------------------------------------------------------------------
3 2 2
11940672267b - 3507953885280a c + 4410589444848a*b*c - 253401457050b c
-----------------------------------------------------------------------
2 2 3
+ 2034064503504a*c + 1752514250076b*c - 10253863858824c )
o19 : Ideal of P2
|
Note: these examples are from the original Macaulay script by David Eisenbud.