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,{14393a + 9256b - 8812c - 4514d + 5521e, - 3378a - 14348b - 2414c + 12378d + 6852e, - 1151a + 7087b + 12722c - 1311d - 6203e, 1454a + 6480b + 8943c - 10493d - 9185e})
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}))
6 3 7 7 2 5
o15 = map(P3,P2,{-a + --b + 2c + -d, 6a + b + --c + -d, 4a + -b + 2c + d})
7 10 9 10 9 4
o15 : RingMap P3 <--- P2
|
i16 : N = pushForward(F,M)
o16 = cokernel {0} | 18526308747975ab+12141126354915b2-29264725179900ac-40447495667970bc+33669836787240c2 25936832247165a2-11052058961205b2-9697613936220ac+33208272972648bc-24058193658012c2 82816263529609029547161415680b3+377789701863445517618853027840b2c-96603363550376655976850490000ac2-1695927579402762359050068657360bc2+1435060371259755646920339000480c3 0 |
{1} | 8392802493103a+5002704113965b-13361138647129c -4844142582370a-2137964553088b+13319533964837c -11156617784503667737699121724215a2-3938238720137508804434387066824ab+2328123457090439415175814537305b2+16389750281967831562338275970631ac-934189373195860800214674467188bc-6227823387101803833805084863598c2 564434974145a3+569142865725a2b+14686822185ab2-76057970155b3-1477184975595a2c-713225540601abc+170121298050b2c+1246028852124ac2+121012714098bc2-329357691972c3 |
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(564434974145a + 569142865725a b + 14686822185a*b -
-----------------------------------------------------------------------
3 2 2
76057970155b - 1477184975595a c - 713225540601a*b*c + 170121298050b c
-----------------------------------------------------------------------
2 2 3
+ 1246028852124a*c + 121012714098b*c - 329357691972c )
o19 : Ideal of P2
|
Note: these examples are from the original Macaulay script by David Eisenbud.