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,{11391a + 6277b + 13956c - 1350d + 671e, - 8840a + 865b + 6112c - 2496d + 15416e, - 162a - 339b - 3736c + 4392d + 4317e, - 11987a + 13289b - 5612c + 10478d - 7623e})
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 2 10 3 5
o15 = map(P3,P2,{-a + -b + 3c + -d, 4a + 3b + 5c + 4d, --a + -b + -c + 2d})
2 2 7 9 5 6
o15 : RingMap P3 <--- P2
|
i16 : N = pushForward(F,M)
o16 = cokernel {0} | 16048134592ab+19257307076b2-65003354640ac-143894609460bc+271699425900c2 64192538368a2+198055857988b2-119788678800ac-1409021771940bc+2546580375900c2 17089012308465587375750000b3-180865977878220379430287500b2c-1129642721668146651411600ac2+638123549105739544407272700bc2-750018322517946021059512500c3 0 |
{1} | 210928043452a-24393477519b-93822468750c 1323140720364a+12906267685b-1224747933030c -49634306620523232505821952a2+143138247836627312358952016ab-6128731359520713130609892b2-404381323287546858594748260ac-88761140500131206752101615bc+341291099468611603105315650c2 67690374976a3-113982835680a2b+80429954472ab2-25395432137b3+273556427760a2c-416907795780abc+206127294840b2c+546295995000ac2-548238285900bc2+469117332000c3 |
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(67690374976a - 113982835680a b + 80429954472a*b - 25395432137b
-----------------------------------------------------------------------
2 2
+ 273556427760a c - 416907795780a*b*c + 206127294840b c +
-----------------------------------------------------------------------
2 2 3
546295995000a*c - 548238285900b*c + 469117332000c )
o19 : Ideal of P2
|
Note: these examples are from the original Macaulay script by David Eisenbud.