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,{3759a + 14630b - 9165c + 14521d - 13912e, - 15035a + 7993b - 8348c - 3736d + 376e, - 10656a + 3313b + 1194c + 12621d - 11793e, 738a - 1830b - 1219c - 7764d - 4854e})
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}))
5 4 2 7 3 6 4 4 2 3
o15 = map(P3,P2,{5a + -b + -c + -d, a + --b + -c + -d, -a + -b + -c + -d})
4 5 3 10 5 5 9 7 7 4
o15 : RingMap P3 <--- P2
|
i16 : N = pushForward(F,M)
o16 = cokernel {0} | 4561102375200ab-32281351807600b2-7211334427920ac+66977086421700bc-21599492481000c2 82099842753600a2-2283944542188400b2-279507071211480ac-3583360027100700bc+12378722508432000c2 44864292221079939827463300000b3-307026642658736907668665773800b2c-1590019188366147302146396860ac2+714301550339170069038618305100bc2-546000722514896592712322893500c3 0 |
{1} | -634037311260a+21876744992505b-36689511723186c 84915577430370a+7304969116106565b-12224581138037856c 280013877735901663452593400a2-5596323497287082248703031900ab+110849926804954778999126251350b2+10977604797596428931033173905ac-401971188948258265199668139055bc+362424931345922086749293596893c2 205140891600a3-5551812446400a2b+114796032910200ab2-732411811798700b3+8759158555140a2c-389778459236460abc+3462026874390945b2c+330700395607344ac2-5433820631976240bc2+2830407616612800c3 |
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(205140891600a - 5551812446400a b + 114796032910200a*b -
-----------------------------------------------------------------------
3 2
732411811798700b + 8759158555140a c - 389778459236460a*b*c +
-----------------------------------------------------------------------
2 2 2
3462026874390945b c + 330700395607344a*c - 5433820631976240b*c +
-----------------------------------------------------------------------
3
2830407616612800c )
o19 : Ideal of P2
|
Note: these examples are from the original Macaulay script by David Eisenbud.