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];
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i2 : R5 = ZZ/32003[a..e];
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i3 : R6 = ZZ/32003[a..f];
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i4 : M = coker genericMatrix(R6,a,2,3)
o4 = cokernel | a c e |
| b d f |
2
o4 : R6-module, quotient of R6
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i5 : pdim M
o5 = 2
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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
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i7 : F = map(R5,R4,random(R5^1, R5^{4:-1}))
o7 = map(R5,R4,{- 13755a - 45b - 8124c + 11314d + 4735e, - 12581a - 2072b + 522c + 13684d + 3781e, 2204a - 4370b - 2355c + 4688d + 11504e, - 1606a + 6106b - 11324c - 3742d + 1499e})
o7 : RingMap R5 <--- R4
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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
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i9 : pdim P
o9 = 1
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i10 : Q = pushForward(F,P)
3
o10 = R4
o10 : R4-module, free, degrees {0, 1, 0}
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i11 : pdim Q
o11 = 0
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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];
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i13 : M = comodule monomialCurveIdeal(P3,{1,2,3})
o13 = cokernel | c2-bd bc-ad b2-ac |
1
o13 : P3-module, quotient of P3
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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];
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i15 : F = map(P3,P2,random(P3^1, P3^{-1,-1,-1}))
4 7 1 8 1 6 5 4 4 9 1
o15 = map(P3,P2,{-a + b + -c + -d, -a + -b + -c + -d, -a + -b + -c + -d})
5 5 3 5 2 5 7 3 5 8 3
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 4131192367680ab+400751588160b2-3737599719840ac-2061579703680bc+1164914684640c2 206559618384a2+279772913504b2-403227768720ac-847075074720bc+781192292832c2 8334200611197057768115280000b3-31930123329998994452610888000b2c+550586835497452809208900320ac2+41250951861319069868852965440bc2-18253615395831319339692423840c3 0 |
{1} | -2802564562959a-364919412977b+2350406364024c 153377727629a-64993866525b-32116759736c 4145500884629428440590293218a2+4675111442576074371310021548ab-1975116379092124264307766798b2-13969439638748607767992644483ac+294973369522436386311183171bc+6216644527312976305168570200c2 16453412307a3+20151510561a2b+16400808522ab2+2161442626b3-62211123657a2c-77509350765abc-24032948562b2c+97652445696ac2+64419695928bc2-53249580576c3 |
2
o16 : P2-module, quotient of P2
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i17 : hilbertPolynomial M
o17 = - 2*P + 3*P
0 1
o17 : ProjectiveHilbertPolynomial
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i18 : hilbertPolynomial N
o18 = - 2*P + 3*P
0 1
o18 : ProjectiveHilbertPolynomial
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i19 : ann N
3 2 2 3
o19 = ideal(16453412307a + 20151510561a b + 16400808522a*b + 2161442626b -
-----------------------------------------------------------------------
2 2 2
62211123657a c - 77509350765a*b*c - 24032948562b c + 97652445696a*c +
-----------------------------------------------------------------------
2 3
64419695928b*c - 53249580576c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.