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,{9020a + 2758b + 4394c + 7937d + 8682e, - 15446a + 10967b - 2668c + 980d - 1926e, - 2424a + 14985b - 12804c + 1968d + 13304e, - 4216a + 10792b + 3171c - 1158d + 5090e})
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}))
1 3 7 3 9 8 7 6
o15 = map(P3,P2,{-a + --b + 2c + 7d, --a + b + -c + 6d, -a + -b + --c + -d})
7 10 10 4 4 5 10 5
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 723024241062130ab-960411894406000b2-181600226552780ac+534560752458360bc-76101191027600c2 7591754531152365a2-18502569087270000b2+148289511228880ac+11073265188180500bc-1694169297840200c2 19734520575732448059294770886541740000b3-16887367956538161128861357949157509000b2c-98370215212553048755395162227984640ac2+4816238643654539460844383397922344800bc2-451864610933392516961799883449210400c3 0 |
{1} | -3896237897291754a+5229643056079532b-1758493984171225c -90964760037567171a+151276710400651656b-46119970405014940c -158077526040353393702630677344103183674a2+353684149109846301161220795956989719354ab-302691659121392868960683819561915994032b2-98581776411625487361761000012710428957ac+148315090733069830564040302343285797552bc-17208018323912336779056569501408858580c2 10371905088678a3-36983469509238a2b+43708211679104ab2-17015569092800b3+10676184903195a2c-23519815376680abc+12172690277088b2c+3141032496000ac2-2747006158180bc2+186052988200c3 |
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
o19 = ideal(10371905088678a - 36983469509238a b + 43708211679104a*b -
-----------------------------------------------------------------------
3 2
17015569092800b + 10676184903195a c - 23519815376680a*b*c +
-----------------------------------------------------------------------
2 2 2
12172690277088b c + 3141032496000a*c - 2747006158180b*c +
-----------------------------------------------------------------------
3
186052988200c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.