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,{- 11890a + 3897b - 185c + 4675d - 4630e, 4363a - 12664b - 9666c + 10059d + 1478e, 6680a + 388b - 3280c - 6251d - 1101e, 5266a - 14019b - 5670c + 15754d + 1332e})
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}))
7 3 9 5 4 1 2 10 1 7 4
o15 = map(P3,P2,{-a + -b + -c + -d, -a + -b + -c + --d, 2a + -b + -c + -d})
2 2 2 3 5 8 3 3 3 8 3
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 9244316835840ab+48619178736000b2-3696837365664ac-54975066897600bc+14209401766104c2 924431683584a2+40463479017600b2-3245822352864ac-33505043849280bc+9776977983432c2 16872838111892727106562248704000b3-20496711598189212755582890185600b2c+228994088286892833386784ac2+8298432174192572751528105316800bc2-1119761053867544096600422040952c3 0 |
{1} | -39156196749082a-138767696881155b+122652428103208c -9840098644350a-114671184872305b+63913740439864c 1000508336282949207819371184128a2-4230593235369442087264221806080ab-47948856649210342053266426835200b2-1795815591730591972398399233870ac+45710166983014343487503954329335bc-7572941228550955146167733824904c2 43159213832a3+44493480780a2b+54753998000ab2+47460834500b3-243356243630a2c-246367017025abc-239506773300b2c+482700802560ac2+385282234620bc2-346962345000c3 |
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(43159213832a + 44493480780a b + 54753998000a*b + 47460834500b
-----------------------------------------------------------------------
2 2
- 243356243630a c - 246367017025a*b*c - 239506773300b c +
-----------------------------------------------------------------------
2 2 3
482700802560a*c + 385282234620b*c - 346962345000c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.