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,{- 14948a + 13728b - 12472c + 8901d - 7996e, - 1147a + 8717b + 3664c - 11216d + 15261e, - 13512a + 15347b + 4532c - 10659d - 5573e, 4301a - 15293b + 10054c - 7588d - 12446e})
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}))
9 8 3 7 2 8 8 6
o15 = map(P3,P2,{-a + -b + 9c + -d, --a + -b + 3c + 2d, -a + -b + 3c + -d})
2 5 5 10 3 9 7 5
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 8866610298000ab-18740207473200b2-6494790543955ac-24119502001455bc+28147160221740c2 8045627863000a2+266049758125800b2-71539036963555ac-597033854481855bc+461137472496540c2 23670613403556415770105290880000b3-64608189872616090228467356099200b2c-30189282701957693807243716280ac2+58261021283926998092375910548220bc2-17220637856019338697699519883785c3 0 |
{1} | -14635776266304a-26966926109484b+124222798991487c 5542108808616a-304265763096804b+298718499458127c 1040958811166807253773671164000a2+4552543472503440276545198963850ab-19795076478859402983626713083150b2-13966029023039432004732321913014ac+13342155767003547571344543572706bc+23696447395864840979379894407742c2 62285308064a3+140755016070a2b+577701565020ab2+6810976073430b3-1101512992986a2c-3266840093580abc-25076677165218b2c+7326811581138ac2+37465925325570bc2-25852526611443c3 |
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(62285308064a + 140755016070a b + 577701565020a*b +
-----------------------------------------------------------------------
3 2
6810976073430b - 1101512992986a c - 3266840093580a*b*c -
-----------------------------------------------------------------------
2 2 2
25076677165218b c + 7326811581138a*c + 37465925325570b*c -
-----------------------------------------------------------------------
3
25852526611443c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.