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,{7319a - 6993b - 751c - 10655d - 11865e, - 12440a + 14992b - 2546c - 675d - 12822e, - 11107a + 12674b + 3345c - 15423d - 13269e, - 11345a - 5565b + 4496c + 3989d - 4669e})
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}))
5 1 1 3 7 5 5 5 10 5
o15 = map(P3,P2,{-a + -b + -c + -d, a + -b + -c + -d, -a + --b + 3c + -d})
9 2 8 5 9 3 2 9 7 4
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 71627269219400ab-24458851300500b2-51353466527720ac-23736233360000bc+44396280194420c2 114603630751040a2-1012229745300b2-46658522083040ac-90861293766600bc+98884844486652c2 1859789260817907982585882963078500b3-10623278726313474037982197442262300b2c-3567130525260779866444097460633600ac2+22013352115753269460930152297632460bc2-12483771178929488634608726402848788c3 0 |
{1} | -585104004148250a+202531149508390b-53213530512793c -1018532196400424a+331967181911085b-71441314421757c 140965298040593490863453092927286000a2-154026231097053704163376973902068240ab+33324624591148982075501633802754275b2+146612004959821332764725477653381600ac-50467794998233439648868002501409270bc+9880219458598972395417686170103583c2 10990448540000a3-15761705043600a2b+7883315658000ab2-1297206104625b3+5097928234000a2c-6835068979200abc+2002241908800b2c+2265906327720ac2-1381439849265bc2+342601353738c3 |
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(10990448540000a - 15761705043600a b + 7883315658000a*b -
-----------------------------------------------------------------------
3 2
1297206104625b + 5097928234000a c - 6835068979200a*b*c +
-----------------------------------------------------------------------
2 2 2
2002241908800b c + 2265906327720a*c - 1381439849265b*c +
-----------------------------------------------------------------------
3
342601353738c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.