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,{- 10822a - 14699b + 5607c - 4780d - 6612e, - 15153a + 6304b + 1501c + 5918d + 4868e, 7661a + 6286b - 10578c + 15216d + 12205e, 5813a + 4176b + 2829c + 8396d + 12181e})
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 2 5 1 3 1 8
o15 = map(P3,P2,{a + -b + c + 2d, -a + 2b + -c + d, -a + -b + -c + -d})
7 3 3 2 2 7 7
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 387639ab+12658905b2-516852ac-34532358bc+23538424c2 1335201a2+270421515b2-5340804ac-735891240bc+505779764c2 2010584177091b3-8059860206244b2c+10769844938832bc2-4796982045632c3 0 |
{1} | 11717720a+7484409b-30144772c 246217531a+174023595b-655904522c -47894422842a2+1825643393625ab+1205510914611b2-2255978644150ac-6348165602136bc+6156242884652c2 4101202a3-22399335a2b-4083786ab2-46922544b3+6403026a2c+48161736abc+171165438b2c-12318992ac2-203791140bc2+50419432c3 |
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 2
o19 = ideal(4101202a - 22399335a b - 4083786a*b - 46922544b + 6403026a c +
-----------------------------------------------------------------------
2 2 2
48161736a*b*c + 171165438b c - 12318992a*c - 203791140b*c +
-----------------------------------------------------------------------
3
50419432c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.