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,{- 3252a - 12059b - 5126c + 10781d - 15310e, - 14959a - 12870b + 7459c + 3464d - 10168e, - 14123a + 13794b + 4875c + 7539d - 6758e, 7998a + 12159b - 1624c - 1964d - 14939e})
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}))
3 5 3 9 5 3 3 10 4 1
o15 = map(P3,P2,{-a + -b + -c + 7d, -a + -b + -c + -d, --a + -b + c + -d})
8 2 2 8 7 8 5 7 3 8
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 17791178675200ab-104601284838400b2-26288439579840ac+81446120105280bc+3953266140600c2 18108878294400a2-962061374601600b2-21829683520560ac+1436330085269520bc-529999090777800c2 23610722993933951633772220416000b3-188251637027189808137530310092800b2c-7442407029447135519423718671360ac2+204906241217095140358251975139200bc2-54195444565558516607378646336696c3 0 |
{1} | -8230148266560a+37748557458640b+117820570701897c -193743926189010a+1395580630757110b-819811073772327c -588506545090677587501755994688a2+12850035078690670836719647868160ab-82171928388572734831653146916800b2-13395699988982542587036225001896ac+259833061876396190066303953009560bc-60633556281919544379919122551913c2 666361436355a3-18467794945635a2b+166220566021825ab2-490310458689625b3+12429397204560a2c-243259212412020abc+1155989161553100b2c+84930707648745ac2-941050515045060bc2+264704118692328c3 |
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(666361436355a - 18467794945635a b + 166220566021825a*b -
-----------------------------------------------------------------------
3 2
490310458689625b + 12429397204560a c - 243259212412020a*b*c +
-----------------------------------------------------------------------
2 2 2
1155989161553100b c + 84930707648745a*c - 941050515045060b*c +
-----------------------------------------------------------------------
3
264704118692328c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.