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,{- 1737a - 10707b + 1125c + 9064d + 1624e, - 12586a + 11967b - 9439c - 8543d + 14474e, 117a + 3069b - 560c - 288d - 1927e, - 5551a + 11497b - 14608c + 2501d + 11551e})
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 8 8 1 10 1 8 7 2 1
o15 = map(P3,P2,{-a + -b + -c + -d, --a + -b + -c + d, -a + 2b + -c + --d})
8 7 5 3 7 9 5 2 3 10
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 9031792397939640ab-16005322306277640b2-2435232860338800ac+12846785525668350bc-2800574124703500c2 2007064977319920a2-2783838808183545b2-702773481172500ac+2288159031428550bc-408668100708000c2 24548983519979724968421510923616564375b3-46218140900010951939847460510186017500b2c+775557900534309507370822936549740000ac2+26323801307669378689764032532100860000bc2-4852389288109179689345489378991825000c3 0 |
{1} | 12283423223337216a+9576003143641989b-18512510937651410c 3301137952997186a+1673125947601419b-4139771162158410c -19041890649801602001911428004674195968a2-25560566235450165893485756848938531064ab-14635003912887622895160740539372452843b2+47143300709091407926974606710163702720ac+38050754581250480362815546518542882530bc-29370077607034529842640805987246079400c2 35577423802368a3-15290314624704a2b+5093216327304ab2+174427639407b3-54475575221760a2c+29484737711520abc-20643987041010b2c+10323919396000ac2+16733343676500bc2-5813149607000c3 |
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(35577423802368a - 15290314624704a b + 5093216327304a*b +
-----------------------------------------------------------------------
3 2
174427639407b - 54475575221760a c + 29484737711520a*b*c -
-----------------------------------------------------------------------
2 2 2
20643987041010b c + 10323919396000a*c + 16733343676500b*c -
-----------------------------------------------------------------------
3
5813149607000c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.