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,{14684a - 988b + 3410c - 4965d + 9253e, - 2988a - 6323b + 1710c + 1672d + 14612e, 7033a + 4471b + 14275c - 5350d + 11962e, - 11468a - 10331b - 3524c - 15839d - 11008e})
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}))
8 4 8 1 3 4 3 2
o15 = map(P3,P2,{-a + 3b + -c + d, -a + -b + -c + 2d, -a + --b + c + -d})
3 3 7 8 5 3 10 5
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 78286520605300ab+79581739036800b2-70396527979275ac-352821337219400bc+250537664673750c2 25163524480275a2+53203330073600b2-80154850155300ac+281104953427200bc-220379661418500c2 9365070707757276606703329588224000b3+12717585227938475671621072487424000b2c+90301487320182767476405218750000ac2-40802884755737386398412425417792000bc2+19552229001659121301306150539012000c3 0 |
{1} | -139838293998488a-136222737738500b+519283566842795c 22682920709349a-111375164651800b-488676782582760c 5314488545470378206348664216861356a2+2456611372245892570233312104535960ab-18431322597668108936785298870189200b2-19981955745489168648437015543608260ac-23767443408368310171363767312644200bc+43917164541246029437397075630677875c2 424896909516a3+628334375256a2b+63494415600ab2-195122547200b3-3124098624060a2c-1211007533400abc+1336826663200b2c+6144909562035ac2-3801439160400bc2-965080128150c3 |
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(424896909516a + 628334375256a b + 63494415600a*b -
-----------------------------------------------------------------------
3 2
195122547200b - 3124098624060a c - 1211007533400a*b*c +
-----------------------------------------------------------------------
2 2 2
1336826663200b c + 6144909562035a*c - 3801439160400b*c -
-----------------------------------------------------------------------
3
965080128150c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.