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,{- 11432a + 6161b + 9946c - 9955d + 13367e, - 10344a + 3595b - 6227c - 6898d - 7708e, - 1670a + 4467b + 12401c - 1059d + 9169e, - 7791a + 4214b + 6048c + 14259d + 1476e})
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 10 2 1 1 3 5 6 6 4
o15 = map(P3,P2,{-a + 2b + --c + d, -a + -b + -c + -d, -a + -b + -c + -d})
5 7 3 4 4 4 3 5 7 5
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 33616606310ab-575288735168b2+33170176375ac+832374455710bc-246497602925c2 84041515775a2+11284822253824b2-942937229325ac-16474188432800bc+4896046082200c2 1456535640163170809865000000b3-4746647017736253464478000000b2c+165132645988649002803657000ac2+4430729163236474475036710400bc2-1125862340945870047866003000c3 0 |
{1} | -45237927080a+302742956594b-314542725275c 538382232845a-5856787453856b+6159809593160c 4588812278847871250577325a2+69664792283698669738380820ab-763176056730027375195711344b2-189134180391001616729826550ac+2184976876055228590882480000bc-1440168506774685324771354500c2 4874776625a3-9417032700a2b+4989598800ab2-60336311744b3+2098146375a2c+1469805300abc+151677294480b2c-4481351250ac2-119408563200bc2+27743598500c3 |
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
o19 = ideal(4874776625a - 9417032700a b + 4989598800a*b - 60336311744b +
-----------------------------------------------------------------------
2 2 2
2098146375a c + 1469805300a*b*c + 151677294480b c - 4481351250a*c -
-----------------------------------------------------------------------
2 3
119408563200b*c + 27743598500c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.