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,{- 14103a + 13418b + 14313c + 9247d + 2155e, 2605a - 2801b - 8096c + 7157d - 1859e, - 3647a + 11134b - 5504c - 6804d + 8846e, - 2557a + 12361b + 10375c + 11514d + 1964e})
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}))
2 3 10 9 4 6 3 5 1 1
o15 = map(P3,P2,{-a + -b + 2c + --d, --a + -b + -c + -d, -a + 2b + -c + --d})
5 2 7 10 5 7 4 3 6 10
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 1298598260643600ab-4461726431902600b2-610297838918040ac+4838301414850680bc-1333470189715650c2 324649565160900a2-2466796234680400b2-174050210996760ac+2597772833047980bc-660407312127150c2 840539953800774827934287571882250000b3-2062895672182207848213691316642845000b2c+6937595167097906672553093065256960ac2+1517895642480580459472458121686544940bc2-352143075056570344683272398332208950c3 0 |
{1} | 1631631739316019a-1745395666974368b-1658423944791060c 772585204496841a-574133125219598b-1114351298408535c -251070186420654319298159830738808460a2-118520029464586137228721702027282230ab+569390336200348432016985923477932640b2+696340054833837744591038728093246449ac-353992522718254498000267107235085144bc-353198838749326788434981528499784980c2 350173138602a3-1037823046456a2b+110880253316ab2+1152842704888b3-231958948896a2c+1256753891610abc-1743399834540b2c-290049577200ac2+972061168050bc2-258291622125c3 |
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(350173138602a - 1037823046456a b + 110880253316a*b +
-----------------------------------------------------------------------
3 2
1152842704888b - 231958948896a c + 1256753891610a*b*c -
-----------------------------------------------------------------------
2 2 2
1743399834540b c - 290049577200a*c + 972061168050b*c -
-----------------------------------------------------------------------
3
258291622125c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.