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,{11370a - 1108b - 1131c - 14993d + 8166e, 339a + 9802b - 5800c + 3237d - 3416e, 2747a - 10137b + 13528c - 893d + 15183e, - 9600a + 5959b + 12615c + 2391d - 14685e})
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 1 10 3 4 1 3 10 5
o15 = map(P3,P2,{2a + -b + -c + --d, -a + -b + -c + 2d, -a + b + --c + -d})
3 2 7 2 3 8 2 7 2
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 52815637161ab-89514010404b2-55182969849ac+118076502012bc-25406048024c2 17605212387a2-38037379632b2-51684794504ac+95025005152bc-19372721536c2 660645486945716338285664256b3-1760909263982857176625317888b2c-4297751049944563521171456ac2+1557073071328343592478328832bc2-451078959557943336110446592c3 0 |
{1} | 57795650360a-5142325162b-39855607222c 88470262866a-58496544696b-25051780448c -1015498091938913877358847531a2+3032645755202169550815298712ab-2930305328206278778996521392b2-718842922073407015270711108ac+2780742562856055980891764496bc-1001672888054528396883345568c2 4334673777a3-20291483532a2b+37533480240ab2-21473724480b3+5083277668a2c-28088277280abc+24248225344b2c+7691690496ac2-9439524864bc2+577860864c3 |
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(4334673777a - 20291483532a b + 37533480240a*b - 21473724480b +
-----------------------------------------------------------------------
2 2 2
5083277668a c - 28088277280a*b*c + 24248225344b c + 7691690496a*c -
-----------------------------------------------------------------------
2 3
9439524864b*c + 577860864c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.