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,{- 15354a - 3997b - 13560c + 3756d - 15942e, - 10216a + 7332b - 4077c - 14676d + 7245e, 6393a + 8864b - 393c - 5985d - 11978e, 9039a + 13994b + 1128c - 1359d - 10693e})
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 1 7 3 3 10 3 5
o15 = map(P3,P2,{-a + -b + -c + -d, -a + --b + 6c + 10d, 4a + 10b + -c + -d})
7 9 5 7 2 9 8 7
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 17204144045544288ab-5774322894024384b2-2697657876908708ac-1123598094985912bc+563024846132896c2 13440737535581475000a2-1937575103660291808b2-2263935745906740500ac-156337878834790744bc+306777496930334752c2 694359225120095489221264260359224160256b3-351375813382031369571119806056432780288b2c+313685062724422837245079955179150000000ac2-228461233504184332250300164823943107392bc2+42453396786506386581634399345221063424c3 0 |
{1} | 99330069002847550a+45371082993090401b+22377097020756100c 91674030280511668750a+14784403569439816685b+8132766901613167540c 127245753502655754305705995295333700468750a2-49490365429220449185929500418415782671875ab-4734577328292792561845847316643251633170b2+45810743672410391062924419613377866050000ac-2848123353062908423820456068337643275560bc+643420092155774583104217813744492628480c2 599311116122031250a3-434243384359640625a2b+37053446536798860ab2-1140185501758564b3+145080950716100000a2c-51747054806337520abc+5890641457374128b2c-22383629337811840ac2+5043115665480256bc2+1026110944257280c3 |
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
o19 = ideal(599311116122031250a - 434243384359640625a b +
-----------------------------------------------------------------------
2 3 2
37053446536798860a*b - 1140185501758564b + 145080950716100000a c -
-----------------------------------------------------------------------
2 2
51747054806337520a*b*c + 5890641457374128b c - 22383629337811840a*c +
-----------------------------------------------------------------------
2 3
5043115665480256b*c + 1026110944257280c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.