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,{3016a - 2761b + 3886c + 14018d - 2623e, 9225a + 11354b - 13067c + 14700d + 12345e, - 1031a + 3152b - 14053c - 9086d + 13398e, - 10724a - 8614b + 12986c - 9248d + 9009e})
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}))
7 5 1 1 5 1 5 8 9 8 7 1
o15 = map(P3,P2,{-a + -b + -c + -d, -a + -b + -c + -d, -a + -b + --c + -d})
3 7 5 8 3 3 3 5 7 5 10 8
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 1506843225034200ab-234906774438240b2-21251063190000ac-4555679096593200bc+2793904760815000c2 1953315291711000a2-61771992738240b2-3565942035621000ac-4271999330816200bc+5679754947125000c2 200543283353252815515480312423360b3+41859893172715652765326447697448600b2c+49618435872779551596607031290005000ac2-171944243616047749485992028810099000bc2+62064876700226542492216043001125000c3 0 |
{1} | -425949932642925a+100235241003684b+8485385270582200c 483518395974670a-33052885978816b+7531456859737375c -7742855612825851826120665619163915a2-6165335571552979323031700483697090ab+747545040493091638902791269944624b2+145050749325214115636388587606701575ac-79387222828578615219920179260760845bc+186575619440386974933791459019907250c2 894942329715a3+573092341542a2b-204046561152ab2+15011618400b3-19473068417565a2c+8265646114320abc-716638227408b2c+36430117533425ac2+1373584930050bc2-29206380578375c3 |
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(894942329715a + 573092341542a b - 204046561152a*b +
-----------------------------------------------------------------------
3 2
15011618400b - 19473068417565a c + 8265646114320a*b*c -
-----------------------------------------------------------------------
2 2 2
716638227408b c + 36430117533425a*c + 1373584930050b*c -
-----------------------------------------------------------------------
3
29206380578375c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.