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,{- 249a - 5385b - 5756c + 8895d + 11949e, 8077a + 3350b + 9087c - 133d - 10894e, - 9081a - 6123b - 6013c - 12528d + 3887e, 3178a - 13128b - 13246c - 5205d + 2052e})
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 9 10 5 1 7 10 1
o15 = map(P3,P2,{-a + -b + --c + -d, a + b + -c + -d, 2a + --b + -c + d})
3 7 3 6 4 9 7 7
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 968782248ab-3855990096b2-438680340ac+3033434628bc-590812110c2 10172213604a2-1080350940816b2-2990726172ac+1075696619760bc-272332340091c2 28480166968689720345600b3-42522466436040214108800b2c-2558930678094228480ac2+21163194186415621925760bc2-3508868913052227288480c3 0 |
{1} | -310364502a+2364795324b-1278885587c 6163090416a+226146849408b-96374623604c -100970434639935037548a2+176493267622997170272ab-3944716689719842996752b2-165728184147989240736ac+3254313263354632610152bc-676917918194242614813c2 1475695368a3-8453099088a2b-65429900064ab2+786572617536b3+4383136044a2c+78084551664abc-1340485263504b2c-23235860466ac2+744952370652bc2-136179961267c3 |
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(1475695368a - 8453099088a b - 65429900064a*b + 786572617536b +
-----------------------------------------------------------------------
2 2 2
4383136044a c + 78084551664a*b*c - 1340485263504b c - 23235860466a*c +
-----------------------------------------------------------------------
2 3
744952370652b*c - 136179961267c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.