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,{5353a - 7539b + 6159c + 7396d + 4013e, 6697a + 3502b - 3687c + 1132d - 10226e, - 6559a + 8736b - 10340c - 9499d - 5857e, 9930a + 6793b - 4373c + 6772d + 1192e})
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}))
4 9 7 5 1 8 1 5 1 9
o15 = map(P3,P2,{-a + -b + c + -d, -a + -b + -c + -d, -a + -b + -c + d})
3 7 9 3 9 5 7 2 8 5
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 132676187392035ab-209315906258256b2-88351872814800ac+203986120423512bc-43014211961280c2 41792999028491025a2-75278579360773824b2-43162080746493240ac+38180193957239040bc+19135673752210176c2 169759975299559077071810093084769600b3-668935446824323074787766234084534400b2c-5805058288463019466369157793600ac2+665822054495811402608903703038856960bc2-196872882540364707355601365721886720c3 0 |
{1} | 97752210784011a-142254883351112b+16414044421856c 24730353221432418a-22589127574880265b-30773067934149944c -181863458599770966031101596844901845a2+123805130721566772383699285983958082ab+284121359105865912869853341411146539b2+148564270550060204031410113415713536ac-543272076955339328791224819508464936bc+167438732096429412851459443558845056c2 753473402862075a3-1701647018198190a2b+150284585822427ab2+1102996097051184b3-248656457727720a2c+1275283136616552abc-1567745579177064b2c-90927402650112ac2+592060813976448bc2-186200521114112c3 |
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(753473402862075a - 1701647018198190a b + 150284585822427a*b +
-----------------------------------------------------------------------
3 2
1102996097051184b - 248656457727720a c + 1275283136616552a*b*c -
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2 2 2
1567745579177064b c - 90927402650112a*c + 592060813976448b*c -
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3
186200521114112c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.