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,{14928a - 13794b - 13481c - 14821d - 8341e, 4303a - 12263b + 14011c + 10640d - 3840e, 11290a - 5393b + 6053c - 1286d + 5338e, - 10272a - 4205b - 8950c + 4640d - 4957e})
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}))
2 3 7 5 1 8 6 3
o15 = map(P3,P2,{3a + -b + -c + 7d, --a + -b + -c + d, -a + 4b + -c + -d})
7 2 10 3 3 5 5 8
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 8316321691776ab-62954629044480b2-3818381493752ac+55074253910640bc-11707593957360c2 4851187653536a2+2722282661001600b2-28898247665560ac-1455138165138000bc+152690289444000c2 621506298491181938110283771904000b3-679983058667648604806497890662400b2c-37188146020963025950193920000ac2+241324037424416806927966504041600bc2-27401614377478715961706902690600c3 0 |
{1} | -7102109547771a+53365396017990b-21049241004486c 98398264275913a-3398093483473890b+931673113635780c 858198771172500529428280609314a2+20431192893085202750660946141432ab-752728480617545138698342534083960b2-14627994338912732952400201262980ac+594152052597295158304533415627560bc-111767994098056192027013932496625c2 313799249631a3+5095175648598a2b-137713748952780ab2+625254956368200b3-4361270879840a2c+158022663281280abc-1009557273686400b2c-38556799394400ac2+476493928851600bc2-70072576156800c3 |
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(313799249631a + 5095175648598a b - 137713748952780a*b +
-----------------------------------------------------------------------
3 2
625254956368200b - 4361270879840a c + 158022663281280a*b*c -
-----------------------------------------------------------------------
2 2 2
1009557273686400b c - 38556799394400a*c + 476493928851600b*c -
-----------------------------------------------------------------------
3
70072576156800c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.