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,{- 7906a - 12921b + 14118c - 7681d + 1660e, - 14782a - 11593b - 12287c + 14172d - 2764e, - 15773a + 6342b + 2583c + 4296d + 4482e, 1143a - 9267b + 12056c + 5338d + 6602e})
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 10 5 1 1 4 9 3 6 6 5
o15 = map(P3,P2,{-a + --b + -c + 8d, -a + -b + -c + -d, -a + -b + -c + -d})
6 9 6 5 4 7 4 2 5 5 6
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 235506571847568ab-315171328497480b2-30513239197182ac-100879447638480bc+18363254486850c2 1362573737118072a2-8269138824496200b2-2137028199781230ac+2239610458192800bc+686252227484250c2 22953648431193674339167002252228600000b3-9173252160197645709867544344366315000b2c+48558149871237105685340530953870ac2+1221910075722630571000343871897928800bc2-54287720609270046362625812919606450c3 0 |
{1} | -212668086172635a-328695343593484b+212596882366086c -15182309245308831a+35272932766857940b+7587830843673990c -4845412072126175191536128218106918832a2+45813905673339897904386838431219885576ab-148925692570548190370456622524086525920b2+1781222817151178628757160569192512675ac+1837724469103832304698945745691005020bc-810712645611244924899883142248069110c2 49930768057923a3-538921758270294a2b+1952846918778420ab2-2383891419856800b3-48400540989075a2c+347397084101970abc-595434721284200b2c+15498014193900ac2-63460524645000bc2-1129965256350c3 |
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(49930768057923a - 538921758270294a b + 1952846918778420a*b -
-----------------------------------------------------------------------
3 2
2383891419856800b - 48400540989075a c + 347397084101970a*b*c -
-----------------------------------------------------------------------
2 2 2
595434721284200b c + 15498014193900a*c - 63460524645000b*c -
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3
1129965256350c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.