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,{- 11600a - 89b - 8421c - 5651d - 13068e, - 857a - 10194b - 6c + 3569d + 11111e, - 4587a - 14840b - 14368c - 1518d + 2481e, - 8267a - 14759b + 10736c - 3608d - 6374e})
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}))
3 9 5 9 3 7 2 3 3
o15 = map(P3,P2,{-a + 3b + -c + -d, 4a + -b + -c + -d, -a + -b + -c + d})
8 8 7 7 4 6 5 7 2
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 51277252992ab-5836268448b2-602335749120ac-8915200920bc+756744372000c2 957175389184a2+148662359856b2-249959082240ac-1635387885240bc+880712074500c2 2425311778200746078445984b3-48586677317542448357429520b2c+43087634724505346539200000ac2+281456982250241381242495200bc2-421608526503138817508484000c3 0 |
{1} | 74627866880a+487177391226b-972569960085c -1185864322048a+512220333678b+764537434705c -20976934510722260856762368a2-157183885259153384405458272ab-7873005136864842390496788b2+456922283801554615699541040ac-219108590200429593002958900bc+206404478471913881241492975c2 6456082497536a3+47641751861760a2b-1996686901440ab2+6816018755160b3-150381333607680a2c+1349666327520abc-84260273912820b2c+150640755652800ac2+148115112212250bc2-129706808020875c3 |
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(6456082497536a + 47641751861760a b - 1996686901440a*b +
-----------------------------------------------------------------------
3 2
6816018755160b - 150381333607680a c + 1349666327520a*b*c -
-----------------------------------------------------------------------
2 2 2
84260273912820b c + 150640755652800a*c + 148115112212250b*c -
-----------------------------------------------------------------------
3
129706808020875c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.