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,{- 3626a + 9690b - 9520c - 8319d - 13301e, 15037a + 10628b - 4674c - 14136d + 7968e, 2553a - 8891b + 7582c - 1676d - 1439e, - 3482a + 13363b - 5632c - 8168d + 104e})
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 2 10 1 3 3 2 4
o15 = map(P3,P2,{-a + 2b + -c + --d, -a + -b + -c + d, -a + -b + 4c + 3d})
4 7 7 5 5 8 5 7
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 1221284471120ab-11247672260160b2-842209433576ac+7589045013830bc-258956948075c2 20517579114816a2-1372266951686400b2-158533856709200ac+1392925352078400bc-58344619444475c2 32632243447542342761633257600b3-23860736470273103304287889120b2c+557799710607904061567010048ac2-993764099889042899823651840bc2+639911640920304238382734600c3 0 |
{1} | -8728194192732a+25124602919470b-6659942473975c -1545450173246280a+4168262400299040b-1201441921345855c 13002411425647454143637993640a2-54719110227632214452101550880ab+24908751967412069309200176800b2+5741634825948892997211699411ac+14015458222969251713285162640bc-1831443454939528542108682350c2 214329697344a3-2875894909440a2b+12561836057600ab2-14847357180800b3+65256251760a2c-676830362880abc+1209563264000b2c-183113232015ac2+652858777200bc2-92694439250c3 |
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(214329697344a - 2875894909440a b + 12561836057600a*b -
-----------------------------------------------------------------------
3 2
14847357180800b + 65256251760a c - 676830362880a*b*c +
-----------------------------------------------------------------------
2 2 2 3
1209563264000b c - 183113232015a*c + 652858777200b*c - 92694439250c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.