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,{- 6811a + 2635b + 2203c - 14432d - 6758e, - 5365a - 6803b + 15472c - 9336d + 4098e, - 8700a + 2771b - 13911c + 313d + 168e, - 9325a - 14378b - 600c + 12729d + 6878e})
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}))
1 5 7 3 1 8 4
o15 = map(P3,P2,{-a + 5b + -c + d, -a + -b + c + -d, -a + -b + c + d})
3 2 9 5 2 7 3
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 38865643760ab+435287053200b2-24705564506ac-577639878270bc+191001416263c2 3886564376a2-36011899800b2-12171459482ac-99548256990bc+87646623057c2 1834145675036595753272064000b3-3754949441441291087033894400b2c+120904252336891582044808ac2+2564230994634040777143573720bc2-584138822271059687904748084c3 0 |
{1} | -18637650332a+163757364580b-68971529760c 10569116716a+75878549820b-69093016132c 7571236013856773847209548a2-108384682996780201026528240ab+1053295607485675215301865700b2+35640648994985212709818140ac-1097193641924126753057354520bc+307624947262643397009829283c2 64157144a3-199883640a2b+6092298600ab2+34477011000b3-194767020a2c-9324049560abc-53219936700b2c+4003147218ac2+29392060530bc2-6334317857c3 |
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 3
o19 = ideal(64157144a - 199883640a b + 6092298600a*b + 34477011000b -
-----------------------------------------------------------------------
2 2 2
194767020a c - 9324049560a*b*c - 53219936700b c + 4003147218a*c +
-----------------------------------------------------------------------
2 3
29392060530b*c - 6334317857c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.