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,{- 7598a + 11297b + 452c + 8729d - 10326e, - 8368a - 14443b + 12530c - 14013d - 729e, - 14755a - 15028b + 13176c - 12902d + 15357e, 1400a - 13022b - 2793c + 172d - 6230e})
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}))
8 7 3 1 4 1 3 9 7
o15 = map(P3,P2,{-a + -b + -c + -d, -a + b + 4c + -d, a + -b + -c + -d})
9 4 2 4 3 2 2 4 6
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 44482824ab+19837674b2-43672464ac-85924932bc+65399184c2 533793888a2-111128454b2-1170437760ac+204063024bc+544103712c2 105309073780654968b3-417821047353840744b2c+40982973225349536ac2+569617304957311128bc2-302747584624042656c3 0 |
{1} | 3117252a+54569215b-55490202c 626064420a-301284803b-454027200c 793232248164184152a2+256247077963029978ab+196253810130336329b2-1640579719946973768ac-52029225290985052bc+415860553392993176c2 224442144a3+172596960a2b+85775346ab2-11789197b3-799468704a2c-323807976abc+56794968b2c+791948016ac2-126879804bc2-67363056c3 |
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(224442144a + 172596960a b + 85775346a*b - 11789197b -
-----------------------------------------------------------------------
2 2 2
799468704a c - 323807976a*b*c + 56794968b c + 791948016a*c -
-----------------------------------------------------------------------
2 3
126879804b*c - 67363056c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.