Description
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, {107a + 4376b - 5570c + 3187d + 3783e, - 5307a + 8570b - 15344c + 8444d - 10480e, 10359a - 7464b - 8251c + 2653d + 5071e, - 6203a + 12365b - 13508c - 9480d - 11950e})
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 4 4 9 8 10 3 1 7 7 5 1
o15 = map (P3, P2, {-a + -b + -c + -d, -a + --b + -c + -d, --a + -b + -c + -d})
9 7 7 7 7 7 7 3 10 4 6 5
o15 : RingMap P3 <-- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 535385198313ab-695748977868b2-493315895835ac+1151963974140bc-509732823600c2 688352397831a2-902430802056b2-1830852961245ac+3042226879860bc-1346494597200c2 29088245077146491639979555b3-143608211076624058241486520b2c+5607746090069702101228125ac2+229835668530158883094188300bc2-126157077745523335926798000c3 0 |
{1} | -75747159432a-120614807243b+279465785185c -1103453648829a+1601687633644b+845235467735c 10893884734768346843578641a2-91927878026702297684757261ab+201706160712458664246153855b2+34169553202851256473159660ac-185460283186370525276628580bc+58137602981654014626547225c2 205355516247a3-1999754948079a2b+6131787984357ab2-4817629849725b3+840072427740a2c-6298181745660abc+7389192046320b2c+2186513229600ac2-3368812507200bc2-214825024000c3 |
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(205355516247a - 1999754948079a b + 6131787984357a*b -
-----------------------------------------------------------------------
3 2
4817629849725b + 840072427740a c - 6298181745660a*b*c +
-----------------------------------------------------------------------
2 2 2
7389192046320b c + 2186513229600a*c - 3368812507200b*c -
-----------------------------------------------------------------------
3
214825024000c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.