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,{6495a - 15328b - 1181c + 12125d - 7168e, - 207a - 3593b - 11160c + 7200d + 4715e, 13551a - 10491b - 4143c + 14856d - 9853e, - 10922a - 14238b - 10043c - 9369d - 3185e})
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 9 5 2
o15 = map(P3,P2,{2a + -b + 4c + d, 2a + --b + 7c + d, -a + 4b + -c + d})
4 10 7 3
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 5367555612900ab-5485989700350b2-5185573529850ac+5227607485485bc+810828169830c2 5367555612900a2-5573653391025b2-8041052670840ac+7805761568640bc+2274621666660c2 231216493555050191354322000b3-2057059103319438934841943300b2c-5131490476866941168299951290ac2+11536641450338447028916031625bc2-6882743822216276961174266010c3 0 |
{1} | -49471945857878a+26241594285065b+22495923534798c -79780697700952a+43720488681160b+34226976233457c 342644958197973662742712181396a2-384109886564991243425809259960ab+117049331892824118440428652175b2-130667618425568637473754473822ac+55094926982968459733366572835bc+2291723375302799827102134351c2 96333260360a3-206449794540a2b+143880857550ab2-33951867125b3-38035790856a2c+49741725180abc-14027323800b2c+18590885082ac2-7586595765bc2-8495356086c3 |
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(96333260360a - 206449794540a b + 143880857550a*b -
-----------------------------------------------------------------------
3 2 2
33951867125b - 38035790856a c + 49741725180a*b*c - 14027323800b c +
-----------------------------------------------------------------------
2 2 3
18590885082a*c - 7586595765b*c - 8495356086c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.