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,{12383a - 8968b + 11714c - 2292d + 6850e, 3201a + 7871b + 281c + 845d - 3251e, - 4480a + 14231b + 1989c + 14400d + 10043e, 12626a - 4513b - 5162c + 15688d + 9868e})
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 4 3 5 4 1 3 1 5 3
o15 = map(P3,P2,{-a + -b + -c + -d, 7a + -b + -c + -d, -a + b + -c + -d})
6 3 8 2 5 6 5 4 9 2
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 83355292469340ab-5964391956375b2-922693580046864ac-47613746597760bc+1305086168219760c2 200052701926416a2-1044204415425b2-1013205805921728ac+34910824205880bc+995698071151920c2 234234354175335482795908573500b3-19520433405586704430068458497200b2c+1330513812454929729686830954708992ac2+444393171717839206541472081948480bc2-6715947602492918460612768353921280c3 0 |
{1} | 1039251928305792a-164113357087295b-1370692096995036c 333841147215132a-61733987518265b-346489719098700c 29293528309377421920719030106422256a2-6450662843774918902308876397155000ab+394043452567313093356042426418675b2-103662162686478478582524086168705928ac+10401399399192619529775135984452610bc+97961857744490776816535794097188872c2 14752787625792a3-4304297343600a2b+429432607500ab2-13967457125b3-72318348435744a2c+13509029828880abc-660207186450b2c+118372681905408ac2-10444405524480bc2-64757138044320c3 |
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(14752787625792a - 4304297343600a b + 429432607500a*b -
-----------------------------------------------------------------------
3 2
13967457125b - 72318348435744a c + 13509029828880a*b*c -
-----------------------------------------------------------------------
2 2 2
660207186450b c + 118372681905408a*c - 10444405524480b*c -
-----------------------------------------------------------------------
3
64757138044320c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.