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,{8971a + 6397b - 13944c + 7470d - 4040e, - 3707a + 8028b + 12533c + 11536d - 10322e, - 3855a + 10595b + 9300c + 7441d - 6369e, 9573a - 2209b + 8095c + 6383d - 11082e})
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}))
7 2 2 7 7 9 2 1
o15 = map(P3,P2,{--a + -b + c + -d, 4a + -b + -c + 3d, -a + 6b + -c + -d})
10 5 3 8 3 4 7 2
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 267984253916820ab-45518792495904b2-117034840157130ac+24640939029672bc-11751863516484c2 239271655282875a2-6842666261088b2-50363029549350ac+9572874399648bc-2882873870184c2 903223818310699152002259086976b3-6743670090754053248332109009280b2c+128731499679622343145620711859000ac2-3868741522091929807357202832768bc2-16933613148711236539829789631312c3 0 |
{1} | -954282515604195a+179003716591764b-44268021378686c -368042203631625a+72055286752488b-24828538677796c 12700883474098304144328190150189875a2-5711312008984002976058243557707000ab+669420911816735923656515378735424b2+86107386431966306532422626826700ac-195312751566445994254242569069424bc+27151429505500263823255370561272c2 204684047674875a3-126808838463600a2b+26473945327200ab2-1842167311488b3+5024738341350a2c-5383185287760abc+797089773312b2c+2017357322400ac2-465035911200bc2+147159962656c3 |
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(204684047674875a - 126808838463600a b + 26473945327200a*b -
-----------------------------------------------------------------------
3 2
1842167311488b + 5024738341350a c - 5383185287760a*b*c +
-----------------------------------------------------------------------
2 2 2
797089773312b c + 2017357322400a*c - 465035911200b*c +
-----------------------------------------------------------------------
3
147159962656c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.