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,{10251a + 13010b - 10446c + 7292d - 11582e, 10505a - 14875b - 2827c + 3012d - 6844e, - 6245a - 10604b - 10242c - 4304d + 5997e, - 2102a - 4575b + 10193c - 10701d + 6558e})
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}))
3 1 5 4 7 7 9 4 5
o15 = map(P3,P2,{-a + -b + -c + -d, -a + -b + -c + -d, a + 9b + -c + 3d})
2 9 2 3 4 2 5 5 8
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 43942600947240ab-48334967683600b2-18627572658870ac+39568930194140bc-8627757806920c2 19774170426258a2-23946928542600b2+2367892767546ac+16388456814460bc-3386053373992c2 942578819694355830966717952000b3-4893842880983552703620023142400b2c-10381881038012519617378054944000ac2+19815597110203566242026676102400bc2-9168712924624769624230533913600c3 0 |
{1} | 8695076297229a-78326419074280b+30774883422908c -27166770011907a-60163602901320b+15398602929148c 167210196811544509467349298822217a2-161374400019769687146611584415730ab+914238665324265524723824401269200b2+47100501004956760315023458531976ac-623334917918003159221252801491000bc+92859808291579710302343830988624c2 101004041313a3-208578996540a2b+659359743300ab2-606436116000b3+72306125172a2c-449610412320abc+648707862000b2c+56324159280ac2-210782813120bc2+16830305216c3 |
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(101004041313a - 208578996540a b + 659359743300a*b -
-----------------------------------------------------------------------
3 2 2
606436116000b + 72306125172a c - 449610412320a*b*c + 648707862000b c +
-----------------------------------------------------------------------
2 2 3
56324159280a*c - 210782813120b*c + 16830305216c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.