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,{13685a - 68b - 2606c - 10000d - 5443e, 10093a - 3395b + 1356c + 12782d + 3788e, 10721a - 385b - 2731c - 4289d - 6885e, 13332a + 14747b - 1580c - 5868d - 1516e})
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 2 1 1 3 9 4 1 4
o15 = map(P3,P2,{3a + -b + -c + d, -a + --b + 3c + --d, -a + -b + -c + -d})
5 9 2 10 10 4 9 4 5
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 25893501193210ab+8012777532780b2-5754105424260ac-38084009977800bc+8067412674720c2 349562266108335a2-4931189226540b2-932191619815440ac-5941801785600bc+623042047031040c2 90848685348499192098453811188756000b3-86378049629595312446210175979704000b2c+18676458176309447332608000ac2+24931185722859884546430432028656000bc2-2271648561562936807432890192288000c3 0 |
{1} | 167257754036439a+5069725725994b-210859461834864c -76508003031192a+3813350168632b+93347446354608c -7186875489999285708043080578372392521a2+1762631898395624010618929773562625624ab+79548409800729858131997893920179236b2+18203837624224651666449232270306993656ac-2233828012345500344950566525992909232bc-11524815955244638837275947162268727104c2 1071910600281a3+68810491494a2b-5692198148ab2-1018528728b3-4217920177248a2c-202194853728abc+7460307648b2c+5535361077696ac2+145492126848bc2-2422370631168c3 |
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 3
o19 = ideal(1071910600281a + 68810491494a b - 5692198148a*b - 1018528728b
-----------------------------------------------------------------------
2 2
- 4217920177248a c - 202194853728a*b*c + 7460307648b c +
-----------------------------------------------------------------------
2 2 3
5535361077696a*c + 145492126848b*c - 2422370631168c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.