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,{- 3961a - 8328b - 7631c + 6726d + 13730e, 9579a + 15708b - 12559c - 12632d + 11726e, 7747a + 9839b + 11837c + 9865d + 8589e, 515a - 12524b - 1579c + 15164d - 14808e})
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}))
4 1 5 9 4 3 9 1 8
o15 = map(P3,P2,{-a + -b + 5c + 3d, -a + --b + -c + -d, -a + 2b + -c + -d})
7 2 8 10 3 2 4 3 7
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 886979537418870ab-3207406586271300b2-211956468770739ac+1557421146271200bc-193875207023127c2 620885676193209a2-8981223541233300b2-219584313795462ac+4991775689050710bc-677888317580082c2 54197741592302780649321837730548000b3-25108141374343171405247026763979900b2c-213915166161351678316541011103256ac2+1398264992574602183284450701177660bc2+441631610397828784555941420342717c3 0 |
{1} | -142241475779669a+1325281887606510b-22639463352387c 1683259149498164a-3045009669120b+2323569368155707c -6257355121806385602353283472737991a2-70110043042036166966189842543931480ab+211012052544982311001362879475153500b2+32075524776435656257611798072301248ac-222278809046678103124814869801190490bc+36134710563249189310202674476662958c2 1082776497313a3+8216456108770a2b-81887736200900ab2+146049780267000b3-4584800484523a2c+69689920693540abc-175030599741900b2c-11282326838909ac2+56774615591190bc2-5711207467119c3 |
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(1082776497313a + 8216456108770a b - 81887736200900a*b +
-----------------------------------------------------------------------
3 2
146049780267000b - 4584800484523a c + 69689920693540a*b*c -
-----------------------------------------------------------------------
2 2 2
175030599741900b c - 11282326838909a*c + 56774615591190b*c -
-----------------------------------------------------------------------
3
5711207467119c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.