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,{12066a - 8407b + 15585c - 12790d + 1140e, 6299a + 10424b - 3738c + 6199d + 11774e, - 12410a - 9799b + 4878c + 14070d - 14100e, 12115a - 7680b - 733c - 2896d + 12203e})
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}))
6 6 1 2 7 3 1 2 4
o15 = map(P3,P2,{7a + 5b + -c + -d, 10a + -b + -c + --d, -a + -b + -c + -d})
5 5 6 3 10 5 8 5 5
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 22083620653260ab-18541307515950b2-393027948681000ac+314265552060300bc+203892716570000c2 220836206532600a2-48814738394670b2-4856159120103000ac+2175381520711500bc+3900219165370000c2 160169296703633349461528483453760b3-5305603330344795275581107692688000b2c+1137089361119071567020096000000000ac2+45480173809808596588314813544800000bc2-39014758279058523117990706360000000c3 0 |
{1} | 313172545849143a-181989719054055b-233815586714830c 3540616053695265a-992008048625991b-3626074635157550c -4694172147017041976848721357932335a2+4093717099687307283348013459767480ab+803369412283586567542671310137648b2+8936475477197692786432038245237200ac-34072989737059479199663203964766400bc+22157975765721636403012214715704000c2 374235266898a3-640571190309a2b+391050502056ab2-51456635856b3-978823398420a2c+771291787920abc-757355443776b2c+1135439608000ac2+667606046400bc2-933976896000c3 |
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(374235266898a - 640571190309a b + 391050502056a*b -
-----------------------------------------------------------------------
3 2 2
51456635856b - 978823398420a c + 771291787920a*b*c - 757355443776b c +
-----------------------------------------------------------------------
2 2 3
1135439608000a*c + 667606046400b*c - 933976896000c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.