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,{4951a - 14093b - 6886c - 3024d - 11691e, - 12524a + 11815b + 3365c - 13785d - 14976e, - 10186a - 4151b - 1688c + 4323d - 4562e, 11600a + 8533b - 11796c + 14165d - 586e})
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}))
9 5 2 3 7 8 2 10 3 1
o15 = map(P3,P2,{--a + -b + -c + --d, -a + b + -c + -d, --a + -b + -c + 2d})
10 9 3 10 4 7 3 9 4 9
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 1733615976067290ab-950139984081372b2-2685663821441790ac+1575878123047788bc-161339931888408c2 14859565509148200a2-4489460840907168b2-23749556123026710ac+14049250261370436bc-1503120834083592c2 251663301146999069956837600554240000b3-934709885374265166099942430604448000b2c-25131843351610696778581702570422960ac2+1078691845654655554610191291236390528bc2-343162739726986250924626842475020684c3 0 |
{1} | 3734419923875280a-2110547423579399b+258927035200689c 34634991084674340a-19681782333932012b+2459264629505461c 310292013882609667665278136122731736400a2-341048451400537389524686775379842238840ab+94679420542302841069834147919648049441b2+27104343166468638584438521622058100080ac-16821281668385652783258228557223069668bc+1256167773554724644150191747395940272c2 204887706138000a3-337488828646200a2b+185582431310885ab2-34061499450846b3+30182615938800a2c-33644290668660abc+9373360194996b2c+1587941399880ac2-889159356456bc2+27983152350c3 |
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(204887706138000a - 337488828646200a b + 185582431310885a*b -
-----------------------------------------------------------------------
3 2
34061499450846b + 30182615938800a c - 33644290668660a*b*c +
-----------------------------------------------------------------------
2 2 2
9373360194996b c + 1587941399880a*c - 889159356456b*c +
-----------------------------------------------------------------------
3
27983152350c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.