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,{6571a - 10565b + 1159c - 3807d - 8369e, 4377a + 8241b - 6283c + 12379d - 12617e, - 13042a - 5879b + 4337c + 4520d + 8862e, 9135a + 10529b + 1350c - 14395d - 13654e})
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 1 1 3 3 5 4
o15 = map(P3,P2,{-a + -b + -c + 3d, 5a + b + -c + -d, -a + -b + -c + -d})
8 5 9 2 7 4 4 8 3
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 3388837662000ab-689361362613b2+6018047098560ac+6245365458528bc-25302636030000c2 116188719840000a2+5048095798809b2-394862385358080ac-55115971117704bc+311464006890000c2 1159888752521249087488855725b3-26140868557083044067045895488b2c+155614169577514069512042946560ac2+187084960505924468687612630928bc2-506894516001229041542171280000c3 0 |
{1} | -17415304438320a-16004442964461b+54610990612156c -171642516600240a+59737627983873b+372619600230892c 70268335638095118795111000000a2+58670289694880468642780851500ab+20166196532530516945844142825b2-649905962826885115482824494320ac-350281226349773927872142923836bc+1312398504522197037423088519456c2 755362602144000a3+477030487096800a2b+152560719959445ab2+14786925274359b3-5321340344361600a2c-2511999527267160abc-388819591186908b2c+12533064631393680ac2+3288930374117584bc2-9869217797568000c3 |
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(755362602144000a + 477030487096800a b + 152560719959445a*b +
-----------------------------------------------------------------------
3 2
14786925274359b - 5321340344361600a c - 2511999527267160a*b*c -
-----------------------------------------------------------------------
2 2 2
388819591186908b c + 12533064631393680a*c + 3288930374117584b*c -
-----------------------------------------------------------------------
3
9869217797568000c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.