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,{9829a - 13115b + 4775c - 15284d + 5816e, 4416a + 15114b + 3028c - 2792d + 14471e, - 10032a + 8776b - 5774c - 2618d - 11245e, 7126a + 14345b + 6076c + 12244d - 9272e})
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 4 5 7 2 6 7 1 5 9
o15 = map(P3,P2,{-a + -b + -c + 2d, -a + -b + -c + d, -a + -b + -c + --d})
2 3 2 4 3 7 4 3 2 10
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 14910701470ab+6621861470b2-15525839630ac-24091032840bc+17996432650c2 1356873833770a2+76979333690b2-2481102477560ac+333288055660bc+719504978850c2 294903184053306697897770b3-268336739536267622371830b2c-7089110188318861859520ac2-325756012495525568819730bc2+305265948139902660601950c3 0 |
{1} | -15342953678a+3547784751b+28606876265c -1347359005291a-613891083799b+1541511780285c 1194887714230602159186601a2-1960029297847540759762585ab-402669742562498540616221b2-1834093459593002684006697ac+3042260867207892126217422bc+130763780342309543982495c2 9032774023a3-10805408432a2b-7330229284ab2-1882274926b3-24282041420a2c+34229689340abc+9377243680b2c+14680434900ac2-26150757150bc2+3106135500c3 |
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(9032774023a - 10805408432a b - 7330229284a*b - 1882274926b -
-----------------------------------------------------------------------
2 2 2
24282041420a c + 34229689340a*b*c + 9377243680b c + 14680434900a*c -
-----------------------------------------------------------------------
2 3
26150757150b*c + 3106135500c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.