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,{13098a - 15449b + 8421c + 4103d - 9360e, - 3417a - 6109b + 15530c + 2808d - 3415e, - 13510a - 15153b + 11258c + 14130d + 6315e, 12290a + 12045b + 4882c + 8292d - 8308e})
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}))
5 10 1 9 1 7 2
o15 = map(P3,P2,{-a + b + 6c + --d, 9a + -b + -c + -d, -a + -b + 2c + 5d})
8 9 4 2 7 6 3
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 520831926256572ab+110796112297080b2-4266749603364696ac-2021145461276823bc+9131535365831046c2 4687487336309148a2-515911094845020b2-2466668988906003ac+8295548547577158bc-33316088329064331c2 329437653204994683976663654882635b3-7640202115854430899467422420042074b2c+94434190050642595633980297068544ac2+59075105561804048117886652027631652bc2-152340660741708327389644794341807928c3 0 |
{1} | 10049215663896948a+5216207651160135b-43027288987266578c 44296624008191835a-19155953493698490b+157684426814309551c 98080232494196431378471891961614581a2-17991565028130137479476836166637425ab+9943477233026243806555776113040495b2+131656358067798988839258752336762292ac-169795396563964446832688610567913293bc+724047197464703374066854022564306171c2 283520191144428a3+8304842797020a2b-694352277360ab2-3422871045125b3-225557622833571a2c-23074498431621abc+55865347582905b2c+238569971109624ac2-223251276384297bc2-38520237296421c3 |
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(283520191144428a + 8304842797020a b - 694352277360a*b -
-----------------------------------------------------------------------
3 2
3422871045125b - 225557622833571a c - 23074498431621a*b*c +
-----------------------------------------------------------------------
2 2 2
55865347582905b c + 238569971109624a*c - 223251276384297b*c -
-----------------------------------------------------------------------
3
38520237296421c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.