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,{3122a - 873b + 1793c + 15013d - 2890e, - 8709a + 10991b + 8355c - 10675d + 3323e, - 7557a + 10765b - 13422c - 1484d + 4230e, - 8037a - 14778b - 4123c - 1800d - 6418e})
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 5 1 10 8 2 8
o15 = map(P3,P2,{-a + 2b + -c + -d, a + 2b + --c + 2d, a + -b + -c + -d})
8 4 3 7 3 7 9
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 12734904ab-813792b2-10792152ac-5360355bc+4959927c2 101879232a2+3133452b2-135087120ac-10135944bc+51635367c2 1292663252790b3-5677839304578b2c+486915819264ac2+7721146760418bc2-3640293095670c3 0 |
{1} | -15711760a+4996800b-1330734c 30444672a+25356104b-75900600c 45775770098240a2-545388835264ab+8303992163012b2-44468716154832ac-22832436243888bc+20126022668469c2 21859840a3-1657344a2b+4743744ab2-495616b3-31620672a2c-12122496abc-264024b2c+21872808ac2+3565944bc2-4202739c3 |
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 2
o19 = ideal(21859840a - 1657344a b + 4743744a*b - 495616b - 31620672a c -
-----------------------------------------------------------------------
2 2 2 3
12122496a*b*c - 264024b c + 21872808a*c + 3565944b*c - 4202739c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.