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,{11841a + 6907b - 1285c - 1217d + 13328e, - 14554a - 15469b - 1853c - 13928d + 10035e, - 1332a - 1985b + 1192c - 53d - 7859e, 14084a - 5401b - 1508c + 2384d + 3386e})
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 1 2 5 10 3 1
o15 = map(P3,P2,{4a + -b + -c + -d, 2a + -b + --c + 2d, 3a + -b + -c + d})
6 3 9 4 7 8 3
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 5932726443ab-4712614641b2-5731516602ac-1034693856bc+5152778820c2 847532349a2+547384635b2-1843406280ac-868739508bc+1287028476c2 76948980624835031893824b3-150062972279033582955648b2c+2490766032475240128000ac2+96655066861439654706432bc2-23862776234270358852096c3 0 |
{1} | 10421940348a+19234908952b-26381834144c -171669624a-1363231499b+1393316194c -105867312293067665915196a2-291692249148324926919892ab-252141983310034203478309b2+410340529837440680224152ac+646557142343116519097716bc-426204937744985894881604c2 322328052a3+632059680a2b+473788203ab2+149780211b3-1552910544a2c-2217555228abc-948618146b2c+2587846236ac2+2048672948bc2-1501032824c3 |
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(322328052a + 632059680a b + 473788203a*b + 149780211b -
-----------------------------------------------------------------------
2 2 2
1552910544a c - 2217555228a*b*c - 948618146b c + 2587846236a*c +
-----------------------------------------------------------------------
2 3
2048672948b*c - 1501032824c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.