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,{6302a + 9490b + 3856c + 14217d + 4020e, - 13916a + 6190b + 693c + 8988d - 9974e, 15898a + 2968b + 12363c + 1746d + 6442e, 14975a - 14121b + 1363c + 8738d + 1023e})
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}))
1 10 5 1 1 7 9 1 5 1
o15 = map(P3,P2,{-a + --b + -c + d, 2a + -b + -c + -d, -a + -b + -c + -d})
2 9 7 3 2 9 2 2 4 7
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 409758426ab-387260433b2-190832544ac+381520818bc-92100288c2 2816679420324a2-13645893339729b2-1224175840776ac+17813030235336bc-5120171504812c2 47046089980263048227784b3-81266367360367604594088b2c+46102755735601663392ac2+43919630374756737479760bc2-7602589058271147741024c3 0 |
{1} | -278865414a+257669973b-134587028c -4503120658470a+12277095174531b-9151362188818c -8102733826490214514218a2+18863506523467085716467ab-44818894088753209826445b2-5275776553594612906806ac+52225382299016355156288bc-14395694369542910286556c2 1117054599444a3-3656272433688a2b+4222555196265ab2-1761030487449b3+1275730867152a2c-3594998964060abc+2561963195898b2c+808514276724ac2-1287665118972bc2+218631865592c3 |
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(1117054599444a - 3656272433688a b + 4222555196265a*b -
-----------------------------------------------------------------------
3 2
1761030487449b + 1275730867152a c - 3594998964060a*b*c +
-----------------------------------------------------------------------
2 2 2
2561963195898b c + 808514276724a*c - 1287665118972b*c +
-----------------------------------------------------------------------
3
218631865592c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.