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,{- 10967a - 11042b - 15524c - 3693d + 7020e, - 2531a + 14862b - 2823c - 8057d + 9301e, 12034a - 12971b + 4672c + 10483d - 9906e, - 7325a + 186b - 2456c + 5437d - 15978e})
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}))
4 1 10 1 5 1 1 4 5
o15 = map(P3,P2,{-a + -b + c + --d, -a + -b + -c + d, -a + -b + 9c + -d})
5 5 7 6 3 4 3 7 2
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 17456147434880ab+13573869149232b2-9019682730240ac-55715987879664bc+25164388283244c2 21820184293600a2+141599710030800b2-67557543027520ac-110061371999376bc+56084600226900c2 5356822358175634113229172072208b3-8035170524336059734707990805312b2c+14893615464303875803980000ac2+4017566389598831455832483403156bc2-669629103164469387694820061276c3 0 |
{1} | 992015096903005a-1107197842980452b-133700843302978c 2077039686640355a-2423159608129500b-225707609853854c -42912406820245961731522343199800a2+153142132945804383425599353992655ab-121099510807481543136996926750988b2-18219928819567998702004662331985ac+16359092631304848633716162608254bc+2265432667735154667450444760618c2 1766534469875a3-4930603305600a2b+12614893781040ab2-10110485373600b3-4178552900200a2c+5176154568900abc-2368112399520b2c+845975394530ac2+83271697992bc2-148430061864c3 |
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(1766534469875a - 4930603305600a b + 12614893781040a*b -
-----------------------------------------------------------------------
3 2
10110485373600b - 4178552900200a c + 5176154568900a*b*c -
-----------------------------------------------------------------------
2 2 2 3
2368112399520b c + 845975394530a*c + 83271697992b*c - 148430061864c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.