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,{- 1334a - 14140b - 757c - 5364d + 2886e, - 4751a - 6763b + 2143c - 3946d - 8746e, - 4413a + 5726b + 11978c - 2052d + 15702e, - 7334a - 14515b - 13832c - 9352d + 5937e})
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}))
2 2 9 8 9 2 9 10 3 7
o15 = map(P3,P2,{-a + -b + -c + -d, 6a + -b + -c + -d, a + --b + -c + -d})
5 3 8 7 7 5 2 9 5 8
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 66732125822040840ab-5995381058641618b2-63816571001463840ac+59151623776154280bc-273706498118139912c2 8007855098644900800a2+200505623667698188b2-14643047626932543840ac-2746228810479141456bc+13835132645827547376c2 223016527094740383803991680593661902920b3-4598543416135635386347447615060940715720b2c+22470228779877785098125638241110371224000ac2+32024710117302120387030774110761954097280bc2-83760359087344885354588988827052978338080c3 0 |
{1} | -298717961929285248a-34665359839237785b+361214105448351339c -2109238981686989964a+940274757335593953b-1687263574115085390c 30150908657472866945739225122069434690560a2+4433636685912493103270165004859300094028ab+1105235929955402867104391213707875042983b2-114031224371232847696427087739517615457592ac-14650010345797297106156853498446018790744bc+97210313727709708229158016869451803533468c2 14045473593446400a3+803481064601040a2b+794354661536016ab2+7711180076971b3-41876992674588960a2c-9543275945177688abc-598260033928872b2c+59344972880678496ac2+6543421756630668bc2-26631551528997264c3 |
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(14045473593446400a + 803481064601040a b + 794354661536016a*b +
-----------------------------------------------------------------------
3 2
7711180076971b - 41876992674588960a c - 9543275945177688a*b*c -
-----------------------------------------------------------------------
2 2 2
598260033928872b c + 59344972880678496a*c + 6543421756630668b*c -
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3
26631551528997264c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.