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 which generate
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,{- 2136a + 9349b + 8735c - 5609d - 9489e, 13529a - 15802b - 371c - 545d - 2519e, - 11250a - 14212b - 1270c - 1415d + 626e, 1414a - 3327b - 4035c + 11874d - 13874e})
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 9 2 7 7 6 3 1 10 7 5
o15 = map(P3,P2,{-a + -b + -c + -d, -a + -b + -c + -d, --a + -b + -c + 8d})
4 2 5 6 6 5 4 4 3 4 6
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 821503652433360ab-2378892593212200b2-180967843812504ac+659996719611660bc+20456077852224c2 1643007304866720a2-21543463552689000b2+502985806533000ac+3727516078503900bc+914776078031280c2 4618110968829401670197894687164320000b3-3163611576589491698363463849180648000b2c+59947856690696232009186202915649040ac2+125160281453177557124256750158202600bc2+123254365576028904442854898163719860c3 0 |
{1} | 1418396941154817a-5265014690542360b-171202052055577c -6579999349220375a-29839829515772016b-7056057635649837c 1686927222636153255177394681421200005a2+2536043553776579117535040486969212200ab-388195627484498156269005416257189380b2-828317411634297959485490273243340420ac-1183265507822376924552857453091129228bc-952957461124128652576034889254455386c2 287958450810a3-400961287925a2b-2680942780560ab2+5244174813700b3-93738370860a2c+911678082564abc-1609862603460b2c+8536409703ac2-23117264574bc2-1516413743c3 |
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(287958450810a - 400961287925a b - 2680942780560a*b +
-----------------------------------------------------------------------
3 2 2
5244174813700b - 93738370860a c + 911678082564a*b*c - 1609862603460b c
-----------------------------------------------------------------------
2 2 3
+ 8536409703a*c - 23117264574b*c - 1516413743c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.