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 iff 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 calling the routine pushForward1.
All optional arguments are passed to
pushForward1.
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
|
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}))
4 7 1 4 1 1 2 1 5
o15 = map(P3,P2,{- -*a - -*b + -*c - -*d, -*a - 10b + -*c + -*d, -*a - -*b +
3 4 3 7 3 2 3 6 2
-----------------------------------------------------------------------
4 3
-*c - -*d})
5 7
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 2656917701760ab-10469713102485b2-19480974893520ac+
{1} | 174463172216509a+34145265519502b-424591228267800c
-----------------------------------------------------------------------
124761250805520bc-320980765121100c2
-----------------------------------------------------------------------
15941506210560a2-263964231567435b2-235748051285040ac+3190889781614640bc
4452003890453283a+850560504667634b-10664548918301800c
-----------------------------------------------------------------------
-8232163444715700c2 47556746041783198147247976600b3-3266000804987747522
3030691636179136091868892505088a2-35394118550240374
-----------------------------------------------------------------------
49259019025b2c+537312166448676125440515750000ac2-
5325431808344ab-125231488021456451390895785744b2-
-----------------------------------------------------------------------
1601588554558040514436183155600bc2+8427620794366323456215544574500c3
12432772069619775056984451518055ac+381273630948409356717609893750bc+
-----------------------------------------------------------------------
15922339816835248068146389280200c2
-----------------------------------------------------------------------
0
100977092876a3-409697805174a2b-61747335300ab2-3921307047b3+
-----------------------------------------------------------------------
1409849608320a2c+1383660568605abc+118321299740b2c-4170251969850ac2-
-----------------------------------------------------------------------
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1175517119000bc2+2620973787000c3 |
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(100977092876a - 409697805174a b - 61747335300a*b - 3921307047b
-----------------------------------------------------------------------
2 2
+ 1409849608320a c + 1383660568605a*b*c + 118321299740b c -
-----------------------------------------------------------------------
2 2 3
4170251969850a*c - 1175517119000b*c + 2620973787000c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.