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,{2177a - 15557b - 8275c - 8058d - 15667e, - 2631a + 3893b - 2200c + 10167d - 6404e, 7554a + 15342b + 978c + 13913d + 8573e, 9889a - 6279b + 7267c + 14754d - 4716e})
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}))
3 1 9 1 1 8 3 4 2
o15 = map(P3,P2,{-a + 4b + -c + --d, -a + -b + -c + -d, -a + 2b + c + -d})
4 9 10 4 3 5 7 5 3
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 1589223090954240ab-962501261433600b2-340729913599950ac+247526271838650bc-129516104565750c2 2383834636431360a2+14734281512966400b2-6225883686656550ac-22004051789994750bc+9178970300687250c2 172789712607501466023499588105666560b3-323202920718498953084380831502803200b2c+2004079122623031679831059178173750ac2+165980987226979274221006448748828750bc2-27458221013169925457183749403021250c3 0 |
{1} | 12217825382709200a+15247699596166184b-26663538715258915c -76903153298583024a-95272704000005560b+167024480943104385c -202878908676687794447109596349726720a2-1227388518853245511291050210629713920ab-1208021733989513279501754664043642880b2+1160392754046279386172517829272903792ac+3011069886998035486165403223497444600bc-1564230449025343451142051662071218905c2 8161842898080a3+44434827697680a2b+72134898400320ab2+37260751649744b3-46471323524688a2c-180087907503280abc-152456315698400b2c+91103853460140ac2+188591229921500bc2-62523040383675c3 |
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(8161842898080a + 44434827697680a b + 72134898400320a*b +
-----------------------------------------------------------------------
3 2
37260751649744b - 46471323524688a c - 180087907503280a*b*c -
-----------------------------------------------------------------------
2 2 2
152456315698400b c + 91103853460140a*c + 188591229921500b*c -
-----------------------------------------------------------------------
3
62523040383675c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.