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,{69a - 1937b + 12856c - 7235d - 8412e, - 15275a + 15388b + 2526c - 10562d - 4211e, 12023a - 6579b + 8343c - 10844d + 9628e, 3072a + 12176b + 2550c + 4123d - 7080e})
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 9 4 3 5 5 2 7 9 9
o15 = map(P3,P2,{-a + -b + -c + -d, -a + -b + -c + -d, a + b + -c + -d})
5 4 5 5 6 2 5 3 8 2
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 152368826400ab-119587982400b2-272348422000ac+299731357440bc-108579161200c2 55022076200a2-29310994800b2-156938432960ac+174193714920bc-70834730608c2 439038228061986457464000b3-565918827202250707708800b2c-1560153008356340320000000ac2+1760304909483483814998000bc2-703933417650231407770000c3 0 |
{1} | 965753678565a-849147920835b+361456935888c 707558800845a-638544026835b+274283994360c -33410389014353341367292825a2+70939471753484412732169425ab-38117440742903845113977550b2-19518290209398180273407685ac+25440480000974063961064515bc-4245427353138349024852224c2 13815306375a3-40176768125a2b+39935171250ab2-13382435250b3+15866368050a2c-32832664975abc+17076643575b2c+6861552000ac2-7391888240bc2+1086193104c3 |
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(13815306375a - 40176768125a b + 39935171250a*b - 13382435250b
-----------------------------------------------------------------------
2 2 2
+ 15866368050a c - 32832664975a*b*c + 17076643575b c + 6861552000a*c -
-----------------------------------------------------------------------
2 3
7391888240b*c + 1086193104c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.