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,{- 5927a - 217b + 6939c + 10643d + 13729e, 8998a + 1337b - 13139c + 15367d + 2297e, 13892a + 8287b + 7707c - 3588d - 3157e, 10548a + 7326b + 14903c - 3377d - 10457e})
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}))
10 5 4 2 5 9 10 3 9
o15 = map(P3,P2,{--a + 2b + c + -d, -a + -b + -c + -d, --a + -b + -c + 2d})
9 6 3 3 4 7 3 7 4
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 402603282984ab-3210037304040b2-278785526544ac+2875369548968bc-597293779824c2 301952462238a2-7545412813050b2-429772008708ac+6893094127500bc-1440264538258c2 403663435448627333338412550000b3-429824474413763235177574854000b2c+3779092091976379244083390464ac2+135929535170758045974309144960bc2-12694055394805376509664548032c3 0 |
{1} | 501879162798a+396268215295b-296341276503c 1821973626957a+266118985305b-422439276779c -13041962636288053474269690822a2-51155356794756607595866856661ab-72883696265855841172248697815b2+3688822437345108707332676169ac+71725608473018840365899398742bc-13307037539112102785959332583c2 6812839314a3-27597691683a2b+87870539610ab2-96015626175b3+9115573635a2c-76667844330abc+127958851125b2c+16858785552ac2-55737660573bc2+7844398583c3 |
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(6812839314a - 27597691683a b + 87870539610a*b - 96015626175b +
-----------------------------------------------------------------------
2 2 2
9115573635a c - 76667844330a*b*c + 127958851125b c + 16858785552a*c -
-----------------------------------------------------------------------
2 3
55737660573b*c + 7844398583c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.