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,{8686a + 4596b + 11861c + 11011d + 4379e, 6552a - 8453b + 10988c + 4273d + 11835e, 8653a - 10429b + 13776c + 14791d + 15790e, - 11244a - 5929b + 15414c + 9744d - 13811e})
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 8 1 6 9 10 1 1 10
o15 = map(P3,P2,{-a + 4b + -c + d, -a + 10b + -c + -d, --a + -b + --c + --d})
9 3 4 7 5 7 4 10 9
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 306268465755413048175ab+942166304609048051400b2-31917635434226356462935ac+12299762292932230530120bc+2775315959414521210800c2 2756416191798717433575a2+7648764878950489627200b2-242980740624462905017485ac+93652057604612816343720bc+21106952126037949222800c2 663681375982742813837546763855099017908164430080000b3+67748285923593366262042478463165011379662231680224000b2c-2039289818413241240293362543070264146296846754590075820ac2+787562070439364713243305165651833023953333913711045440bc2+177321150141386469318457252316737234726208136178407300c3 0 |
{1} | 34845546846159846485568a-16618921325776590246455b-987472564950665396961c 264410540443558523547153a-127623947528476564700920b-5048396455480328913291c -6169942469449413218469693993050953780095427515817911a2+27125079059010936998522591766821068515993370189508352ab-11563228474291744845256921553818100903822568962936865b2+2227761568378760463695090289547146783979260128149229646ac-1059929230125505030281409510723126685758224889504243790bc-68138821810960909553045219296823383915060682448370572c2 2427555194978315391a3-3204494663555425464a2b+1427836461557170329ab2-173664909604796680b3-762254812941846030a2c-572720143744543410abc+266681086447325760b2c+2033672409793536120ac2-681461580113153280bc2-172664106799870980c3 |
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
o19 = ideal(2427555194978315391a - 3204494663555425464a b +
-----------------------------------------------------------------------
2 3 2
1427836461557170329a*b - 173664909604796680b - 762254812941846030a c
-----------------------------------------------------------------------
2
- 572720143744543410a*b*c + 266681086447325760b c +
-----------------------------------------------------------------------
2 2
2033672409793536120a*c - 681461580113153280b*c -
-----------------------------------------------------------------------
3
172664106799870980c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.