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,{14741a - 710b + 976c - 8854d - 6618e, - 7950a - 13414b + 15743c + 14542d + 13852e, 2471a + 6613b - 11767c + 2820d - 9685e, - 3112a + 13788b - 4180c - 7010d + 13732e})
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}))
1 4 5 2 9 2 2 7
o15 = map(P3,P2,{-a + -b + -c + -d, 5a + 4b + -c + -d, 3a + 2b + -c + --d})
6 3 6 7 5 5 3 10
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 11191182660ab-6486462150b2-14197778700ac+12833788730bc-3619152600c2 235014835860a2-225729109725b2+152992594860ac+424070549350bc-88983887160c2 157537536058455122397600b3-629883359365144127457600b2c+50096609818136979216000ac2+694954612730103276316000bc2-140705279351960505312000c3 0 |
{1} | 7001523858a-7078483011b+2298970180c -127820087346a-220782162405b+51469278332c 157810322354966349756828a2+69112729272871639223260ab+144493813044304723983015b2-84428756721424891300896ac-364054862574644345085920bc+82495602330339571507280c2 84018286296a3-11901746220a2b+3128258770ab2+8461994895b3-10380003984a2c-9396060240abc-18044589380b2c-160757856ac2+7414321680bc2-934211520c3 |
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(84018286296a - 11901746220a b + 3128258770a*b + 8461994895b -
-----------------------------------------------------------------------
2 2 2
10380003984a c - 9396060240a*b*c - 18044589380b c - 160757856a*c +
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2 3
7414321680b*c - 934211520c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.