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,{4414a - 7095b - 12251c - 6091d + 4571e, 9100a - 10547b + 6336c + 15463d - 11194e, - 613a + 10516b + 11906c - 6057d + 10203e, 5572a + 14759b - 3894c - 9086d + 5523e})
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 5 3 3 4 2 3
o15 = map(P3,P2,{-a + -b + 2c + 2d, -a + -b + -c + -d, -a + -b + -c + -d})
7 8 8 3 4 2 5 9 3 2
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 2654785728ab+7050311216b2-7184963520ac-27485330250bc+28325344050c2 669006003456a2-9524059125712b2+1621408236840ac+34723230782550bc-33122515687650c2 14468228361988925608512b3-96865878230989583849640b2c+4233384994672388324700ac2+209022930120846205264050bc2-149769359748874839420000c3 0 |
{1} | 4230390528a+16481389311b-24897041955c -4121605942524a-17553850298721b+29907508452075c 3209492841324603672384a2+16389150966781370370096ab+19640988983805726385268b2-34532761235411892211290ac-113017024986330998223075bc+138111100565654723043600c2 29150034732a3+226267235712a2b+364301976864ab2-342052279304b3-405921828060a2c-1495423329120abc+1195573162815b2c+1556719512075ac2-939075753150bc2-231619780125c3 |
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(29150034732a + 226267235712a b + 364301976864a*b -
-----------------------------------------------------------------------
3 2
342052279304b - 405921828060a c - 1495423329120a*b*c +
-----------------------------------------------------------------------
2 2 2
1195573162815b c + 1556719512075a*c - 939075753150b*c -
-----------------------------------------------------------------------
3
231619780125c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.