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,{15518a + 11673b - 2780c - 15637d + 4010e, 15889a + 1771b - 13085c - 4796d - 3566e, - 1127a - 10588b - 1507c + 2113d + 81e, - 8195a + 1161b + 6680c + 11051d - 5034e})
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}))
5 4 5 5 7 5 2 5
o15 = map(P3,P2,{9a + -b + 4c + d, 2a + -b + -c + -d, -a + -b + -c + -d})
6 9 4 2 4 3 7 7
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 24312912970878ab-53558417980368b2+11035767679344ac-208332755100360bc+108392666235840c2 4052152161813a2-38625686802360b2-2919735355128ac-117096169265820bc+92114562082032c2 2435938617634788214728183914585712b3-384290528615247451220104789394568b2c+4109195066488308651105245818994688ac2-30440791323378185934086788025411712bc2+10522249813613678477603918549958144c3 0 |
{1} | 15433754268565a+113739461997426b+396218376009121c 52224662825987a+80538179318172b+207167633670926c 330449348561425599845250340874015a2-3545842063890162762779841278514936ab-4981757290597360529004206215380456b2+7701131063978961709869293192952655ac+4055323340826415842573611121695460bc+46017069495501479964492760999389548c2 8875074335a3-114783504864a2b+34457756400ab2-11319830352b3+139659196075a2c+399497485128abc-106198855056b2c-1090185713176ac2-211239154644bc2+1992712156176c3 |
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(8875074335a - 114783504864a b + 34457756400a*b - 11319830352b
-----------------------------------------------------------------------
2 2
+ 139659196075a c + 399497485128a*b*c - 106198855056b c -
-----------------------------------------------------------------------
2 2 3
1090185713176a*c - 211239154644b*c + 1992712156176c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.