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,{- 714a + 9595b - 1176c - 15199d - 100e, 13768a + 9076b + 15846c - 3851d + 222e, 2291a + 10681b + 14430c - 778d + 5838e, - 81a - 5358b + 996c - 7295d - 76e})
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 2 5 7 1 8 5 6 1
o15 = map(P3,P2,{a + -b + -c + -d, -a + -b + 2c + -d, 4a + -b + -c + -d})
8 3 4 5 8 3 7 5 2
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 2248388483527200ab-467923545828000b2-780699687907020ac-237223330799700bc+138789729811950c2 4721615815407120a2-173733050202000b2-2352082619730060ac+117873726009900bc+272946161015850c2 9660116687820647541867278131200000b3-10342951612346861878298561418240000b2c-2514145713213442500297346500000ac2+3691050736888663961387181428052000bc2-438405151960548709126424939334000c3 0 |
{1} | 678690434716054a-1675473777460590b+443865448371345c -1595244163469298a-1251126779545670b+865907448548485c -447465457492055964238965146574805068a2+510691399224102024480568623708555000ab-162903786885767677810193538772878700b2+24619198248670715837047455556778700ac-6566232528462369944860245018794300bc-638360255984703141901835524984675c2 349754579853672a3-471963378944280a2b+212684658535800ab2-32126241837000b3-81419512091820a2c+78449945964600abc-18142664487500b2c+4218964305150ac2-2830882573750bc2+106122013375c3 |
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(349754579853672a - 471963378944280a b + 212684658535800a*b -
-----------------------------------------------------------------------
3 2
32126241837000b - 81419512091820a c + 78449945964600a*b*c -
-----------------------------------------------------------------------
2 2 2
18142664487500b c + 4218964305150a*c - 2830882573750b*c +
-----------------------------------------------------------------------
3
106122013375c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.