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,{9909a - 15775b + 1711c + 1911d - 14960e, 10130a - 2212b + 11465c - 13910d - 12051e, - 7677a + 10011b + 4589c - 9376d + 13343e, - 5040a + 15871b - 15205c + 785d + 1936e})
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}))
9 9 2 1 5 3 10
o15 = map(P3,P2,{--a + -b + -c + -d, -a + b + 2c + 10d, 6a + -b + 7c + --d})
10 2 7 8 4 4 3
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 75930685971840ab+70320549973248b2-406138918920ac-120836152809072bc+19810178199408c2 14173728048076800a2+11237035927707648b2+8819184672529560ac-72652874355365040bc+13006510335488688c2 9193760706527214185437444085760b3-48179890906455656236112964493056b2c-1406013863695918826035551742080ac2+26189122376912903646703103732256bc2-3237156224116884639492876191508c3 0 |
{1} | -733588949560320a-646244076881904b+970529677941569c 60097989534705920a-109516930166266176b+671190909727747501c 53816328574589050592406046918400a2+21187185314925255816497358095520ab-86946239243522151030975308978337b2+47537676214603258824578843446720ac+434657343462643498279046887020372bc-158664465297156218984150816317720c2 1062229861952000a3+1401940958040000a2b+424352265023610ab2-44891515509012b3-746870100412800a2c-393474460627080abc+263037099360996b2c+194115256891920ac2-445063482803736bc2+68774969781297c3 |
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(1062229861952000a + 1401940958040000a b + 424352265023610a*b -
-----------------------------------------------------------------------
3 2
44891515509012b - 746870100412800a c - 393474460627080a*b*c +
-----------------------------------------------------------------------
2 2 2
263037099360996b c + 194115256891920a*c - 445063482803736b*c +
-----------------------------------------------------------------------
3
68774969781297c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.