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,{- 9617a + 7296b + 1479c + 1148d - 5448e, - 6772a + 4485b - 1407c - 4187d + 8298e, - 2295a - 2059b + 14367c + 2576d + 14590e, 8583a - 6610b - 10218c - 7058d + 3696e})
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 1 5 5 1 7 9 5 1
o15 = map(P3,P2,{-a + -b + -c + -d, -a + -b + -c + 10d, a + -b + -c + d})
7 9 8 8 6 4 8 2 2
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 397360829880ab+25994750664b2+46591802124ac-44613030560bc-9403515186c2 16689154854960a2-1013523765744b2-3706898319264ac+533700664880bc+128165232836c2 1168488403635902645173518720b3-500872416198937879782777216b2c-198739261495539859667091456ac2+32911312796881836819800160bc2+31409558408152420135029184c3 0 |
{1} | -4630133290524a-171973396209b-42999040024c -2047258789926a+9124805490014b+1716296870649c -241195744595373149405540127600a2+9359336612202234406132907880ab-10731444694783701110738170320b2+22700847541337453575591906206ac-2393905411705033539194364374bc-20744875562906142057924569c2 16297083720288a3+433740342672a2b-236237939280ab2+13265338560b3-3488583333564a2c+639474165792abc-66280115548b2c+112462965468ac2-24756771940bc2-2241524455c3 |
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(16297083720288a + 433740342672a b - 236237939280a*b +
-----------------------------------------------------------------------
3 2 2
13265338560b - 3488583333564a c + 639474165792a*b*c - 66280115548b c +
-----------------------------------------------------------------------
2 2 3
112462965468a*c - 24756771940b*c - 2241524455c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.