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,{14093a - 10348b - 10489c + 3644d - 1057e, - 14312a - 9957b + 15866c + 15785d - 15473e, - 10137a - 8781b - 2282c + 15001d + 15844e, 4634a + 4209b + 11452c + 8488d + 2738e})
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 4 2 9 1 5 7 7 3
o15 = map(P3,P2,{-a + -b + -c + -d, a + 2b + -c + -d, -a + -b + 9c + --d})
2 5 5 2 2 3 6 4 10
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 442945955550ab-4182312211245b2-396525898800ac+5260562798760bc-1371311181000c2 1476486518500a2-69839818928565b2-11141726242800ac+122680323986520bc-32834949757200c2 1960979221252371662228120220b3-4230256983212990483904674160b2c+1448483054674745424913200ac2+2946947778853385820094337400bc2-658500705787372584262948800c3 0 |
{1} | -22426407279335a+65834706340977b-8272639958325c -527525800049485a+1530208451873584b-185747868765965c 2372352080571574737194287450a2-1305528786893762860882404120ab-17195295963937645566512250177b2-10162656847040344568929206210ac+35292905094659093486357594100bc-4322901769864012947145959360c2 26661602900a3-266411991150a2b+1061093109330ab2-1450888043913b3-22929171300a2c+16639231950abc+138677823144b2c-25351795800ac2+81355458240bc2-13618611000c3 |
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(26661602900a - 266411991150a b + 1061093109330a*b -
-----------------------------------------------------------------------
3 2 2
1450888043913b - 22929171300a c + 16639231950a*b*c + 138677823144b c -
-----------------------------------------------------------------------
2 2 3
25351795800a*c + 81355458240b*c - 13618611000c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.