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,{15049a - 2218b + 10322c - 15425d + 9077e, 2109a + 2693b - 8761c + 12097d - 199e, 13086a + 10220b + 6057c - 3501d - 5965e, 8039a + 1189b + 722c - 13795d - 7780e})
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}))
4 7 3 1 10 5 7 1 1
o15 = map(P3,P2,{-a + -b + 2c + -d, -a + --b + -c + d, a + -b + -c + -d})
9 2 8 2 7 3 3 5 5
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 17215985792880ab-61582238435760b2-8071317143640ac+48883768731280bc-9284847313500c2 88539355506240a2+1070474553057840b2-31085837277480ac-1518255573554440bc+487835820783000c2 787635611585806028832094757400b3-1214953661099744579661329028900b2c-179218509495729694461504000ac2+625912330514839798901335336850bc2-107592548759822517164031031875c3 0 |
{1} | 40580657556064a+39476526440638b-41840201643375c -1071466975177432a-566988323397664b+861660794193375c -74604445060233360798937152576a2-722456210640154222232034980400ab-462284103827260908452648925396b2+369405182184482958289907795720ac+788031096396133456873986479570bc-260827973946963596692024868625c2 460356715008a3+2811299213184a2b-1637650527072ab2+940731449784b3-1744332183360a2c-3827435151120abc-3067964120340b2c+2319559853000ac2+4535237705250bc2-1590147571875c3 |
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(460356715008a + 2811299213184a b - 1637650527072a*b +
-----------------------------------------------------------------------
3 2
940731449784b - 1744332183360a c - 3827435151120a*b*c -
-----------------------------------------------------------------------
2 2 2
3067964120340b c + 2319559853000a*c + 4535237705250b*c -
-----------------------------------------------------------------------
3
1590147571875c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.