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,{5232a + 5302b + 5809c + 2734d - 664e, - 12885a + 15214b + 13391c - 12115d - 2976e, 1921a + 6426b - 4993c - 6438d + 12260e, - 14422a + 11178b - 15403c - 5975d + 14203e})
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 3 9 7 7 7 1 9 1 7
o15 = map(P3,P2,{-a + 3b + -c + -d, -a + -b + 4c + -d, -a + -b + -c + -d})
5 5 8 6 3 8 6 5 4 3
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 173784975762680ab-92602456157232b2-229147184032200ac-80133568092960bc+359186626083600c2 868924878813400a2-247821194838384b2-3378720553593600ac+743092372262880bc+3139421390629200c2 367863893180035810043939118322200b3-7276839282658992572612622995506200b2c+154418531078887932868865881405593600ac2-37030162660199881830700858111557240bc2-251610001067724551742702814770659400c3 0 |
{1} | 496527819783450a+104142528086573b-831575818847895c 3589197987581515a-1305175600785024b-5805524937983040c 828677322582551922179180820589693555a2-582660176907931931823500069990635905ab+528267873136434012532415012545423719b2-2066043133452073814080456299566000515ac+150283848932645457227922674023892166bc+1226308095080975051863677391676257935c2 219508319975a3-271307280915a2b+221895980595ab2-74622533427b3-802719192725a2c+516176191890abc-182509865793b2c+889826291775ac2-146224876185bc2-267310017675c3 |
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(219508319975a - 271307280915a b + 221895980595a*b -
-----------------------------------------------------------------------
3 2 2
74622533427b - 802719192725a c + 516176191890a*b*c - 182509865793b c +
-----------------------------------------------------------------------
2 2 3
889826291775a*c - 146224876185b*c - 267310017675c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.