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,{- 4245a - 4130b + 7979c - 9749d + 14209e, - 113a + 13101b + 12397c - 14742d + 13430e, - 7860a + 14193b - 2039c - 1731d + 15000e, - 13976a + 3176b + 1452c - 1998d + 2152e})
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 1 5 3 7 7 5 4 3
o15 = map(P3,P2,{a + --b + -c + -d, -a + b + -c + -d, -a + -b + -c + -d})
10 3 3 9 4 6 4 8 3 5
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 27420845398200ab+8755239013740b2-8751698111400ac-21242907095760bc+5888085151200c2 15233802999000a2-7291032628500b2-23900711154600ac+2547077769360bc+8609448040800c2 301644303265754657180217676620b3-287280291894110114199748635060b2c-3481099611423705582600ac2+91200092522558114059903276560bc2-9650801319118760053183799200c3 0 |
{1} | -9537964417935a-10882342285704b+12992560343630c -3507622695215a+7381070411184b+1145009990270c -278559735766807565504115788350a2-44486541437471344395715336375ab-286542276373645413066721233144b2+570227176140272459327516987585ac+227304127944101178453171889414bc-320828482619721335654103020730c2 81022463625a3+38809220475a2b+56957664240ab2-8208490428b3-220368845250a2c-108451966650abc-13563200520b2c+201845412000ac2+44462105400bc2-58615676000c3 |
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 3
o19 = ideal(81022463625a + 38809220475a b + 56957664240a*b - 8208490428b -
-----------------------------------------------------------------------
2 2 2
220368845250a c - 108451966650a*b*c - 13563200520b c + 201845412000a*c
-----------------------------------------------------------------------
2 3
+ 44462105400b*c - 58615676000c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.