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];
|
i2 : R5 = ZZ/32003[a..e];
|
i3 : R6 = ZZ/32003[a..f];
|
i4 : M = coker genericMatrix(R6,a,2,3)
o4 = cokernel | a c e |
| b d f |
2
o4 : R6-module, quotient of R6
|
i5 : pdim M
o5 = 2
|
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
|
i7 : F = map(R5,R4,random(R5^1, R5^{4:-1}))
o7 = map(R5,R4,{- 3900a + 6634b - 6992c - 11714d + 9926e, 6104a - 6692b - 13549c + 8025d + 14201e, - 9113a + 2561b - 10676c + 6481d + 11324e, 8351a - 14204b + 12736c - 7759d + 4756e})
o7 : RingMap R5 <--- R4
|
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
|
i9 : pdim P
o9 = 1
|
i10 : Q = pushForward(F,P)
3
o10 = R4
o10 : R4-module, free, degrees {0..1, 0}
|
i11 : pdim Q
o11 = 0
|
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];
|
i13 : M = comodule monomialCurveIdeal(P3,{1,2,3})
o13 = cokernel | c2-bd bc-ad b2-ac |
1
o13 : P3-module, quotient of P3
|
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];
|
i15 : F = map(P3,P2,random(P3^1, P3^{-1,-1,-1}))
3 1 5 5 1 4 3 9 4 6 5
o15 = map(P3,P2,{-a + -b + -c + -d, -a + -b + -c + --d, -a + -b + -c + d})
2 4 4 8 5 3 5 10 5 5 8
o15 : RingMap P3 <--- P2
|
i16 : N = pushForward(F,M)
o16 = cokernel {0} | 323172485760ab+1547896968000b2-1394726444960ac-3922993650000bc+3347629833600c2 21544832384a2-258541884000b2+172524335840ac+619562794800bc-537958512000c2 107772364027688070041280000b3+245429194202438313955104000b2c-464745251188642939787763200ac2-1042428853140880876151124000bc2+1114981291438340710423248000c3 0 |
{1} | 849434825668a+418753596975b-1288587591140c -148740560092a-90947564325b+248342920620c -25890889337494842267814936a2+43241254319440386910152090ab+19552631827008634568845875b2+346341263479089977757931680ac+111568181589967154353863600bc-510615104091310685777656800c2 1827724208a3+5701693380a2b+3996054000ab2+3247003125b3-8855184640a2c-19184080200abc-14840561250b2c+16481175200ac2+25474332000bc2-13797468000c3 |
2
o16 : P2-module, quotient of P2
|
i17 : hilbertPolynomial M
o17 = - 2*P + 3*P
0 1
o17 : ProjectiveHilbertPolynomial
|
i18 : hilbertPolynomial N
o18 = - 2*P + 3*P
0 1
o18 : ProjectiveHilbertPolynomial
|
i19 : ann N
3 2 2 3
o19 = ideal(1827724208a + 5701693380a b + 3996054000a*b + 3247003125b -
-----------------------------------------------------------------------
2 2 2
8855184640a c - 19184080200a*b*c - 14840561250b c + 16481175200a*c +
-----------------------------------------------------------------------
2 3
25474332000b*c - 13797468000c )
o19 : Ideal of P2
|
Note: these examples are from the original Macaulay script by David Eisenbud.