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,{- 9859a - 15643b + 7653c - 2929d + 11349e, 1021a + 2006b + 2415c - 3390d - 6470e, 4205a + 4310b + 11127c + 2954d + 3348e, - 659a + 9986b - 7105c + 12664d + 6323e})
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}))
9 3 1 1 5 5 9 8
o15 = map(P3,P2,{7a + -b + -c + 4d, --a + -b + c + -d, -a + -b + 5c + -d})
2 2 10 7 4 4 8 7
o15 : RingMap P3 <--- P2
|
i16 : N = pushForward(F,M)
o16 = cokernel {0} | 21885461380020ab+944826272689600b2-1677333506464ac-271227698467600bc+15239708670144c2 6409313689863a2-12432987828755200b2-73626246730736ac+1926552752887840bc+136757806251024c2 75807466930034608486096470336000000b3-17915537527845439489064512408240000b2c+102500264386057210021464012352ac2+1414473188942563459226108799612160bc2-37885839485933903929077415837440c3 0 |
{1} | -1336734208129a-1040839875434800b+83920767456484c -287181668763860a+13647855888177280b+716308028545466c -60875401668140285835402839980761a2-114344626502599972116764665474320ab-75957164837874706147593009698284800b2+545430078306179373128785363757344ac+180356799349040173510938293736520bc-646954666939051368271565032895536c2 122062589161a3+5498889233600a2b+152917332198400ab2-653894058496000b3-2202511025428a2c-48176453975720abc+338067788419200b2c+11162982813808ac2-80183128055520bc2-7985410615616c3 |
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
o19 = ideal(122062589161a + 5498889233600a b + 152917332198400a*b -
-----------------------------------------------------------------------
3 2
653894058496000b - 2202511025428a c - 48176453975720a*b*c +
-----------------------------------------------------------------------
2 2 2
338067788419200b c + 11162982813808a*c - 80183128055520b*c -
-----------------------------------------------------------------------
3
7985410615616c )
o19 : Ideal of P2
|
Note: these examples are from the original Macaulay script by David Eisenbud.