i1 : R = QQ[x] o1 = R o1 : PolynomialRing |
i2 : X = R^1 / x^9
o2 = cokernel | x9 |
1
o2 : R-module, quotient of R
|
i3 : A = apply(5,k->image map(X,,{{x^(8-2*k)}}))
o3 = {subquotient (| x8 |, | x9 |), subquotient (| x6 |, | x9 |), subquotient
------------------------------------------------------------------------
(| x4 |, | x9 |), subquotient (| x2 |, | x9 |), subquotient (| 1 |, | x9
------------------------------------------------------------------------
|)}
o3 : List
|
i4 : W = coker map(R^1,,{{x^3}})
o4 = cokernel | x3 |
1
o4 : R-module, quotient of R
|
We build the exact couple coming from applying Hom(W,-) to a filtered module
i5 : couple = prune covariantExtCouple(W,A)
warning: clearing value of symbol f to allow access to subscripted variables based on it
: debug with expression debug 3406 or with command line option --debug 3406
o5 = cokernel {-1, 1, 8} | x e_1^2 f_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{-1, 3, 6} | 0 0 -x2 e_1^2 0 0 0 0 x2e_1 e_1f_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1, 1, 5} | 0 0 0 0 x 0 0 0 0 0 e_1^2 f_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1, 3, 3} | 0 0 0 0 0 -x 0 0 0 0 0 -x2 e_1^2 f_1 0 0 0 0 0 0 0 0 0 0 0 x2e_1 0 0 0 0 |
{1, 5, 1} | 0 0 0 0 0 0 -x 0 0 0 0 0 0 -x2 e_1^2 f_1 0 0 0 0 0 0 0 0 0 0 x2e_1 0 0 0 |
{1, 7, -1} | 0 0 0 0 0 0 0 -x 0 0 0 0 0 0 0 -x2 e_1^2 f_1 0 0 0 0 0 0 0 0 0 x2e_1 0 0 |
{1, 9, -3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x2 e_1^2 0 0 0 0 0 0 0 0 0 x2e_1 e_1f_1 |
{0, 4, 4} | 0 0 0 0 0 e_1 0 0 0 0 0 0 0 0 0 0 0 0 0 x2 f_1 0 0 0 0 0 0 0 0 0 |
{0, 6, 2} | 0 0 0 0 0 0 e_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x2 f_1 0 0 0 0 0 0 0 |
{0, 8, 0} | 0 0 0 0 0 0 0 e_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x2 f_1 0 0 0 0 0 |
10
o5 : R[e , f ]-module, quotient of (R[e , f ])
1 1 1 1
|
i6 : expectExactCouple couple |
We check that {0,4} is an even degree, and then use excerptCouple
i7 : Q = ring couple
warning: clearing value of symbol e to allow access to subscripted variables based on it
: debug with expression debug 3903 or with command line option --debug 3903
warning: clearing value of symbol f to allow access to subscripted variables based on it
: debug with expression debug 3406 or with command line option --debug 3406
o7 = Q
o7 : PolynomialRing
|
i8 : Q.isEvenDegree({0,4})
o8 = true
|
i9 : excerptCouple({0,4},2,couple)
o9 = .- -> 0
0 (
\
0 \
.- -> cokernel {1} | x3 | - - - - - -> cokernel {1} | x2 | - - -> 0 - -'
{1} | x2 |( {1} | 1 |
\
{3} | x | \
.- -> cokernel {6} | x3 | - - -> cokernel {4} | x2 | - - - - - -> cokernel {3} | x3 | - -'
{6} | 1 |( 0
\
\
cokernel {6} | x3 | - -'
|
The middle column of the displayed long exact sequence is in even degrees; in other words, its entries appear on the E_1 page of the couple. Specifically, $E_1^{pq} = Ext^p(W,A_q/A_{q-1})$ can be found in degree $(2p,2q)$. The bottom element of the middle column is degree $(0,4)$, which is then $Ext^0(W,A_2/A_1)$. The top element of the middle column is $Ext^1(W,A_2/A_1)$.
i10 : prune Ext^0(W,A_2/A_1)
o10 = cokernel {4} | x2 |
1
o10 : R-module, quotient of R
|
i11 : prune Ext^1(W,A_2/A_1)
o11 = cokernel {1} | x2 |
1
o11 : R-module, quotient of R
|
The object excerptCouple is a method function.