The filtered versions are described in detail at their individual pages TorCouple, contravariantExtCouple, and covariantExtCouple. We invite the reader to check these first, since the corresponding spectral sequences are easier to describe.
The remainder of this page describes an alternative, and more-general way of using these three functions. Specifically, we explain how to use a graded R[t]-module in place of a filtered module.
The idea is to replace a sequence of inclusions with a general sequence of maps. The first page then consists of the homology of mapping cones, which play the role of the associated graded. If the variable t acts by inclusions, the mapping cone is quasi-isomorphic to the associated graded, and we recover the filtered case.
How to use the R[t] versions
The schematic for using these generalized versions: replace the "submods" argument, which is an increasing list of submodules of a fixed R-module, with a "seqmod" argument which is a module for a ring of the form R[t].
Such a seqmod can be built directly, or from a sequence of maps using sequenceModule.
In the case of contravariantExtCouple, the appropriate sequence module (the one generalizing the filtered case) is not the one corresponding to the filtration, but rather the cofiltration $X/A_i$. So, in recovering the filtered case, the variable t acts by surjections.
These generalized versions also allow a symbol as the first argument, which will be used to name the variable "e" in the resulting couple ring. (The variable "f" in the couple ring will be named after the variable (ring seqmod)_0).
i1 : R = QQ[x,y,z] o1 = R o1 : PolynomialRing |
i2 : A = coker matrix {{x^2+y^2+z^2}};
|
i3 : B = coker vars R; |
i4 : toSeqMod = cm -> ( -- This short function converts chain modules to sequence modules
amb := ambient ring cm; -- by forgetting that the action of d squares to zero.
pres := lift(presentation cm, amb); -- It can be replaced by pushForward once that
coker(pres | (id_(target pres) ** (matrix {{(amb_0)^2}}))) -- function is fixed.
);
|
i5 : B' = toSeqMod chainModule(res B)
o5 = cokernel {-3, 3} | d 0 0 0 0 0 0 0 d2 0 0 0 0 0 0 0 |
{-2, 2} | -z d 0 0 0 0 0 0 0 d2 0 0 0 0 0 0 |
{-2, 2} | y 0 d 0 0 0 0 0 0 0 d2 0 0 0 0 0 |
{-2, 2} | -x 0 0 d 0 0 0 0 0 0 0 d2 0 0 0 0 |
{-1, 1} | 0 y z 0 d 0 0 0 0 0 0 0 d2 0 0 0 |
{-1, 1} | 0 -x 0 z 0 d 0 0 0 0 0 0 0 d2 0 0 |
{-1, 1} | 0 0 -x -y 0 0 d 0 0 0 0 0 0 0 d2 0 |
{0, 0} | 0 0 0 0 -x -y -z d 0 0 0 0 0 0 0 d2 |
8
o5 : R[d]-module, quotient of (R[d])
|
i6 : torCouple = prune TorCouple(A,B')
warning: clearing value of symbol d to allow access to subscripted variables based on it
: debug with expression debug 4400 or with command line option --debug 4400
o6 = cokernel {1, -5, 3} | x2+y2+z2 e_1^2 d_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 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1, -3, 2} | 0 0 -z -z x2+y2+z2 e_1^2 0 0 0 0 0 ze_1 0 d_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 0 0 |
{1, -3, 2} | 0 0 y y 0 0 x2+y2+z2 e_1^2 0 0 0 -ye_1 0 0 0 0 0 0 0 0 0 0 d_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{2, -2, 3} | 0 0 0 e_1 0 0 0 0 x2+y2+z2 d_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 0 0 0 0 0 0 |
{1, -3, 2} | 0 0 -x -x 0 0 0 0 0 0 x2+y2+z2 xe_1 e_1^2 0 0 0 0 0 0 0 d_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1, -1, 1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 y y 0 0 0 0 x 0 0 z z e_1^2 0 -ye_1 0 0 0 0 0 -ze_1 0 0 0 0 d_1 0 0 0 0 0 |
{2, 0, 2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e_1 -x y -z 0 0 0 0 0 0 0 d_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1, -1, 1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 -x -x 0 0 0 0 y z z 0 0 0 0 xe_1 e_1^2 0 0 0 -ze_1 0 0 0 0 d_1 0 0 0 0 0 0 |
{2, 0, 2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y x 0 -z e_1 0 0 0 0 0 0 0 0 d_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{2, 0, 2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 z 0 -x y 0 0 e_1 0 0 0 0 0 0 0 d_1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{2, 0, 2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 z y x 0 0 0 0 e_1 0 0 0 0 0 0 d_1 0 0 0 0 0 0 0 0 0 0 0 0 |
{1, -1, 1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 z -y -y -x -x 0 0 0 0 0 0 0 ye_1 xe_1 e_1^2 0 d_1 0 0 0 0 0 0 0 |
{2, 2, 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 0 0 0 0 0 0 0 0 x2+y2+z2 -ze_1 -ye_1 -xe_1 d_1 ze_1^2 ye_1^2 xe_1^2 e_1^3 |
13
o6 : R[e , d ]-module, quotient of (R[e , d ])
1 1 1 1
|
i7 : plotPages((-3..3,-4..2,1..3), prune @@ evaluateInDegree, torCouple)
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 d to allow access to subscripted variables based on it
: debug with expression debug 4400 or with command line option --debug 4400
page 1, with differential of degree {-1, -1}:
+----++----+----+----+-------------------------------------------+---------------------------+---+---+
|q=2 ||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+-------------------------------------------+---------------------------+---+---+
|q=1 ||0 |0 |0 |0 |cokernel | x2+y2+z2 | |0 |0 |
+----++----+----+----+-------------------------------------------+---------------------------+---+---+
|q=0 ||0 |0 |0 |cokernel | z y x | |cokernel {2} | x y z 0 ||0 |0 |
| || | | | | {2} | -y x 0 -z || | |
| || | | | | {2} | -z 0 x y || | |
| || | | | | {2} | 0 z -y x || | |
+----++----+----+----+-------------------------------------------+---------------------------+---+---+
|q=-1||0 |0 |0 |cokernel {1} | -y 0 -z x2+y2+z2 xy xz ||cokernel {3} | x2+y2+z2 | |0 |0 |
| || | | | {1} | x -z 0 0 y2+z2 yz || | | |
| || | | | {1} | 0 y x 0 0 z2 || | | |
+----++----+----+----+-------------------------------------------+---------------------------+---+---+
|q=-2||0 |0 |0 |cokernel {2} | z x2+y2+z2 0 -xz ||0 |0 |0 |
| || | | | {2} | -y 0 x2+y2+z2 xy || | | |
| || | | | {2} | x 0 0 y2+z2 || | | |
+----++----+----+----+-------------------------------------------+---------------------------+---+---+
|q=-3||0 |0 |0 |cokernel {3} | x2+y2+z2 | |0 |0 |0 |
+----++----+----+----+-------------------------------------------+---------------------------+---+---+
|q=-4||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+-------------------------------------------+---------------------------+---+---+
| ||p=-3|p=-2|p=-1|p=0 |p=1 |p=2|p=3|
+----++----+----+----+-------------------------------------------+---------------------------+---+---+
page 2, with differential of degree {-1, -2}:
+----++----+----+----+-------------------------------------------+-------------------------------------------+---+---+
|q=2 ||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+-------------------------------------------+-------------------------------------------+---+---+
|q=1 ||0 |0 |0 |0 |cokernel {1} | y x 0 z | |0 |0 |
| || | | | | {1} | -z 0 x y | | | |
| || | | | | {1} | 0 -z -y x | | | |
+----++----+----+----+-------------------------------------------+-------------------------------------------+---+---+
|q=0 ||0 |0 |0 |0 |cokernel {2} | z x2+y2+z2 -xz 0 ||0 |0 |
| || | | | | {2} | x 0 y2+z2 0 || | |
| || | | | | {2} | -y 0 xy x2+y2+z2 || | |
+----++----+----+----+-------------------------------------------+-------------------------------------------+---+---+
|q=-1||0 |0 |0 |cokernel {1} | y x 0 z | |cokernel {3} | x2+y2+z2 | |0 |0 |
| || | | | {1} | -z 0 x y | | | | |
| || | | | {1} | 0 -z -y x | | | | |
+----++----+----+----+-------------------------------------------+-------------------------------------------+---+---+
|q=-2||0 |0 |0 |cokernel {2} | z x2+y2+z2 -xz 0 ||0 |0 |0 |
| || | | | {2} | x 0 y2+z2 0 || | | |
| || | | | {2} | -y 0 xy x2+y2+z2 || | | |
+----++----+----+----+-------------------------------------------+-------------------------------------------+---+---+
|q=-3||0 |0 |0 |cokernel {3} | x2+y2+z2 | |0 |0 |0 |
+----++----+----+----+-------------------------------------------+-------------------------------------------+---+---+
|q=-4||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+-------------------------------------------+-------------------------------------------+---+---+
| ||p=-3|p=-2|p=-1|p=0 |p=1 |p=2|p=3|
+----++----+----+----+-------------------------------------------+-------------------------------------------+---+---+
page 3, with differential of degree {-1, -3}:
+----++----+----+----+---+---+---+---+
|q=2 ||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+---+---+---+---+
|q=1 ||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+---+---+---+---+
|q=0 ||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+---+---+---+---+
|q=-1||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+---+---+---+---+
|q=-2||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+---+---+---+---+
|q=-3||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+---+---+---+---+
|q=-4||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+---+---+---+---+
| ||p=-3|p=-2|p=-1|p=0|p=1|p=2|p=3|
+----++----+----+----+---+---+---+---+
|
i8 : covExtCouple = prune covariantExtCouple(A,B')
o8 = cokernel {1, -5, 1} | x2+y2+z2 e_1^2 d_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 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1, -3, 0} | 0 0 -z -z x2+y2+z2 e_1^2 0 0 0 0 0 ze_1 0 d_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 0 0 |
{1, -3, 0} | 0 0 y y 0 0 x2+y2+z2 e_1^2 0 0 0 -ye_1 0 0 0 0 0 0 d_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, -2, 1} | 0 0 0 e_1 0 0 0 0 x2+y2+z2 d_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 0 0 0 0 0 0 |
{1, -3, 0} | 0 0 -x -x 0 0 0 0 0 0 x2+y2+z2 xe_1 e_1^2 0 0 0 d_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 |
{1, -1, -1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 y y 0 0 0 z z 0 0 0 -x e_1^2 0 -ye_1 0 0 0 0 -ze_1 0 0 0 0 d_1 0 0 0 0 0 0 |
{0, 0, 0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e_1 -z 0 0 0 0 0 x -y 0 0 d_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1, -1, -1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 -x -x 0 z z 0 0 0 0 0 -y 0 0 xe_1 e_1^2 0 0 -ze_1 0 0 0 0 d_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 -x 0 e_1 0 0 y -z 0 0 0 0 0 0 d_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 0 0 0 0 0 y 0 0 0 e_1 x 0 -z 0 0 0 0 0 0 d_1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1, -1, -1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -y -y -x -x 0 0 0 -z 0 0 0 0 0 0 ye_1 xe_1 e_1^2 0 d_1 0 0 0 0 0 0 0 0 |
{0, 2, -2} | 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 0 0 0 0 0 0 x2+y2+z2 -ze_1 -ye_1 -xe_1 d_1 0 ze_1^2 ye_1^2 xe_1^2 e_1^3 |
{0, 0, 0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 z y x e_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d_1 0 0 0 0 |
13
o8 : R[e , d ]-module, quotient of (R[e , d ])
1 1 1 1
|
i9 : plotPages((-3..3,-4..2,1..3), prune @@ evaluateInDegree, covExtCouple)
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 d to allow access to subscripted variables based on it
: debug with expression debug 4400 or with command line option --debug 4400
page 1, with differential of degree {1, -1}:
+----++----+----+----+--------------------------+--------------------------------------------+---+---+
|q=2 ||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+--------------------------+--------------------------------------------+---+---+
|q=1 ||0 |0 |0 |cokernel {-2} | x2+y2+z2 ||0 |0 |0 |
+----++----+----+----+--------------------------+--------------------------------------------+---+---+
|q=0 ||0 |0 |0 |cokernel | z 0 x -y | |cokernel {-2} | z y x | |0 |0 |
| || | | | | x y -z 0 | | | | |
| || | | | | -y x 0 -z | | | | |
| || | | | | 0 z y x | | | | |
+----++----+----+----+--------------------------+--------------------------------------------+---+---+
|q=-1||0 |0 |0 |cokernel {1} | x2+y2+z2 | |cokernel {-1} | -y 0 -z x2+y2+z2 xy xz ||0 |0 |
| || | | | | {-1} | x -z 0 0 y2+z2 yz || | |
| || | | | | {-1} | 0 y x 0 0 z2 || | |
+----++----+----+----+--------------------------+--------------------------------------------+---+---+
|q=-2||0 |0 |0 |0 |cokernel | z x2+y2+z2 0 -xz | |0 |0 |
| || | | | | | -y 0 x2+y2+z2 xy | | | |
| || | | | | | x 0 0 y2+z2 | | | |
+----++----+----+----+--------------------------+--------------------------------------------+---+---+
|q=-3||0 |0 |0 |0 |cokernel {1} | x2+y2+z2 | |0 |0 |
+----++----+----+----+--------------------------+--------------------------------------------+---+---+
|q=-4||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+--------------------------+--------------------------------------------+---+---+
| ||p=-3|p=-2|p=-1|p=0 |p=1 |p=2|p=3|
+----++----+----+----+--------------------------+--------------------------------------------+---+---+
page 2, with differential of degree {1, -2}:
+----++----+----+----+---------------------------------------+---------------------------------------+---+---+
|q=2 ||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+---------------------------------------+---------------------------------------+---+---+
|q=1 ||0 |0 |0 |cokernel {-1} | y x 0 z | |0 |0 |0 |
| || | | | {-1} | -z 0 x y | | | | |
| || | | | {-1} | 0 -z -y x | | | | |
+----++----+----+----+---------------------------------------+---------------------------------------+---+---+
|q=0 ||0 |0 |0 |cokernel | z x2+y2+z2 -xz 0 ||0 |0 |0 |
| || | | | | x 0 y2+z2 0 || | | |
| || | | | | -y 0 xy x2+y2+z2 || | | |
+----++----+----+----+---------------------------------------+---------------------------------------+---+---+
|q=-1||0 |0 |0 |cokernel {1} | x2+y2+z2 | |cokernel {-1} | y x 0 z | |0 |0 |
| || | | | | {-1} | -z 0 x y | | | |
| || | | | | {-1} | 0 -z -y x | | | |
+----++----+----+----+---------------------------------------+---------------------------------------+---+---+
|q=-2||0 |0 |0 |0 |cokernel | z x2+y2+z2 -xz 0 ||0 |0 |
| || | | | | | x 0 y2+z2 0 || | |
| || | | | | | -y 0 xy x2+y2+z2 || | |
+----++----+----+----+---------------------------------------+---------------------------------------+---+---+
|q=-3||0 |0 |0 |0 |cokernel {1} | x2+y2+z2 | |0 |0 |
+----++----+----+----+---------------------------------------+---------------------------------------+---+---+
|q=-4||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+---------------------------------------+---------------------------------------+---+---+
| ||p=-3|p=-2|p=-1|p=0 |p=1 |p=2|p=3|
+----++----+----+----+---------------------------------------+---------------------------------------+---+---+
page 3, with differential of degree {1, -3}:
+----++----+----+----+---+---+---+---+
|q=2 ||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+---+---+---+---+
|q=1 ||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+---+---+---+---+
|q=0 ||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+---+---+---+---+
|q=-1||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+---+---+---+---+
|q=-2||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+---+---+---+---+
|q=-3||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+---+---+---+---+
|q=-4||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+----+---+---+---+---+
| ||p=-3|p=-2|p=-1|p=0|p=1|p=2|p=3|
+----++----+----+----+---+---+---+---+
|
i10 : contraExtCouple = prune contravariantExtCouple(B' ** (ring B')^{{-4,0}},A) -- see Caveat
warning: clearing value of symbol d to allow access to subscripted variables based on it
: debug with expression debug 4400 or with command line option --debug 4400
o10 = cokernel {-1, -7, 0} | x2+y2+z2 e_1^2 d_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 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{-1, -5, -1} | 0 0 -x x x2+y2+z2 e_1^2 0 0 0 xe_1 0 0 0 d_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 0 0 |
{-1, -5, -1} | 0 0 -y y 0 0 x2+y2+z2 e_1^2 0 ye_1 0 0 0 0 d_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 0 |
{-1, -5, -1} | 0 0 -z z 0 0 0 0 x2+y2+z2 ze_1 e_1^2 0 0 0 0 d_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 |
{-2, -4, 0} | 0 0 0 e_1 0 0 0 0 0 0 0 x2+y2+z2 d_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 0 0 0 |
{-1, -3, -2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 y -x 0 y -x 0 0 0 0 0 z e_1^2 ye_1 0 -xe_1 0 0 0 d_1 0 0 0 0 0 0 0 0 0 0 0 |
{-1, -3, -2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 z 0 -x z 0 0 x 0 0 0 -y 0 ze_1 e_1^2 0 xe_1 0 0 0 d_1 0 0 0 0 0 0 0 0 0 0 |
{-1, -3, -2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 z -y 0 z 0 y 0 0 0 x 0 0 0 ze_1 ye_1 e_1^2 0 0 0 d_1 0 0 0 0 0 0 0 0 0 |
{-2, 0, -3} | 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 0 0 0 x2+y2+z2 ze_1 -ye_1 xe_1 d_1 0 0 0 0 ze_1^2 ye_1^2 xe_1^2 e_1^3 |
{-2, -2, -1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e_1 0 -x 0 y -z 0 0 0 0 0 0 0 0 0 0 0 0 0 d_1 0 0 0 0 0 0 0 |
{-2, -2, -1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e_1 -y 0 -x 0 z 0 0 0 0 0 0 0 0 0 0 0 0 0 d_1 0 0 0 0 0 0 |
{-2, -2, -1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 z e_1 0 -x y 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d_1 0 0 0 0 0 |
{-2, -2, -1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 z y x e_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d_1 0 0 0 0 |
13
o10 : R[e , d ]-module, quotient of (R[e , d ])
1 1 1 1
|
i11 : plotPages((-3..1,-5..1,1..3), prune @@ evaluateInDegree, contraExtCouple)
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 d to allow access to subscripted variables based on it
: debug with expression debug 4400 or with command line option --debug 4400
page 1, with differential of degree {1, -1}:
+----++----+----+----------------------------+--------------------------------------------+---+
|q=1 ||0 |0 |0 |0 |0 |
+----++----+----+----------------------------+--------------------------------------------+---+
|q=0 ||0 |0 |cokernel {-3} | x2+y2+z2 | |0 |0 |
+----++----+----+----------------------------+--------------------------------------------+---+
|q=-1||0 |0 |cokernel {-1} | x -y z 0 ||cokernel {-3} | z y x | |0 |
| || | | {-1} | y x 0 z || | |
| || | | {-1} | -z 0 x y || | |
| || | | {-1} | 0 -z -y x || | |
+----++----+----+----------------------------+--------------------------------------------+---+
|q=-2||0 |0 |cokernel | x2+y2+z2 | |cokernel {-2} | y x 0 z2 0 0 ||0 |
| || | | | {-2} | z 0 x -yz -y2-z2 0 || |
| || | | | {-2} | 0 -z y xz xy x2+y2+z2 || |
+----++----+----+----------------------------+--------------------------------------------+---+
|q=-3||0 |0 |0 |cokernel {-1} | x 0 -y2-z2 0 ||0 |
| || | | | {-1} | y x2+y2+z2 xy 0 || |
| || | | | {-1} | z 0 xz x2+y2+z2 || |
+----++----+----+----------------------------+--------------------------------------------+---+
|q=-4||0 |0 |0 |cokernel | x2+y2+z2 | |0 |
+----++----+----+----------------------------+--------------------------------------------+---+
|q=-5||0 |0 |0 |0 |0 |
+----++----+----+----------------------------+--------------------------------------------+---+
| ||p=-3|p=-2|p=-1 |p=0 |p=1|
+----++----+----+----------------------------+--------------------------------------------+---+
page 2, with differential of degree {1, -2}:
+----++----+----+---------------------------------------------+---------------------------------------------+---+
|q=1 ||0 |0 |0 |0 |0 |
+----++----+----+---------------------------------------------+---------------------------------------------+---+
|q=0 ||0 |0 |cokernel {-2} | y x 0 z | |0 |0 |
| || | | {-2} | -z 0 x y | | | |
| || | | {-2} | 0 -z -y x | | | |
+----++----+----+---------------------------------------------+---------------------------------------------+---+
|q=-1||0 |0 |cokernel {-1} | x 0 -y2-z2 0 ||0 |0 |
| || | | {-1} | y x2+y2+z2 xy 0 || | |
| || | | {-1} | -z 0 -xz x2+y2+z2 || | |
+----++----+----+---------------------------------------------+---------------------------------------------+---+
|q=-2||0 |0 |cokernel | x2+y2+z2 | |cokernel {-2} | y x 0 z | |0 |
| || | | | {-2} | -z 0 x y | | |
| || | | | {-2} | 0 -z -y x | | |
+----++----+----+---------------------------------------------+---------------------------------------------+---+
|q=-3||0 |0 |0 |cokernel {-1} | x 0 -y2-z2 0 ||0 |
| || | | | {-1} | y x2+y2+z2 xy 0 || |
| || | | | {-1} | -z 0 -xz x2+y2+z2 || |
+----++----+----+---------------------------------------------+---------------------------------------------+---+
|q=-4||0 |0 |0 |cokernel | x2+y2+z2 | |0 |
+----++----+----+---------------------------------------------+---------------------------------------------+---+
|q=-5||0 |0 |0 |0 |0 |
+----++----+----+---------------------------------------------+---------------------------------------------+---+
| ||p=-3|p=-2|p=-1 |p=0 |p=1|
+----++----+----+---------------------------------------------+---------------------------------------------+---+
page 3, with differential of degree {1, -3}:
+----++----+----+----+---+---+
|q=1 ||0 |0 |0 |0 |0 |
+----++----+----+----+---+---+
|q=0 ||0 |0 |0 |0 |0 |
+----++----+----+----+---+---+
|q=-1||0 |0 |0 |0 |0 |
+----++----+----+----+---+---+
|q=-2||0 |0 |0 |0 |0 |
+----++----+----+----+---+---+
|q=-3||0 |0 |0 |0 |0 |
+----++----+----+----+---+---+
|q=-4||0 |0 |0 |0 |0 |
+----++----+----+----+---+---+
|q=-5||0 |0 |0 |0 |0 |
+----++----+----+----+---+---+
| ||p=-3|p=-2|p=-1|p=0|p=1|
+----++----+----+----+---+---+
|
It bears mentioning that there are many other spectral sequences arising from Tor and Ext. For example, both coordinates could vary instead of just one, or these functors could be composed in various ways. For such applications, work directly with the function exactCouple. If you obtain anything compelling, the author of this package would appreciate hearing about it!
In the case of contravariantExtCouple, the module seqmod should be concentrated in degrees that are positive multiples of $deg(t)$.