covariantExtCouple -- the exact couple obtained by applying Ext(W,-) to a filtered module

Synopsis

Usage:

covariantExtCouple(W, submods)
Inputs:
- W, a module, for some ring R, giving a functor Hom(W,-)
- submods, a list, of R-modules {A_0, A_1, ..., A_m} with each A_i inside A_{i+1}
Outputs:
- M, a module, an exact couple

Description

For notational convenience, set $X = A_m$, and extend the sequence $A_i$ to all $i \in \ZZ$ by setting $A_i = 0$ for $i < 0$, and $A_i = X$ for $i > m$.

The returned couple $M$ is a module for the ring R[e_1,f_1,Degrees=>{{1,-1},{0,2}}]. We describe the module $M$ in every bidegree $(s,t)$. The description depends on the parity of $s$ and $t$.

If $s$ and $t$ are both even, say $(s,t) = (2p, 2q)$, then

$M_{s,t} = Ext^p(W, A_q/A_{q-1})$;

if $s$ and $t$ are both odd, say $(s,t) = (2p-1,2q+1)$, then

$M_{s,t} = Ext^p(W, A_q)$;

and otherwise, if $s$ and $t$ sum to an odd number, then $M_{s,t} = 0$.

The variables $e_1$ and $f_1$ act by the maps in the various long exact sequences

$Ext^p(W, A_{q-1}) \to Ext^p(W, A_q) \to Ext^p(W, A_q / A_{q-1}) \to Ext^{p+1}(W, A_{q-1})$.

Associated spectral sequence

The spectral sequence associated to this couple converges to $Ext^p(W,X)$. The differential on page $r$ has bidegree $(1,-r)$. The first page has

$E^{pq}_1 = Ext^p(W,A_q/A_{q-1})$.

Setting $F^p_q = image(Ext^p(W,A_q) \to Ext^p(W,X))$, the infinity page has

$E^{p,q}_{\infty} = F^p_{q} / F^p_{q-1}$.

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 : submods = apply(5,k->image map(X,,{{x^(8-2*k)}}));

i4 : for m in submods do print m;
subquotient (| x8 |, | x9 |)
subquotient (| x6 |, | x9 |)
subquotient (| x4 |, | x9 |)
subquotient (| x2 |, | x9 |)
subquotient (| 1 |, | x9 |)

i5 : W = coker map(R^1,,{{x^3}})

o5 = cokernel | x3 |

                            1
o5 : R-module, quotient of R

i6 : couple = prune covariantExtCouple(W,submods)
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

o6 = 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
o6 : R[e , f ]-module, quotient of (R[e , f ])
        1   1                          1   1

i7 : expectExactCouple couple

i8 : plotPages((-1..2,-1..5,1..3), prune @@ evaluateInDegree, 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
page 1, with differential of degree {1, -1}:
+----++----+-------------------+--------------------+---+
|q=5 ||0   |0                  |0                   |0  |
+----++----+-------------------+--------------------+---+
|q=4 ||0   |cokernel | x2 |    |cokernel {-3} | x2 ||0  |
+----++----+-------------------+--------------------+---+
|q=3 ||0   |cokernel {2} | x2 ||cokernel {-1} | x2 ||0  |
+----++----+-------------------+--------------------+---+
|q=2 ||0   |cokernel {4} | x2 ||cokernel {1} | x2 | |0  |
+----++----+-------------------+--------------------+---+
|q=1 ||0   |cokernel {6} | x2 ||cokernel {3} | x2 | |0  |
+----++----+-------------------+--------------------+---+
|q=0 ||0   |cokernel {8} | x | |cokernel {5} | x |  |0  |
+----++----+-------------------+--------------------+---+
|q=-1||0   |0                  |0                   |0  |
+----++----+-------------------+--------------------+---+
|    ||p=-1|p=0                |p=1                 |p=2|
+----++----+-------------------+--------------------+---+

page 2, with differential of degree {1, -2}:
+----++----+-------------------+--------------------+---+
|q=5 ||0   |0                  |0                   |0  |
+----++----+-------------------+--------------------+---+
|q=4 ||0   |cokernel {1} | x | |cokernel {-3} | x2 ||0  |
+----++----+-------------------+--------------------+---+
|q=3 ||0   |cokernel {3} | x | |cokernel {-1} | x | |0  |
+----++----+-------------------+--------------------+---+
|q=2 ||0   |cokernel {5} | x | |cokernel {1} | x |  |0  |
+----++----+-------------------+--------------------+---+
|q=1 ||0   |cokernel {6} | x2 ||cokernel {3} | x |  |0  |
+----++----+-------------------+--------------------+---+
|q=0 ||0   |cokernel {8} | x | |cokernel {5} | x |  |0  |
+----++----+-------------------+--------------------+---+
|q=-1||0   |0                  |0                   |0  |
+----++----+-------------------+--------------------+---+
|    ||p=-1|p=0                |p=1                 |p=2|
+----++----+-------------------+--------------------+---+

page 3, with differential of degree {1, -3}:
+----++----+-------------------+--------------------+---+
|q=5 ||0   |0                  |0                   |0  |
+----++----+-------------------+--------------------+---+
|q=4 ||0   |0                  |cokernel {-3} | x2 ||0  |
+----++----+-------------------+--------------------+---+
|q=3 ||0   |0                  |cokernel {-1} | x | |0  |
+----++----+-------------------+--------------------+---+
|q=2 ||0   |0                  |0                   |0  |
+----++----+-------------------+--------------------+---+
|q=1 ||0   |cokernel {6} | x2 ||0                   |0  |
+----++----+-------------------+--------------------+---+
|q=0 ||0   |cokernel {8} | x | |0                   |0  |
+----++----+-------------------+--------------------+---+
|q=-1||0   |0                  |0                   |0  |
+----++----+-------------------+--------------------+---+
|    ||p=-1|p=0                |p=1                 |p=2|
+----++----+-------------------+--------------------+---+

i9 : A = i -> if i < 0 then image(0*id_X) else if i >= #submods then X else submods#i;

i10 : E1 = (q,p) -> prune Ext^p(W,A(q)/A(q-1));

i11 : netList reverse table(5,2,E1)

      +-------------------+--------------------+
o11 = |cokernel | x2 |    |cokernel {-3} | x2 ||
      +-------------------+--------------------+
      |cokernel {2} | x2 ||cokernel {-1} | x2 ||
      +-------------------+--------------------+
      |cokernel {4} | x2 ||cokernel {1} | x2 | |
      +-------------------+--------------------+
      |cokernel {6} | x2 ||cokernel {3} | x2 | |
      +-------------------+--------------------+
      |cokernel {8} | x | |cokernel {5} | x |  |
      +-------------------+--------------------+

i12 : inc = q -> inducedMap(X,A(q));

i13 : filt = (p,q) -> image Ext^p(W,inc q);

i14 : Einfty = (q,p) -> prune(filt(p,q)/filt(p,q-1));

i15 : netList reverse table(5,2,Einfty)

      +-------------------+--------------------+
o15 = |0                  |cokernel {-3} | x2 ||
      +-------------------+--------------------+
      |0                  |cokernel {-1} | x | |
      +-------------------+--------------------+
      |0                  |0                   |
      +-------------------+--------------------+
      |cokernel {6} | x2 ||0                   |
      +-------------------+--------------------+
      |cokernel {8} | x | |0                   |
      +-------------------+--------------------+

Ways to use `covariantExtCouple`:

covariantExtCouple(Module,List)
"covariantExtCouple(Module,Module)" -- see Exact couples for Tor and Ext -- building couples by applying Tor or Ext to a filtered module or a graded R[t]-module
"covariantExtCouple(Symbol,Module,Module)" -- see Exact couples for Tor and Ext -- building couples by applying Tor or Ext to a filtered module or a graded R[t]-module

For the programmer

The object covariantExtCouple is a method function.

covariantExtCouple -- the exact couple obtained by applying Ext(W,-) to a filtered module

Synopsis

Description

See also

Ways to use covariantExtCouple:

For the programmer

Ways to use `covariantExtCouple`: