Let $S$ be a commutative ring and let $B : \dots \rightarrow B_{i} \rightarrow B_{i - 1} \rightarrow \cdots $ and $C : \dots \rightarrow C_{i} \rightarrow C_{i - 1} \rightarrow \cdots $ be chain complexes.
For all integers $p$ and $q$ let $K_{p,q} := Hom_S(B_{-p}, C_q)$, let $d'_{p,q} : K_{p,q} \rightarrow K_{p - 1, q}$ denote the homorphism $ \phi \mapsto \partial^B_{-p + 1} \phi$, and let $d^{''}_{p,q} : K_{p,q} \rightarrow K_{p, q - 1} $ denote the homorphism $\phi \mapsto (-1)^p \partial^C_q \phi$.
The chain complex $Hom(B, C)$ is given by $ Hom(B, C)_k := \prod_{p + q = k} Hom_S(B_{-p}, C_q) $ and the differentials by $ \partial := d^{'} + d^{''} $; it carries two natural ascending filtrations $F' ( Hom(B, C) )$ and $F''( Hom(B, C))$.
The first is obtained by letting $F'_n (Hom(B, C))$ be the chain complex determined by setting $F'_n (Hom(B, C))_k := \prod_{p + q = k , p \leq n} Hom_S(B_{-p}, C_q)$ and the differentials $\partial := d' + d''$.
The second is obtained by letting $F''_n (Hom(B, C)) := \prod_{p + q = k , q \leq n} Hom_S(B_{-p}, C_q)$ and the differentials $\partial := d' + d''$.
In Macaulay2, using this package, $F'$ and $F''$ as defined above are computed as illustrated in the following example, by using Hom(filteredComplex B, C) or Hom(B,filteredComplex C).
i1 : A = QQ[x,y,z,w]; |
i2 : B = res monomialCurveIdeal(A, {1,2,3});
|
i3 : C = res monomialCurveIdeal(A, {1,3,4});
|
i4 : F' = Hom(filteredComplex B, C)
o4 = -3 : image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0
-3 -2 -1 0 1 2 3 4
-2 : image 0 <-- image {-3} | 1 0 | <-- image {-2} | 0 0 0 0 0 0 0 0 0 0 0 | <-- image {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image 0 <-- image 0 <-- image 0
{-3} | 0 1 | {-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 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
-3 {-2} | 0 0 0 0 0 0 0 0 0 0 0 | {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 2 3 4
-2 {-1} | 0 0 0 1 0 0 0 0 0 0 0 | {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 0 0 0 0 0 0 | {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {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 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} | 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 0 0 0 0 | {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{-1} | 0 0 0 0 0 0 0 1 0 0 0 | {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {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 1 0 0 | {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {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 1 0 | {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {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 1 | {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
-1 {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 1
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
0
-1 : image 0 <-- image {-3} | 1 0 | <-- image {-2} | 1 0 0 0 0 0 0 0 0 0 0 | <-- image {0} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {4} | 0 0 0 0 0 0 0 | <-- image 0 <-- image 0
{-3} | 0 1 | {-2} | 0 1 0 0 0 0 0 0 0 0 0 | {0} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 |
-3 {-2} | 0 0 1 0 0 0 0 0 0 0 0 | {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 | 3 4
-2 {-1} | 0 0 0 1 0 0 0 0 0 0 0 | {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 |
{0} | 0 0 0 0 1 0 0 0 0 0 0 | {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 1 0 0 |
{0} | 0 0 0 0 0 1 0 0 0 0 0 | {0} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 1 0 |
{0} | 0 0 0 0 0 0 1 0 0 0 0 | {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 1 |
{-1} | 0 0 0 0 0 0 0 1 0 0 0 | {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
{0} | 0 0 0 0 0 0 0 0 1 0 0 | {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 | 2
{0} | 0 0 0 0 0 0 0 0 0 1 0 | {0} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
{0} | 0 0 0 0 0 0 0 0 0 0 1 | {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
-1 {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 1
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
0
2 11 21 18 7 1
0 : 0 <-- A <-- A <-- A <-- A <-- A <-- A <-- 0
-3 -2 -1 0 1 2 3 4
o4 : FilteredComplex
|
i5 : F'' = Hom(B,filteredComplex C)
o5 = -1 : image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0
-3 -2 -1 0 1 2 3 4
0 : image 0 <-- image {-3} | 1 0 | <-- image {-2} | 1 0 0 0 0 0 0 0 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image 0 <-- image 0 <-- image 0 <-- image 0
{-3} | 0 1 | {-2} | 0 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 |
-3 {-2} | 0 0 1 0 0 0 0 0 0 0 0 | {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 2 3 4
-2 {-1} | 0 0 0 0 0 0 0 0 0 0 0 | {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} | 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 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} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{-1} | 0 0 0 0 0 0 0 0 0 0 0 | {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} | 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 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} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
-1 {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
0
1 : image 0 <-- image {-3} | 1 0 | <-- image {-2} | 1 0 0 0 0 0 0 0 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {2} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image 0 <-- image 0 <-- image 0
{-3} | 0 1 | {-2} | 0 1 0 0 0 0 0 0 0 0 0 | {0} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
-3 {-2} | 0 0 1 0 0 0 0 0 0 0 0 | {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 2 3 4
-2 {-1} | 0 0 0 1 0 0 0 0 0 0 0 | {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{0} | 0 0 0 0 1 0 0 0 0 0 0 | {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {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 1 0 0 0 0 0 | {0} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {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 1 0 0 0 0 | {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{-1} | 0 0 0 0 0 0 0 1 0 0 0 | {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | {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 1 0 0 | {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 | {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 1 0 | {0} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 | {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 1 | {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
-1 {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
0
2 : image 0 <-- image {-3} | 1 0 | <-- image {-2} | 1 0 0 0 0 0 0 0 0 0 0 | <-- image {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {2} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {4} | 1 0 0 0 0 0 0 | <-- image 0 <-- image 0
{-3} | 0 1 | {-2} | 0 1 0 0 0 0 0 0 0 0 0 | {0} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {4} | 0 1 0 0 0 0 0 |
-3 {-2} | 0 0 1 0 0 0 0 0 0 0 0 | {1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 1 0 0 0 0 | 3 4
-2 {-1} | 0 0 0 1 0 0 0 0 0 0 0 | {1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 1 0 0 0 |
{0} | 0 0 0 0 1 0 0 0 0 0 0 | {1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 |
{0} | 0 0 0 0 0 1 0 0 0 0 0 | {0} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 |
{0} | 0 0 0 0 0 0 1 0 0 0 0 | {1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 |
{-1} | 0 0 0 0 0 0 0 1 0 0 0 | {1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
{0} | 0 0 0 0 0 0 0 0 1 0 0 | {1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 | 2
{0} | 0 0 0 0 0 0 0 0 0 1 0 | {0} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
{0} | 0 0 0 0 0 0 0 0 0 0 1 | {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
-1 {1} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 1
{1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
0
2 11 21 18 7 1
3 : 0 <-- A <-- A <-- A <-- A <-- A <-- A <-- 0
-3 -2 -1 0 1 2 3 4
o5 : FilteredComplex
|
Notice that the display above shows that these are different filtered complexes. The resulting spectral sequences take the form:
i6 : E' = prune spectralSequence F'; |
i7 : E'' = prune spectralSequence F'' ; |
i8 : E' ^0
+-------+-------+------+
| 2 | 3 | 1 |
o8 = |A |A |A |
| | | |
|{-2, 3}|{-1, 3}|{0, 3}|
+-------+-------+------+
| 8 | 12 | 4 |
|A |A |A |
| | | |
|{-2, 2}|{-1, 2}|{0, 2}|
+-------+-------+------+
| 8 | 12 | 4 |
|A |A |A |
| | | |
|{-2, 1}|{-1, 1}|{0, 1}|
+-------+-------+------+
| 2 | 3 | 1 |
|A |A |A |
| | | |
|{-2, 0}|{-1, 0}|{0, 0}|
+-------+-------+------+
o8 : SpectralSequencePage
|
i9 : E' ^ 0 .dd
o9 = {0, -4} : 0 <----- 0 : {0, -3}
0
{0, -3} : 0 <----- 0 : {0, -2}
0
{0, -2} : 0 <----- 0 : {0, -1}
0
1
{0, -1} : 0 <----- A : {0, 0}
0
1 4
{0, 0} : A <----------------------------------- A : {0, 1}
| yz-xw y3-x2z xz2-y2w z3-yw2 |
4 4
{0, 1} : A <--------------------------- A : {0, 2}
{2} | -y2 -xz -yw -z2 |
{3} | z w 0 0 |
{3} | x y -z -w |
{3} | 0 0 x y |
4 1
{0, 2} : A <-------------- A : {0, 3}
{4} | w |
{4} | -z |
{4} | -y |
{4} | x |
1
{0, 3} : A <----- 0 : {0, 4}
0
{-1, -3} : 0 <----- 0 : {-1, -2}
0
{-1, -2} : 0 <----- 0 : {-1, -1}
0
3
{-1, -1} : 0 <----- A : {-1, 0}
0
3 12
{-1, 0} : A <------------------------------------------------------------------------------------------------------------ A : {-1, 1}
{-2} | -yz+xw -y3+x2z -xz2+y2w -z3+yw2 0 0 0 0 0 0 0 0 |
{-2} | 0 0 0 0 -yz+xw -y3+x2z -xz2+y2w -z3+yw2 0 0 0 0 |
{-2} | 0 0 0 0 0 0 0 0 -yz+xw -y3+x2z -xz2+y2w -z3+yw2 |
12 12
{-1, 1} : A <----------------------------------------------- A : {-1, 2}
{0} | y2 xz yw z2 0 0 0 0 0 0 0 0 |
{1} | -z -w 0 0 0 0 0 0 0 0 0 0 |
{1} | -x -y z w 0 0 0 0 0 0 0 0 |
{1} | 0 0 -x -y 0 0 0 0 0 0 0 0 |
{0} | 0 0 0 0 y2 xz yw z2 0 0 0 0 |
{1} | 0 0 0 0 -z -w 0 0 0 0 0 0 |
{1} | 0 0 0 0 -x -y z w 0 0 0 0 |
{1} | 0 0 0 0 0 0 -x -y 0 0 0 0 |
{0} | 0 0 0 0 0 0 0 0 y2 xz yw z2 |
{1} | 0 0 0 0 0 0 0 0 -z -w 0 0 |
{1} | 0 0 0 0 0 0 0 0 -x -y z w |
{1} | 0 0 0 0 0 0 0 0 0 0 -x -y |
12 3
{-1, 2} : A <-------------------- A : {-1, 3}
{2} | -w 0 0 |
{2} | z 0 0 |
{2} | y 0 0 |
{2} | -x 0 0 |
{2} | 0 -w 0 |
{2} | 0 z 0 |
{2} | 0 y 0 |
{2} | 0 -x 0 |
{2} | 0 0 -w |
{2} | 0 0 z |
{2} | 0 0 y |
{2} | 0 0 -x |
3
{-1, 3} : A <----- 0 : {-1, 4}
0
{-1, 4} : 0 <----- 0 : {-1, 5}
0
{-2, -2} : 0 <----- 0 : {-2, -1}
0
2
{-2, -1} : 0 <----- A : {-2, 0}
0
2 8
{-2, 0} : A <-------------------------------------------------------------------- A : {-2, 1}
{-3} | yz-xw y3-x2z xz2-y2w z3-yw2 0 0 0 0 |
{-3} | 0 0 0 0 yz-xw y3-x2z xz2-y2w z3-yw2 |
8 8
{-2, 1} : A <-------------------------------------------- A : {-2, 2}
{-1} | -y2 -xz -yw -z2 0 0 0 0 |
{0} | z w 0 0 0 0 0 0 |
{0} | x y -z -w 0 0 0 0 |
{0} | 0 0 x y 0 0 0 0 |
{-1} | 0 0 0 0 -y2 -xz -yw -z2 |
{0} | 0 0 0 0 z w 0 0 |
{0} | 0 0 0 0 x y -z -w |
{0} | 0 0 0 0 0 0 x y |
8 2
{-2, 2} : A <----------------- A : {-2, 3}
{1} | w 0 |
{1} | -z 0 |
{1} | -y 0 |
{1} | x 0 |
{1} | 0 w |
{1} | 0 -z |
{1} | 0 -y |
{1} | 0 x |
2
{-2, 3} : A <----- 0 : {-2, 4}
0
{-2, 4} : 0 <----- 0 : {-2, 5}
0
{-2, 5} : 0 <----- 0 : {-2, 6}
0
{-3, -1} : 0 <----- 0 : {-3, 0}
0
{-3, 0} : 0 <----- 0 : {-3, 1}
0
{-3, 1} : 0 <----- 0 : {-3, 2}
0
{-3, 2} : 0 <----- 0 : {-3, 3}
0
{-3, 3} : 0 <----- 0 : {-3, 4}
0
{-3, 4} : 0 <----- 0 : {-3, 5}
0
{-3, 5} : 0 <----- 0 : {-3, 6}
0
{-3, 6} : 0 <----- 0 : {-3, 7}
0
o9 : SpectralSequencePageMap
|
i10 : E'' ^0
+-------+-------+-------+-------+
| 1 | 4 | 4 | 1 |
o10 = |A |A |A |A |
| | | | |
|{0, 0} |{1, 0} |{2, 0} |{3, 0} |
+-------+-------+-------+-------+
| 3 | 12 | 12 | 3 |
|A |A |A |A |
| | | | |
|{0, -1}|{1, -1}|{2, -1}|{3, -1}|
+-------+-------+-------+-------+
| 2 | 8 | 8 | 2 |
|A |A |A |A |
| | | | |
|{0, -2}|{1, -2}|{2, -2}|{3, -2}|
+-------+-------+-------+-------+
o10 : SpectralSequencePage
|
i11 : E'' ^1
+--------------------------+-----------------------------------------------------+----------------------------------------------------+-------------------------+
o11 = |cokernel {-3} | z y x ||cokernel {-1} | z y x 0 0 0 0 0 0 0 0 0 ||cokernel {1} | z y x 0 0 0 0 0 0 0 0 0 ||cokernel {2} | z y x ||
| {-3} | -w -z -y || {0} | 0 0 0 z y x 0 0 0 0 0 0 || {1} | 0 0 0 z y x 0 0 0 0 0 0 || {2} | -w -z -y ||
| | {0} | 0 0 0 0 0 0 z y x 0 0 0 || {1} | 0 0 0 0 0 0 z y x 0 0 0 || |
|{0, -2} | {0} | 0 0 0 0 0 0 0 0 0 z y x || {1} | 0 0 0 0 0 0 0 0 0 z y x ||{3, -2} |
| | {-1} | -w -z -y 0 0 0 0 0 0 0 0 0 || {1} | -w -z -y 0 0 0 0 0 0 0 0 0 || |
| | {0} | 0 0 0 -w -z -y 0 0 0 0 0 0 || {1} | 0 0 0 -w -z -y 0 0 0 0 0 0 || |
| | {0} | 0 0 0 0 0 0 -w -z -y 0 0 0 || {1} | 0 0 0 0 0 0 -w -z -y 0 0 0 || |
| | {0} | 0 0 0 0 0 0 0 0 0 -w -z -y || {1} | 0 0 0 0 0 0 0 0 0 -w -z -y || |
| | | | |
| |{1, -2} |{2, -2} | |
+--------------------------+-----------------------------------------------------+----------------------------------------------------+-------------------------+
o11 : SpectralSequencePage
|