# Filtrations and tensor product complexes

Let S be a commutative ring and let B : …→Bi →Bi - 1 →… and C : …→Ci →Ci - 1 →… be chain complexes.

For all integers p and q let Kp,q := BpS Cq, let d’p,q : Kp,q →Kp - 1, q denote the homorphism Bp ⊗1, and let d”p,q : Kp,q →Kp, q - 1 denote the homorphism (-1)p ⊗∂qC .

The chain complex B ⊗S C is given by (B ⊗S C)k := ⊕p + q = k BpS Cq and the differentials by ∂:= d’ + d”. It carries two natural ascending filtrations F’B ⊗S C and F” B ⊗S C.

The first is obtained by letting F’n (B ⊗S C) be the chain complex determined by setting F’n (B ⊗S C)k := ⊕p + q = k , p ≤n BpS Cq and the differentials ∂:= d’ + d”.

The second is obtained by letting F”n (B ⊗S C) be the chain complex determined by setting F”n (B ⊗S C)k := ⊕p + q = k , q ≤n BpS Cq and the differentials ∂:= d’ + d”.

In Macaulay2 we can compute these filtered complexes as follows.

 `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' = (filteredComplex B) ** C o4 = -1 : image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 0 1 2 3 4 5 6 7 0 : image | 1 | <-- image {2} | 1 0 0 0 0 0 0 | <-- image {4} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {5} | 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 0 0 0 0 0 | {4} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 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 1 0 0 0 0 | {4} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 4 5 6 7 {3} | 0 0 0 1 0 0 0 | {4} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 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 | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 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 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 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 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 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 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 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 | {6} | 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 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 2 {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 3 1 : image | 1 | <-- image {2} | 1 0 0 0 0 0 0 | <-- image {4} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {5} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {7} | 1 0 0 0 0 0 0 0 0 0 0 | <-- image 0 <-- image 0 <-- image 0 {3} | 0 1 0 0 0 0 0 | {4} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {7} | 0 1 0 0 0 0 0 0 0 0 0 | 0 {3} | 0 0 1 0 0 0 0 | {4} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {7} | 0 0 1 0 0 0 0 0 0 0 0 | 5 6 7 {3} | 0 0 0 1 0 0 0 | {4} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {7} | 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 1 0 0 | {4} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {7} | 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 1 0 | {5} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {7} | 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 1 | {5} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {7} | 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | {7} | 0 0 0 0 0 0 0 0 0 0 0 | 1 {4} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 | {7} | 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 | {7} | 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 | {7} | 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 1 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 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 | 4 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | {6} | 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 | {6} | 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 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 2 {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 3 1 7 18 21 11 2 2 : A <-- A <-- A <-- A <-- A <-- A <-- 0 <-- 0 0 1 2 3 4 5 6 7 o4 : FilteredComplex``` ```i5 : F'' = B ** (filteredComplex C) o5 = -1 : image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0 0 1 2 3 4 5 6 7 0 : image | 1 | <-- image {2} | 0 0 0 0 0 0 0 | <-- image {4} | 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 <-- image 0 {3} | 0 0 0 0 0 0 0 | {4} | 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 | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 3 4 5 6 7 {3} | 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 1 0 0 | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 1 0 | {5} | 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 1 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 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 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 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 1 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 2 1 : image | 1 | <-- image {2} | 1 0 0 0 0 0 0 | <-- image {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {5} | 0 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 0 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 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 1 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 4 5 6 7 {3} | 0 0 0 1 0 0 0 | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 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 1 0 0 | {4} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 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 1 0 | {5} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 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 1 | {5} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 {4} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 | {6} | 0 0 0 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 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | 2 {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 3 2 : image | 1 | <-- image {2} | 1 0 0 0 0 0 0 | <-- image {4} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image {7} | 0 0 0 0 0 0 0 0 0 0 0 | <-- image 0 <-- image 0 <-- image 0 {3} | 0 1 0 0 0 0 0 | {4} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {7} | 0 0 0 0 0 0 0 0 0 0 0 | 0 {3} | 0 0 1 0 0 0 0 | {4} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {7} | 0 0 0 0 0 0 0 0 0 0 0 | 5 6 7 {3} | 0 0 0 1 0 0 0 | {4} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {7} | 0 0 0 1 0 0 0 0 0 0 0 | {2} | 0 0 0 0 1 0 0 | {4} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {7} | 0 0 0 0 1 0 0 0 0 0 0 | {2} | 0 0 0 0 0 1 0 | {5} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {7} | 0 0 0 0 0 1 0 0 0 0 0 | {2} | 0 0 0 0 0 0 1 | {5} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {7} | 0 0 0 0 0 0 1 0 0 0 0 | {5} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | {7} | 0 0 0 0 0 0 0 1 0 0 0 | 1 {4} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 | {7} | 0 0 0 0 0 0 0 0 1 0 0 | {5} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 | {7} | 0 0 0 0 0 0 0 0 0 1 0 | {5} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 | {7} | 0 0 0 0 0 0 0 0 0 0 1 | {5} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 1 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 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 | 4 {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | 2 {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 3 1 7 18 21 11 2 3 : A <-- A <-- A <-- A <-- A <-- A <-- 0 <-- 0 0 1 2 3 4 5 6 7 o5 : FilteredComplex```

The pages of the resulting spectral sequences take the form:

 `i6 : E' = prune spectralSequence F';` `i7 : E'' = prune spectralSequence F'';` ```i8 : E' ^0 +------+------+------+ | 1 | 3 | 2 | o8 = |A |A |A | | | | | |{0, 3}|{1, 3}|{2, 3}| +------+------+------+ | 4 | 12 | 8 | |A |A |A | | | | | |{0, 2}|{1, 2}|{2, 2}| +------+------+------+ | 4 | 12 | 8 | |A |A |A | | | | | |{0, 1}|{1, 1}|{2, 1}| +------+------+------+ | 1 | 3 | 2 | |A |A |A | | | | | |{0, 0}|{1, 0}|{2, 0}| +------+------+------+ o8 : SpectralSequencePage``` ```i9 : E' ^ 1 +----------------------------------------+----------------------------------------------------------------------------------------------------+------------------------------------------------------------------------+ o9 = |cokernel | yz-xw z3-yw2 xz2-y2w y3-x2z ||cokernel {2} | yz-xw 0 0 z3-yw2 xz2-y2w y3-x2z 0 0 0 0 0 0 ||cokernel {3} | yz-xw 0 z3-yw2 xz2-y2w y3-x2z 0 0 0 || | | {2} | 0 yz-xw 0 0 0 0 z3-yw2 xz2-y2w y3-x2z 0 0 0 || {3} | 0 yz-xw 0 0 0 z3-yw2 xz2-y2w y3-x2z || |{0, 0} | {2} | 0 0 yz-xw 0 0 0 0 0 0 z3-yw2 xz2-y2w y3-x2z || | | | |{2, 0} | | |{1, 0} | | +----------------------------------------+----------------------------------------------------------------------------------------------------+------------------------------------------------------------------------+ o9 : SpectralSequencePage``` ```i10 : E'' ^0 +------+------+------+------+ | 2 | 8 | 8 | 2 | o10 = |A |A |A |A | | | | | | |{0, 2}|{1, 2}|{2, 2}|{3, 2}| +------+------+------+------+ | 3 | 12 | 12 | 3 | |A |A |A |A | | | | | | |{0, 1}|{1, 1}|{2, 1}|{3, 1}| +------+------+------+------+ | 1 | 4 | 4 | 1 | |A |A |A |A | | | | | | |{0, 0}|{1, 0}|{2, 0}|{3, 0}| +------+------+------+------+ o10 : SpectralSequencePage``` ```i11 : E'' ^1 +------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------+ o11 = |cokernel | z2-yw yz-xw y2-xz ||cokernel {2} | z2-yw yz-xw y2-xz 0 0 0 0 0 0 0 0 0 ||cokernel {4} | z2-yw yz-xw y2-xz 0 0 0 0 0 0 0 0 0 ||cokernel {5} | z2-yw yz-xw y2-xz || | | {3} | 0 0 0 z2-yw yz-xw y2-xz 0 0 0 0 0 0 || {4} | 0 0 0 z2-yw yz-xw y2-xz 0 0 0 0 0 0 || | |{0, 0} | {3} | 0 0 0 0 0 0 z2-yw yz-xw y2-xz 0 0 0 || {4} | 0 0 0 0 0 0 z2-yw yz-xw y2-xz 0 0 0 ||{3, 0} | | | {3} | 0 0 0 0 0 0 0 0 0 z2-yw yz-xw y2-xz || {4} | 0 0 0 0 0 0 0 0 0 z2-yw yz-xw y2-xz || | | | | | | | |{1, 0} |{2, 0} | | +------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------+ o11 : SpectralSequencePage```