# FilteredComplex ^ ZZ -- the filtered pieces

## Synopsis

• Usage:
C = K ^ j
• Operator: ^
• Inputs:
• j, an integer, an integer, infinity, or -infinity
• Outputs:
• C,

## Description

Returns the chain complex in (cohomological) filtration degree j. The relationship K j = K (-j) holds.

 `i1 : A = QQ[x,y];` `i2 : C = koszul vars A;` ```i3 : K = filteredComplex C o3 = -1 : image 0 <-- image 0 <-- image 0 0 1 2 0 : image | 1 | <-- image 0 <-- image 0 0 1 2 1 : image | 1 | <-- image {1} | 1 0 | <-- image 0 {1} | 0 1 | 0 2 1 1 2 1 2 : A <-- A <-- A 0 1 2 o3 : FilteredComplex``` ```i4 : K_0 o4 = image | 1 | <-- image 0 <-- image 0 0 1 2 o4 : ChainComplex``` ```i5 : K_1 o5 = image | 1 | <-- image {1} | 1 0 | <-- image 0 {1} | 0 1 | 0 2 1 o5 : ChainComplex``` ```i6 : K_2 1 2 1 o6 = A <-- A <-- A 0 1 2 o6 : ChainComplex``` ```i7 : K^(-1) o7 = image | 1 | <-- image {1} | 1 0 | <-- image 0 {1} | 0 1 | 0 2 1 o7 : ChainComplex``` ```i8 : K^(-2) 1 2 1 o8 = A <-- A <-- A 0 1 2 o8 : ChainComplex``` ```i9 : K_infinity 1 2 1 o9 = A <-- A <-- A 0 1 2 o9 : ChainComplex``` ```i10 : K_(-infinity) o10 = image 0 <-- image 0 <-- image 0 0 1 2 o10 : ChainComplex``` ```i11 : K^(-infinity) 1 2 1 o11 = A <-- A <-- A 0 1 2 o11 : ChainComplex``` ```i12 : K^infinity o12 = image 0 <-- image 0 <-- image 0 0 1 2 o12 : ChainComplex```