The alternating constant complex #
Given an object X : C
and endomorphisms φ, ψ : X ⟶ X
such that φ ∘ ψ = ψ ∘ φ = 0
, this file
defines the periodic chain and cochain complexes
... ⟶ X --φ--> X --ψ--> X --φ--> X --ψ--> 0
and 0 ⟶ X --ψ--> X --φ--> X --ψ--> X --φ--> ...
(or more generally for any complex shape c
on ℕ
where c.Rel i j
implies i
and j
have
different parity). We calculate the homology of these periodic complexes.
In particular, we show ... ⟶ X --𝟙--> X --0--> X --𝟙--> X --0--> X ⟶ 0
is homotopy equivalent
to the single complex where X
is in degree 0
.
Let c : ComplexShape ℕ
be such that i j : ℕ
have opposite parity if they are related by
c
. Let φ, ψ : A ⟶ A
be such that φ ∘ ψ = ψ ∘ φ = 0
. This is a complex of shape c
whose
objects are all A
. For all i, j
related by c
, dᵢⱼ = φ
when i
is even, and dᵢⱼ = ψ
when
i
is odd.
Equations
Instances For
The i, j, k
th short complex associated to the alternating constant complex on φ, ψ : A ⟶ A
is A --ψ--> A --φ--> A
when i ~ j, j ~ k
and j
is even.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The i, j, k
th short complex associated to the alternating constant complex on φ, ψ : A ⟶ A
is A --φ--> A --ψ--> A
when i ~ j, j ~ k
and j
is even.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The j
th homology of the alternating constant complex on φ, ψ : A ⟶ A
is the homology of
A --ψ--> A --φ--> A
when prev(j) ~ j, j ~ next(j)
and j
is even.
Equations
- HomologicalComplex.alternatingConstHomologyIsoEven A hOdd hEven hc hpj hnj h = CategoryTheory.ShortComplex.homologyMapIso (HomologicalComplex.alternatingConstScIsoEven A hOdd hEven hc hpj hnj h)
Instances For
The j
th homology of the alternating constant complex on φ, ψ : A ⟶ A
is the homology of
A --φ--> A --ψ--> A
when prev(j) ~ j, j ~ next(j)
and j
is odd.
Equations
- HomologicalComplex.alternatingConstHomologyIsoOdd A hOdd hEven hc hpj hnj h = CategoryTheory.ShortComplex.homologyMapIso (HomologicalComplex.alternatingConstScIsoOdd A hOdd hEven hc hpj hnj h)
Instances For
The chain complex X ←0- X ←𝟙- X ←0- X ←𝟙- X ⋯
.
It is exact away from 0
and has homology X
at 0
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The n
-th homology of the alternating constant complex is zero for non-zero even n
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The n
-th homology of the alternating constant complex is zero for odd n
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The n
-th homology of the alternating constant complex is X
for n = 0
.
Equations
Instances For
The n
-th homology of the alternating constant complex is X
for n ≠ 0
.
The n
-th homology of the alternating constant complex is X
for n = 0
.
Equations
Instances For
The alternating face complex of the constant complex is the alternating constant complex.
Equations
- One or more equations did not get rendered due to their size.
Instances For
alternatingConst.obj X
is homotopy equivalent to the chain
complex (single₀ C).obj X
.
Equations
- One or more equations did not get rendered due to their size.