Abstract
Let
{\mathcal{K}}
be an abelian category that has enough injective objects,
let
{T\colon\mathcal{K}\to A}
be any left exact covariant additive functor to an abelian category A and let
{T^{(i)}}
be a right derived functor,
{u\geq 1}
, [S. Mardešić,
Strong Shape and Homology,
Springer Monogr. Math.,
Springer, Berlin, 2000].
If
{T^{(i)}=0}
for
{i\geq 2}
and
{T^{(i)}C_{n}=0}
for all
{n\in\mathbb{Z}}
, then there is an exact sequence
0\longrightarrow T^{(1)}H_{n+1}(C_{*})\longrightarrow H_{n}(TC_{*})%
\longrightarrow TH_{n}(C_{*})\longrightarrow 0,
where
{C_{*}=\{C_{n}\}}
is a chain complex in the category
{\mathcal{K}}
,
{H_{n}(C_{*})}
is the homology of the chain complex
{C_{*}}
,
{TC_{*}}
is a chain complex in the category A, and
{H_{n}(TC_{*})}
is the homology of the chain complex
{TC_{*}}
. This exact sequence is the well known Künneth’s correlation.
In the present paper Künneth’s correlation is generalized. Namely, the conditions are found under which the infinite exact sequence
\displaystyle\cdots\longrightarrow T^{(2i+1)}H_{n+i+1}\longrightarrow\cdots%
\longrightarrow T^{(3)}H_{n+2}\longrightarrow T^{(1)}H_{n+1}\longrightarrow H_%
{n}(TC_{*})
\displaystyle\longrightarrow TH_{n}(C_{*})\longrightarrow T^{(2)}H_{n+1}%
\longrightarrow T^{(4)}H_{n+2}\longrightarrow\cdots\longrightarrow T^{(2i)}H_{%
n+i}\longrightarrow\cdots
holds, where
{T^{(2i+1)}H_{n+i+1}=T^{(2i+1)}H_{n+i+1}(C_{*})}
,
{T^{(2i)}H_{n+i}=T^{(2i)}H_{n+i}(C_{*})}
.
The formula makes it possible to generalize Milnor’s formula for the cohomologies of an arbitrary complex, relatively to the Kolmogorov homology to the Alexandroff–Čech homology for a compact space, to a generative result of Massey for a local compact Hausdorff space X and a direct system
{\{U\}}
of open subsets U of X such that
{\overline{U}}
is a compact subset of X.
A direct system for the 3x + 1 dynamics
