On Küneth’s correlation and its applications

Abstract

Abstract Let 𝒦 {\mathcal{K}} be an abelian category that has enough injective objects, let T : 𝒦 → A {T\colon\mathcal{K}\to A} be any left exact covariant additive functor to an abelian category A and let T ( i ) {T^{(i)}} be a right derived functor, u ≥ 1 {u\geq 1} , [S. Mardešić, Strong Shape and Homology, Springer Monogr. Math., Springer, Berlin, 2000]. If T ( i ) = 0 {T^{(i)}=0} for i ≥ 2 {i\geq 2} and T ( i ) ⁢ C n = 0 {T^{(i)}C_{n}=0} for all n ∈ ℤ {n\in\mathbb{Z}} , then there is an exact sequence 0 ⟶ T ( 1 ) ⁢ H n + 1 ⁢ ( C * ) ⟶ H n ⁢ ( T ⁢ C * ) ⟶ T ⁢ H n ⁢ ( C * ) ⟶ 0 , 0\longrightarrow T^{(1)}H_{n+1}(C_{*})\longrightarrow H_{n}(TC_{*})% \longrightarrow TH_{n}(C_{*})\longrightarrow 0, where C * = { C n } {C_{*}=\{C_{n}\}} is a chain complex in the category 𝒦 {\mathcal{K}} , H n ⁢ ( C * ) {H_{n}(C_{*})} is the homology of the chain complex C * {C_{*}} , T ⁢ C * {TC_{*}} is a chain complex in the category A, and H n ⁢ ( T ⁢ C * ) {H_{n}(TC_{*})} is the homology of the chain complex T ⁢ C * {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 ⋯ ⟶ T ( 2 ⁢ i + 1 ) ⁢ H n + i + 1 ⟶ ⋯ ⟶ T ( 3 ) ⁢ H n + 2 ⟶ T ( 1 ) ⁢ H n + 1 ⟶ H n ⁢ ( T ⁢ C * ) \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_{*}) ⟶ T ⁢ H n ⁢ ( C * ) ⟶ T ( 2 ) ⁢ H n + 1 ⟶ T ( 4 ) ⁢ H n + 2 ⟶ ⋯ ⟶ T ( 2 ⁢ i ) ⁢ H n + i ⟶ ⋯ \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 ( 2 ⁢ i + 1 ) ⁢ H n + i + 1 = T ( 2 ⁢ i + 1 ) ⁢ H n + i + 1 ⁢ ( C * ) {T^{(2i+1)}H_{n+i+1}=T^{(2i+1)}H_{n+i+1}(C_{*})} , T ( 2 ⁢ i ) ⁢ H n + i = T ( 2 ⁢ i ) ⁢ H n + i ⁢ ( 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 } {\{U\}} of open subsets U of X such that U ¯ {\overline{U}} is a compact subset of X.