Linearly-compact module

A topological module over a topological ring that has a basis (cf. Base) of neighbourhoods of zero consisting of submodules, and in which every centred system (or filter base, cf. also Centred family of sets) consisting of cosets with respect to closed submodules has a non-empty intersection. Every linearly-compact module is a complete topological group.

A linearly-compact module is called a linearly-compact module in the narrow sense if every continuous homomorphism onto a topological module that has a basis of neighbourhoods of zero consisting of submodules, is open (cf. Open mapping). A topological module is a linearly-compact module in the narrow sense if and only if it is a complete topological group and if every quotient module of it with respect to an open submodule is an Artinian module. In particular, an Artinian module in the discrete topology is a linearly-compact module in the narrow sense. Thus, linearly-compact modules in the narrow sense are the topological analogues of Artinian modules.

Direct products, closed submodules, quotient modules with respect to closed submodules, and continuous homomorphic images having a basis of neighbourhoods of zero consisting of submodules of linearly-compact modules (in the narrow sense) are themselves linearly-compact modules (in the narrow sense).

References

[1]	S. Lefschetz, "Algebraic topology" , Amer. Math. Soc. (1955)
[2]	D. Zelinsky, "Linearly compact modules and rings" Amer. J. Math. , 75 : 1 (1953) pp. 79–90
[3a]	H. Leptin, "Linear kompakte Moduln und Ringe" Math. Z. , 62 (1955) pp. 241–267
[3b]	H. Leptin, "Linear kompakte Moduln und Ringe II" Math. Z. , 66 (1957) pp. 289–327

Comments

A topological module $M$ over a (topological) field (ring) is said to have a linear topology if there is a base of neighbourhoods of zero consisting of submodules. A linear subvariety $V$ of $M$ is linearly compact if for every system $\{V_\alpha\}$ of closed linear subvarieties of $V$ with the finite intersection property, i.e. every finite intersection of elements of $\{V_\alpha\}$ is non-empty, the intersection $\bigcap V_\alpha$ is non-empty. Such a linear subvariety is closed.

A filtration on a group $G$ (indexed by $\mathbf Z$) is an increasing or decreasing collection of subgroups $(G_n)_{n\in\mathbf Z}$ of $G$. Let $A$ be a ring with a filtration $(A_n)_{n\in\mathbf Z}$ of the underlying additive group of $A$. Such a filtration is said to be compatible with the ring structure if $A_nA_m\subset A_{n+m}$ for all $n,m\in\mathbf Z$ and $1\in A_0$. A filtered ring is a ring provided with such a filtration. Now let $M$ be an $A$-module and let $(M_n)_{n\in\mathbf Z}$ be a filtration of the underlying Abelian group. This filtration is said to be compatible with the filtered ring $A$ if $A_nM_m\subset M_{n+m}$ for all $n,m\in\mathbf Z$. Such a module is called a filtered module over the filtered ring $A$. If $A_0=A$, as is often the case, the $M_n$ are all $A$-submodules of $M$. The definitions in the article filtered module correspond to the case that all $M_n$ are indeed $A$-submodules of $M$. Using the $(A_n)$ and $(M_n)$ as a base of open neighbourhoods in $A$ and $M$, linear topologies are defined on $A$ and $M$. These are the more frequently occurring examples of linearly topologized modules and rings.

References

[a1]	N. Bourbaki, "Algèbre commutative" , Eléments de mathématiques , Hermann (1961) pp. Chapt. 3. Graduations, filtrations, et topologies
[a2]	S. Warner, "Topological fields" , North-Holland (1989) pp. Chapt. 5, Sect.31