The Zariski covering number for vector spaces and modules

Abstract

Given a module M over a commutative unital ring R, let $\sigma (M,R)$ denote the covering number, i.e. the smallest (cardinal) number of proper submodules whose union covers M; this includes the covering numbers of Abelian groups, which are extensively studied in the literature. Recently, Khare–Tikaradze [Comm. Algebra, in press] showed in several cases that $\sigma (M,R)={\min }_{\mathfrak{m}\in S_M}|R/\mathfrak{m}|+1,$ where S_M is the set of maximal ideals $\mathfrak{m}$ with ${\dim }_{R/\mathfrak{m}}(M/\mathfrak{m}M)\ge 2.$ Our first main result extends this equality to all R-modules with small Jacobson radical and finite dual Goldie dimension. We next introduce and study a topological counterpart for finitely generated R-modules M over rings R, whose ‘some’ residue fields are infinite, which we call the Zariski covering number ${\sigma }_{\tau }(M,R).$ To do so, we first define the “induced Zariski topology” τ on M, and now define ${\sigma }_{\tau }(M,R)$ to be the smallest (cardinal) number of proper τ-closed subsets of M whose union covers M. We then show our next main result: ${\sigma }_{\tau }(M,R)={\min }_{\mathfrak{m}\in S_M}|R/\mathfrak{m}|+1,$ for all finitely generated R-modules M for which (a) the dual Goldie dimension is finite, and (b) $\mathfrak{m}\notin S_M$ whenever $R/\mathfrak{m}$ is finite. As a corollary, this alternately recovers the aforementioned formula for the covering number $\sigma (M,R)$ of the aforementioned finitely generated modules. Finally, we discuss the notion of κ-Baire spaces, and show that the inequalities ${\sigma }_{\tau }(M,R)\le \sigma (M,R)\le {\kappa }_M:={\min }_{\mathfrak{m}\in S_M}|R/\mathfrak{m}|+1$ again become equalities when the image of M under the continuous map $q:M\to \prod _{\mathfrak{m}\in \text{mSpec}(R)}M/\mathfrak{m}M$ (with appropriate Zariski-type topologies) is a κ_M-Baire subspace of the product space.

Keywords:

• Baire spaces
• covering number
• dual Goldie dimension
• Zariski topology

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13F05
• 13H99
• 13F10
• Secondary: 54E52
• 54H99

Acknowledgments

The author is thankful to Prof. Amartya Kumar Dutta, Indian Statistical Institute, Kolkata for introducing him to the toy problem for vector spaces, and for motivating him to explore this in a broader setting and in greater depth. Further, the author is extremely grateful to Prof. Apoorva Khare, Indian Institute of Science, Bangalore, for several stimulating and motivating interactions, as well as for a careful and detailed reading of previous versions of this manuscript, which helped greatly improve the exposition.