Research

All Modules have Covers, MSc Thesis in Pure Mathematics. Supervisor: Ioannis Emmanouil.

Abstract: In this thesis, we demonstrate detailed proof of Bican, Bachir, and Enochs's results so that each module has a flat cover. Historically, the notion of flat modules was introduced by J-P Serre in 1955-1956. A few years later, when the injective envelopes had already been studied, the dual notion of the injective envelopes, the projective covers, was investigated. H. Bass in his 1959 thesis introduced the projective covers and he described the apt rings, in which each module has a projective cover (left/right perfect rings). After this result arose the question of when a module has a flat cover. After many years, the significant progress which had shown by J. Xu, brought to the fore again this open problem, which was finally solved and presented in Enochs's paper "All modules have flat covers", in which was crucial the use of useful lemmas proved by Eklof and stated at his paper "How to make Ext vanish".

A small sample of my thesis:

Theorem: The flat cotorsion theory \(\left(\mathscr{FL}(R),\mathscr{E}(R)\right)\) is complete. Since \(\mathscr{FL}(R)\) is closed under direct limits, \(\mathscr{FL}(R)\) is a cover class.

We consider the set $$\mathscr{S}=\left\{M\ \text{is flat}\mid |M|\le \kappa\right\}$$ We will show that the flat cotorsion theory \(\big(\mathscr{FL}(R),\mathscr{E}(R)\big)\) is cogenerated by \(\mathscr{S}\), equivalently $$\mathscr{E}(R)=\left(\mathscr{FL}(R)\right)^{\perp}=\mathscr{S}^{\perp}$$ The first relation it is obvious since $$\mathscr{S}\subseteq \mathscr{FL}(R)\Rightarrow \mathscr{E}(R)=\left(\mathscr{FL}(R)\right)^{\perp}\subseteq \mathscr{S}^{\perp}$$ Let \(C\in \mathscr{S}\). We shall show that \(\mathrm{Ext}^{1}(F,C)=0\), for every \(F\in \mathscr{FL}(R)\). Let \(F\in \mathscr{FL}(R)\) and \(\left\{F_{\alpha}\mid \alpha<\lambda\right\}\) a flat family of submodules with the properties of the above corollary. Then $$F_{\alpha+1}/F_{\alpha}\in \mathscr{FL}(R)\quad \text{and}\quad \left|F_{\alpha+1}/F_{\alpha}\right|\le \kappa$$ therefore \(F_{\alpha+1}/F_{\alpha}\in \mathscr{S}\). Since $$\mathrm{Ext}^{1}(F_{0},C)=0\quad \text{and}\quad \mathrm{Ext}^{1}(F_{\alpha+1}/F_{\alpha},C)=0\ \forall \alpha+1<\lambda$$ by Lemma ?? we deduce that $$\mathrm{Ext}^{1}(F,C)=\mathrm{Ext}^{1}\left(\bigcup_{\alpha<\lambda}F_{\alpha},C\right)=0$$