Local splitters for bounded cotorsion theories (Q2769480)

Let \(R\) be a ring. The notion of cotorsion theory was introduced by Salce and it has been important in the solution by Enochs of the flat conjecture. Two classes of modules \({\mathcal C}=({\mathcal A},{\mathcal B})\) form a cotorsion theory provided that \({\mathcal A}={^\perp{\mathcal B}}=\{M\in R\text{-Mod}\mid\text{Ext}_R^1(M,B)=0\;\forall B\in{\mathcal B}\}\) and \({\mathcal B}={\mathcal A}^\perp=\{M\in R\text{-Mod}\mid\text{Ext}_R^1(A,M)=0\;\forall A\in{\mathcal A}\}\). A cotorsion theory \(\mathcal C\) is called bounded if there exists a cardinal \(\rho\) such that for each \(A\in{\mathcal A}\) there exists a cardinal \(\lambda_A\) and a strictly increasing sequence of submodules of \(A\), \((A_\alpha\mid\alpha<\lambda_A)\), such that \(A=\bigcup_{\alpha<\lambda_A}A_\alpha\), \(A_{\alpha+1}/A_\alpha\in{\mathcal A}\) and \(\text{card}(A_{\alpha+1}/A_\alpha)\leq\rho\) for all \(\alpha<\lambda_A\). The kernel of a cotorsion theory is \({\mathcal A}\cap{\mathcal B}\), and a module \(K\) in the kernel of \(\mathcal C\) is said to be a strong splitter for \(\mathcal C\) if \(K^{(\omega)}\) is also in the kernel. Moreover, \(K\) is a local splitter for \(\mathcal C\) if it is a strong splitter and there exists a non-split embedding \(\nu\colon K^{(\omega)}\to K^{(\omega)}\) such that for each \(n<\omega\), there exists a submodule \(C_n\subseteq K^{(\omega)}\) such that \(\nu(K^{(\omega)})\oplus C_n=K^{(\omega)}\) and \(\nu(K^{(\omega-n)})\subseteq C_n\).NEWLINENEWLINENEWLINEIt is shown that if \(\mathcal C\) is a bounded cotorsion theory and \(K\) is a local splitter for \(\mathcal C\), then Shelah's uniformization principle UP implies that \(\mathcal C\) is not generated by \(K\) (i.e. \({\mathcal A}\subset{^\perp K}\)). Some applications of this result for Whitehead modules are given.NEWLINENEWLINENEWLINEIn Section 2, cotorsion theories not cogenerated by any set of modules are constructed assuming UP and GCH.