On modules and submodules with finite projective dimension (Q2738763)

Let \(R\) be an associative ring with identity and \(A\) a submodule of an \(R\)-module \(B\). A family \(\mathcal C\) of submodules of \(B\) containing \(A\) is called a \(G(\aleph_n)\)-family of submodules of \(B\) over \(A\) provided (a) \(A,B\in{\mathcal C}\); (b) \(\mathcal C\) is closed with respect to taking the unions of ascending chains; and (c) for any \(C\in{\mathcal C}\) and any subset \(X\) of \(B\) with cardinality not exceeding \(\aleph_n\), there exists \(D\in{\mathcal C}\) such that \(C\cup X\subseteq D\) and \(D/C\) is \(\aleph_n\)-generated (has a generating set of cardinality no greater than \(\aleph_n\)). The ring \(R\) is called \(\aleph_n\)-Noetherian if every ideal of \(R\) is \(\aleph_n\)-generated.NEWLINENEWLINENEWLINEThe main theorem of the paper is a generalization of Kaplansky's theorem on projective modules to modules of finite projective dimension.NEWLINENEWLINENEWLINETheorem: Let \(n\) be a nonnegative integer and suppose \(R\) is \(\aleph_n\)-Noetherian and \(B\) is an \(R\)-module with \(\text{pd}(B)=n\). Then the following are equivalent for a submodule \(A\) of \(B\): (i) \(\text{pd}(A)\leq n\); (ii) There is a \(G(\aleph_n)\)-family \(\mathcal C\) of submodules of \(B\) over \(A\), each of which has projective dimension \(\leq n\); (iii) There is a smooth well-ordered increasing chain of submodules NEWLINE\[NEWLINEA=A_0<\cdots<A_\alpha<A_{\alpha+1}<\cdots<A_\tau=\bigcup_{\alpha<\tau}A_\alpha=B,NEWLINE\]NEWLINE where \(\tau\) is some ordinal number and, for each \(\alpha<\tau\), \(\text{pd}(A_\alpha)\leq n\) and \(A_{\alpha+1}/A_\alpha\) is \(\aleph_n\)-generated.NEWLINENEWLINENEWLINEA submodule \(S\) of a module \(M\) is called tight if both \(\text{pd}(S)\leq\text{pd}(M)\) and \(\text{pd}(M/S)\leq\text{pd}(M)\). Under additional assumptions, it can be stipulated that the submodules in the theorem are all tight.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00043].