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Flat modules and lifting of finitely generated projective modules. (English) Zbl 1107.16004

Let \(A\) be a set with transitive relation \(<\). A net in \(A\) is defined to be a pair \((\Lambda,\psi)\) such that \(\Lambda\) is a nonempty partially ordered set with neither a greatest nor a least element and both upward and downward directed and \(\psi\colon\Lambda\to A\) a strictly increasing map. Instead of \(\psi(\lambda)\), the notation \(a_\lambda\) is used, and the standard notation used for the net is \((a_\lambda)_{\lambda\in\Lambda}\).
Let \((s_\lambda)_{\lambda\in\Lambda}\) be a net in a ring \(S\) with the transitive relation \(<\) defined by \(s<t\) if \(ts=s\) for \(s,t\in S.\) Then it is shown that \((s_\lambda S)_{\lambda\in\Lambda}\) is a net in the set \(L(S_S)\) of all submodules of \(S_S\) with the transitive relation \(\subseteq\). The canonical projections \(S/s_\lambda S\to S/s_\mu S\) form a direct system of right \(S\)-modules over the upward directed set \(\Lambda\). The direct limit \(S/\bigcup_{\lambda\in\Lambda}s_\lambda S\) is called the upper limit of the net \((s_\lambda)_{\lambda\in\Lambda}\) and is denoted by \(\varlimsup_S(s_\lambda)_{\lambda\in\Lambda}\). Similarly, \((1-s_\lambda)_{\lambda\in\Lambda^{op}}\) is shown to be a net in \(S^{op}\) defined on the opposite partially ordered set \(\Lambda^{op}\) of \(\Lambda\). The canonical projections \(S/S(1-s_\mu)\to S/S(1-s_\lambda)\) give a direct system of left \(S\)-modules over \(\Lambda^{op}\). The direct limit of this system is called the lower limit of the net \((s_\lambda)_{\lambda\in\Lambda}\) and is denoted by \(\varliminf_S(s_\lambda)_{\lambda\in\Lambda}\). The upper limit is shown to be a cyclic flat right \(S\)-module and the lower limit a cyclic flat left \(S\)-module. Tensoring the upper limit or the lower limit by a bimodule from right or left, produces a pure exact sequence.
If \(M_R\) is any flat right \(R\)-module and \(F_R\to M_R\) is an epimorphism with \(F_R\cong R_R^{(I)}\) a free right \(R\)-module, a ring \(S\) is constructed and a net \((A_\lambda)_{\lambda\in\Lambda}\) is formed in \(S\) such that \(\varlimsup_S (A_\lambda)_{\lambda\in\Lambda}\otimes_SF\cong M_R\). A dual construction with respect to the lower limit, right tensor and flat left \(R\)-modules is also made. Thus all flat right and all flat left modules arise from suitable nets.
Two examples involving known flat modules are discussed. The second one is based on work of I. I. Sakhajev [Math. Nachr. 130, 157-175 (1987; Zbl 0617.13009)] and is used in the last two sections to study lifting of projective modules modulo the Jacobson radical. For a finitely generated projective right \(R/J(R)\)-module \(P\), a finitely generated flat right \(R\)-module \(M\) with \(M/MJ(R)\cong P\) is shown to exist if and only if there exists a projective left \(R\)-module \(P'\) with \(P'/J(R)P'\) isomorphic to the dual of \(P\). Various other equivalences of this property are also found.

MSC:

16D40 Free, projective, and flat modules and ideals in associative algebras

Citations:

Zbl 0617.13009
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