Derived functors of Hom relative to \(n\)-flat covers.
A left \(R\)-module \(M\) is called \(n\)-flat if \(\text{Tor}_1(N,M)=0\) for all finitely presented right \(R\)-modules \(N\) with projective dimension \(\leq n\) [\textit{S. B. Lee}, Commun. Algebra 30, No. 3, 1119-1126 (2002; Zbl 1022.16001)]. The category of all \(n\)-flat left \(R\)-modules is denoted by \(\mathfrak F_n\). It is known that every module over any ring has a flat cover [\textit{L. Bican} et al., Bull. Lond. Math. Soc. 33, No. 4, 385-390 (2001; Zbl 1029.16002)]. One of the main aims of this paper is to show that a similar result holds with respect to \(n\)-flat covers. This is done via the concept of the \(n\)-flat \(n\)-cotorsion theory. The construction of an \(n\)-flat resolution of an \(R\)-module \(M\) is subsequently described, for any right \(n\)-coherent ring \(R\). This leads to new derived functors, denoted \(F_n\text{ext}^i(N,M)\). It is shown that these functors are well defined. A canonical map \(\sigma\colon F_n\text{ext}^0(N,M)\to\Hom(N,M)\) is found and \(n\)-flat modules as well as modules with \(\mathfrak F_n\)-\(\dim(M)\leq 1\) and with \(\mathfrak F_n\)-\(\dim(M)\leq m\) (\(m\geq 2\)) are characterized in terms of this canonical map.
scientific article; zbMATH DE number 6484013