Let $𝒯$ be a weak torsion class of left $R$-modules and $n$ a positive integer. A left $R$-module $M$ is called $(𝒯,n)$-injective if ${\mathrm{Ext}}_R^n(C,M)=0$ for each $(𝒯,n+1)$-presented left $R$-module $C$; a right $R$-module $M$ is called $(𝒯,n)$-flat if ${\mathrm{Tor}}_n^R(M,C)=0$ for each $(𝒯,n+1)$-presented left $R$-module $C$; a left $R$-module $M$ is called $(𝒯,n)$-projective if ${\mathrm{Ext}}_R^n(M,N)=0$ for each $(𝒯,n)$-injective left $R$-module $N$; the ring $R$ is called strongly $(𝒯,n)$-coherent if whenever $0\to K\to P\to C\to 0$ is exact, where $C$ is $(𝒯,n+1)$-presented and $P$ is finitely generated projective, then $K$ is $(𝒯,n)$-projective; the ring $R$ is called $(𝒯,n)$-semihereditary...