Pure and finitely presentable modules, duality homomorphisms and the coherence property of a ring

The homological properties of pure modules are considered, showing, in particular, that for coherent rings the pure modules occupy roughly the same position with respect to injective modules as the flat with respect to projective (for arbitrary rings). The duality homomorphisms $\operatorname{Tor}_p(A^*,F)\to\operatorname{Ext}^p(F,A)^*$ are examined in situations where they are not isomorphisms; dependence of the structure of these homomorphisms on the finite presentability or the purity of the modules $F$ and $A$, as well as on the coherence of the base ring, is studied. Characterizations of pure and flat modules in terms of duality, and characterizations of coherence, semihereditariness and noetherianness in terms of duality, purity and finite presentability are given.
Bibliography: 21 titles.