2 - Purity

Summary

This chapter presents basic results on purity and certain kinds of modules (absolutely pure, flat, coherent). Then detailed connections between the shape of pp conditions and direct-sum decomposition of finitely presented modules, especially over RD rings, are established.

Purity

Sections 2.1.1 and 2.1.3 contain the definition of purity (of embeddings, of exact sequences, of epimorphisms) and basic results on this. Pure-projective modules are introduced in Section 2.1.2: the dual concept of pure-injectivity is the theme of Chapter 4.

Homological dimensions based on pure-exact sequences are described in Section 2.2.

For purity and pure-injectivity in more general contexts see, for example, [86], [622].

Pure-exact sequences

Purity is defined first in terms of pp conditions/solutions of systems of equations; there are other characterisations (2.1.7, 2.1.19, 2.1.28). An exact sequence is pure iff it is a direct limit of split exact sequences (2.1.4). There is a characterisation of pure epimorphisms in terms of lifting of pp conditions (2.1.14). A short exact sequence which is pure is split if its last non-zero term is finitely presented (2.1.18). Every subset of a module is contained in a pure submodule which is not too much bigger (2.1.21).