Syzygies and Homotopy Theory pp 117-128 | Cite as

# Finiteness Conditions

## Abstract

As no restrictions were placed on the modules involved, the derived module category \(\mathcal{D}\mathrm{er}\) of Chap. 5 has an absolute character. This is unrealistic, and in practice it is necessary to limit the size of modules. Although our primary interest is in modules which are finitely generated, it is inconvenient to impose this restriction from the outset. We assume that the ring *Λ* is *weakly coherent* and initially restrict attention to modules which are countably generated. The appropriate derived module category is denoted by \(\mathcal {D}\mathrm{er}_{\infty}\). Recalling that any countably generated module *M* has a hyperstabilization \(\widehat{M} = M \oplus\varLambda^{\infty}\), the objects of \(\mathcal{D}\mathrm{er}_{\infty}\) have a convenient representation as hyperstable modules. Thereafter we consider the difficulties involved in imposing the more stringent restriction of finite generation; compare (Serre in Prospects in mathematics, Annals of mathematics studies, vol. 70, pp. 77–169. Princeton University Press, Princeton, 1971).

### Keywords

Coherence### References

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