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Extensions relative to a Serre class

Published online by Cambridge University Press:  20 January 2009

S. Cormack
S. Cormack
Affiliation:
University of Edinburgh
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Consider a class C of projective R-modules, where R is a commutative ring with identity, which satisfies the conditions of (2), namely that C is closed under the operations of direct sum and isomorphism and C contains the zero module. Following (2) a module M is said to have C-cotype n (respectively C-type n) if it has a projective resolution … → Pn → P0 → M → 0 with Pi ∈ C for i>n (respectively Pi ∈ C for i≦n). Let S be the class of modules of C-cotype −1, equivalently of C-type infinity. It is assumed throughout that S is a Serre Class. We define an abelian category of modules with the property that C-cotype is homological dimension in while in the case C = 0, S is just the category of R-modules. It follows that all categorical results on homological dimension also hold for cotype.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 19 , Issue 4 , September 1975 , pp. 375 - 381
Copyright
Copyright © Edinburgh Mathematical Society 1975

References

REFERENCES

(1) Adams, J. F., Lectures on generalised cohomology, lecture 5. Lecture notes vol. 99 (Springer-Verlag, Berlin, 1969).Google Scholar
(2) Cormack, S., An analogue of homological dimension using a general class of projective modules, J. London Math. Soc. (2) 1 (1969), 760–764.CrossRefGoogle Scholar
(3) Serre, J. P., Groupes d'homotopie et classes de groupes abelièns, Ann. of Math. 58 (1953), 258–294.CrossRefGoogle Scholar