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Projective Approximations

Published online by Cambridge University Press:  20 November 2018

K. Varadarajan
K. Varadarajan*
Affiliation:
University of Calgary, Calgary, Alberta
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Let R be an associative ring with 1 ≠ 0. Throughout we will be considering unitary left R-modules. Given a chain complex C over R, a free approximation of C is defined to be a free chain complex F over R together with an epimorphism τ:F → C of chain complexes with the property that H(τ):H(F) ≃ H(C). In Chapter 5, Section 2 of [3] it is proved that any chain complex C over Z has a free approximation τ:F → C. Moreover given a free approximation τ:F → C of C and any chain map f:F’ → C with F’ a free chain complex over Z, there exists a chain map φ:F’→ F with T O φ = f . Any two chain maps φ, ψ of F’ in F with T O φ = T O ψ are chain homotopic.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 1 , 01 February 1984 , pp. 178 - 192
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Cartan, H. and Eilenberg, S., Homological algebra (Princeton University Press, 1956).Google Scholar
2. Dold, A., Zur Homotopietheorie der Kettencomplexe, Math. Annalen 140 (1960), 278–298.Google Scholar
3. Spanier, E. H., Algebraic topology (McGraw-Hill Book Company, 1966).Google Scholar