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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

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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