Syzygies and Homotopy Theory by F.E.A. Johnson

Author:F.E.A. Johnson
Language: eng
Format: epub
Publisher: Springer London, London

A ring Λ is said to have a stable range when there exists an integer n≥0 such that there is no nontrivial finitely generated stably free module S with n<rk(S). We write σ(Λ)=n when there is a nontrivial stably free module of rank =n but none of rank >n. This definition of σ(Λ) is normalized so that σ(F)=0 when F is a field. Moreover, if Λ has no stable range we write σ(Λ)=+∞.

Theorem 7.34

Let J be tame; if max{r(J),σ(Λ)+1}≤n then J⊕Λ n is full.

Proof

In essence the proof repeats Theorem 7.4 with more precise book-keeping. Since J is tame we may first choose a free 0-complex of finite type

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