Stably Free Modules
نویسنده
چکیده
Let R be a commutative ring. When an R-module has a particular module-theoretic property after direct summing it with a finite free module, it is said to have the property stably. For example, R-modules M and N are stably isomorphic if Rk ⊕M ∼= Rk ⊕N for some k ≥ 0. An R-module M is stably free if it is stably isomorphic to a free module: Rk ⊕M is free for some k. When M is finitely generated and stably free, then for some k Rk⊕M is finitely generated and free, so Rk⊕M ∼= R` for some `. Necessarily k ≤ ` (why?). Are stably isomorphic modules in fact isomorphic? Is a stably free module actually free? Not always, and that’s why the concepts are interesting. This “stable mathematics” is part of algebraic K-theory. Our purpose here is to describe the simplest example of a non-free module which is stably free and then discuss what it means for all stably free modules over a ring to be free.
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