[Math] Local-global properties (localization): free, projective, injective, flat, torsion-free, etc

Question

Let $R$ be a commutative unital ring. We say that a property $(\ast)$ of modules is local-global when the following conditions are equivalent for any $R$-module $M$:

1. $M$ is a $(\ast)$ $R$-module;
2. $S^{-1}M$ is a $(\ast)$ $S^{-1}R$-module for any multiplicatively closed subset $S\!\subseteq\!R$;
3. $M_\mathfrak{p}$ is a $(\ast)$ $R_\mathfrak{p}$-module for any prime ideal $\mathfrak{p}\!\unlhd\!R$;
4. $M_\mathfrak{m}$ is a $(\ast)$ $R_\mathfrak{m}$-module for any maximal ideal $\mathfrak{m}\!\unlhd\!R$.

I'm asking for examples of local-global properties. I've heard that flat is such a property and that injective isn't. What about free, projective, torsion-free $(\{m\in M: \exists \text{ non-zero-divisor }r\in R\text{ with }rm=0\}=0)$, divisible $(\forall$ non-zero-divisors $r\in R\, \forall m\in M\, \exists x\in M: rx=m)$, …?

Answer 1

The very first local–global property on which many others are built is the following:

$M = 0$ if and only if $M_\mathfrak{m} = 0$ for all maximal ideals $\mathfrak{m}$.

Since localisation is exact, we may now derive the following local–global properties:

• Whether a module homomorphism is injective, surjective, or an isomorphism.
• Exactness of a sequence of modules.
• Flatness of modules.

Freeness is not a local–global property. Indeed, for a finitely-presented module $M$, $M$ is projective if and only if each $M_\mathfrak{p}$ is free, but there are certainly projective modules that are not free.

Nonetheless, the localisation of a free (resp. projective) module is free (resp. projective), so free (resp. projective) resolutions can be localised as well. This implies:

• $\mathrm{Tor}^R_n (M, N) = 0$ if and only if $\mathrm{Tor}^{R_\mathfrak{m}}_n (M_\mathfrak{m}, N_\mathfrak{m}) = 0$ for all maximal ideals $\mathfrak{m}$.

This may be regarded as a generalisation of the fact that flatness is a local–global property.