Injectivity, Projectivity, and $P$-injectivity of Localization

Question

Let $R$ be a commutative ring with unity.

I have read that if $M$ is an injective $R$-module, then $S^{-1}M$ is not necessarily an injective $S^{-1}R$-module. I need an example...

Does last statement true for $P$-injectivity and projectivity?

Answer 1

The localisation of a projective module is projective since a projective module is a direct summand of a free module, and localisation preserves direct summands and freeness.

A counter-example for injective modules was built by E. Dade in 1981, and you can find it in his paper Localization of injective modules, in which he also gives a sufficient condition for this property to be true.

This condition consists of the following $3$ subconditions on the ring $R$ and the multiplicative set $S$:

1. $R$ is a coherent ring.
2. For any finitely generated ideal $I$ of $R$ and $s\in S$, the chain: $$I\subset (I:s)\subseteq (I:s^2)\subseteq (I:s^3)\subseteq\cdots$$ stabilises.
3. Ideals of $R$ are countably generated.

In particular, it is true for noetherian rings.