I am working my way through Bosch's book "Algebraic Geometry and Commutative Algebra" and found an exercise in chapter 4.4, exercise 3, which I think is missing some prerequisite. The original text is as follows:

Consider a ring morphism $R\rightarrow R'$, an $R'$-module $M'$, as well as its restriction $M'_{/ R}$ on $R$, and assume that the latter is a faithfully flat $R$-module. Show:

  1. For any $R$ - module $M$, the canonical map $M\rightarrow M\otimes_R R'$, $x\mapsto x\otimes 1$ is injective.
  2. Any ideal $\mathcal{I}\subset R$ satisfies $\mathcal{I}R'\cap R=\mathcal{I}$.
  3. If $R'$ is Noetherian (Artinian), the same is true for $R$.

I think there must be some condition imposed on $R\rightarrow R'$ or $M'$ as an $R'$-module, because even the first exercise is only true for flat $R$ - modules $M$. The statements look rather standard, so I guess there is some well known condition.


1 Answer 1


I think I got it. The restriction $R'_{/R}$ is simply $R'$ with $r\cdot r'=f(r)r'$ for $r\in R, r'\in R'$, and this is faithfully flat. Now, the restriction $(M\otimes_RR')_{/R}$ is faitfully flat, so Bosch 4.4 Prop 1(iv) tells us that $(M\otimes_R R')_{/R}\otimes_R R'\cong M\otimes_R R'$ is faithfully flat. Same proposition but (v) tells us that $M$ is faithfully flat.

  • $\begingroup$ It is not true in general that $R\rightarrow R'$ is faithfully flat and it is not assumed in the prerequesite of the exercise. Also, $M$ in the first part is supposed to be an arbitrary $R$ - module, so $(M\otimes_R R')_{/R}$ will not even be flat. $\endgroup$
    – Teddyboer
    Mar 17, 2020 at 17:18
  • $\begingroup$ The exercise says that the restrictions of $R'$-modules are faithfully flat over $R$, does it not? $\endgroup$
    – user759746
    Mar 17, 2020 at 19:12
  • $\begingroup$ The way I see it, it says that we have a given $R'$ - module $M'$ such that its restriciton $M'_{/R}$ on $R$ via $R\rightarrow R'$ is faithfully flat. Though, I think in order for the exercise to give valid statements, we should assume something about $R\rightarrow R'$ or rephrase the exercises... $\endgroup$
    – Teddyboer
    Mar 18, 2020 at 15:12

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .