Proving that free modules are flat (without appealing projective modules)

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Solution 1

You can just show that the functor $M\otimes_R(-)$ is left-exact. Write $M=\bigoplus_{i\in I} R$ and let $$0\to N'\xrightarrow{\quad\iota\quad} N\to N''\to 0$$ be an exact sequence of $R$-modules. Note that $M\otimes N=\bigoplus_{i\in I} R\otimes_R N = \bigoplus_{i\in I} N$, so the sequence

$$0 \to M\otimes N' \xrightarrow{\quad\mathrm{id}\otimes\iota\quad} M\otimes N \to M\otimes N'' \to 0$$

is the same as

$$0 \to \bigoplus_{i\in I} N' \xrightarrow{\textstyle\quad\bigoplus_{i\in I} \iota\quad} \bigoplus_{i\in I}N \to \bigoplus_{i\in I}N'' \to 0$$

and the morphism $\bigoplus_{i\in I} \iota$ is clearly injective if $\iota$ is injective.

Solution 2

  1. The identity functor is exact.

  2. If $\{F_i\}$ is a family of exact functors, then $\oplus_i F_i$ is also exact.

Now apply this to $M \otimes - $ for a free module $M$. 1. deals with the case $M=R$, and 2. generalizes this to $M = \oplus_i R$.

Solution 3

Assume our free module $M$ is of the form $\oplus_{i\in S}R_{i}$, then the statement just follows. The details may be best to leave out to you.

For your question on the projective module, you can consider the projective module as a direct summand of a free module and "piece" up the maps together.

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Updated on August 01, 2022

Comments

  • Prism
    Prism over 1 year

    Suppose $R\neq 0$ is a commutative ring with $1$. Let $M$ be a free $R$-module. I would like to prove that $M$ is a flat $R$-module. Everywhere I have looked (mostly online) this is proved by first proving that every free module is projective, and then proving that every projective module is flat. Unfortunately, Atiyah & Macdonald's "Introduction to Commutative Algebra" (Chapter 2) does not discuss projective modules. But the result that every free module is flat comes very handy in the exercises.

    So my question is,

    Is it possible to prove that every free module is flat just by definitions and without appealing to projective modules?

    Thanks!

    • Martin
      Martin over 10 years
      The way people usually prove that projective modules are flat is to reduce to the free case by using that every projective module is a direct summand of a free module, so I'm not sure I understand the question. In fact, it is an somewhat challenging exercise to show that a projective module is flat without appealing to free modules.
    • Prism
      Prism over 10 years
      @Martin: You are right. I just wanted to see a proof that doesn't involve discussion of projective modules... But I was ignorant enough not to look at the proof of "projective implies flat" and to see that it does indeed use the result of seemingly weaker statement "free implies flat". Thanks for the heads-up!
  • gen
    gen almost 5 years
    If flatness is defined by asserting that $M \otimes (\_)$ is exact, should you not also show right-exactness?
  • Jesko Hüttenhain
    Jesko Hüttenhain almost 5 years
    Dear @gen, you are correct. However, it is a rather well-known lore result that taking the tensor product with a fixed module is right-exact in general, so I will weasel out of this by arguing that you may define flatness over general rings by requiring that the tensor product preserves monomorphisms.
  • gen
    gen almost 5 years
    Understood, thanks for clarifying. It's just that not all of us did phd's in algebra, as you can imagine
  • gen
    gen almost 5 years
    For 2. is the reverse implication also true?
  • Jesko Hüttenhain
    Jesko Hüttenhain almost 5 years
    @gen I am honestly very sorry if my comment came across as patronizing. I firmly believe in asking every question and clarifying every detail in Mathematics. Shouting "well-known" is a last resort; Please forgive me for using it here. I answered this question 5 years ago and it seemed to me at the time that the OP would know the result about right-exactness, so I only went for left-exactness. I have little time these days, but if I can clear my schedule a bit, I will try to include more detail in my answer.
  • Martin Brandenburg
    Martin Brandenburg almost 4 years
    @gen Yes it is.
  • gen
    gen almost 4 years
    Thank you for this
  • Babai
    Babai over 3 years
    @JeskoHüttenhain The world needs more humble Mathematicians like you :)