Lemma 10.38.5. Let $M$ be an $R$-module. The following are equivalent:

1. $M$ is flat over $R$.

2. for every injection of $R$-modules $N \subset N'$ the map $N \otimes _ R M \to N'\otimes _ R M$ is injective.

3. for every ideal $I \subset R$ the map $I \otimes _ R M \to R \otimes _ R M = M$ is injective.

4. for every finitely generated ideal $I \subset R$ the map $I \otimes _ R M \to R \otimes _ R M = M$ is injective.

Proof. The implications (1) implies (2) implies (3) implies (4) are all trivial. Thus we prove (4) implies (1). Suppose that $N_1 \to N_2 \to N_3$ is exact. Let $K = \mathop{\mathrm{Ker}}(N_2 \to N_3)$ and $Q = \mathop{\mathrm{Im}}(N_2 \to N_3)$. Then we get maps

$N_1 \otimes _ R M \to K \otimes _ R M \to N_2 \otimes _ R M \to Q \otimes _ R M \to N_3 \otimes _ R M$

Observe that the first and third arrows are surjective. Thus if we show that the second and fourth arrows are injective, then we are done1. Hence it suffices to show that $- \otimes _ R M$ transforms injective $R$-module maps into injective $R$-module maps.

Assume $K \to N$ is an injective $R$-module map and let $x \in \mathop{\mathrm{Ker}}(K \otimes _ R M \to N \otimes _ R M)$. We have to show that $x$ is zero. The $R$-module $K$ is the union of its finite $R$-submodules; hence, $K \otimes _ R M$ is the colimit of $R$-modules of the form $K_ i \otimes _ R M$ where $K_ i$ runs over all finite $R$-submodules of $K$ (because tensor product commutes with colimits). Thus, for some $i$ our $x$ comes from an element $x_ i \in K_ i \otimes _ R M$. Thus we may assume that $K$ is a finite $R$-module. Assume this. We regard the injection $K \to N$ as an inclusion, so that $K \subset N$.

The $R$-module $N$ is the union of its finite $R$-submodules that contain $K$. Hence, $N \otimes _ R M$ is the colimit of $R$-modules of the form $N_ i \otimes _ R M$ where $N_ i$ runs over all finite $R$-submodules of $N$ that contain $K$ (again since tensor product commutes with colimits). Notice that this is a colimit over a directed system (since the sum of two finite submodules of $N$ is again finite). Hence, (by Lemma 10.8.4) the element $x \in K \otimes _ R M$ maps to zero in at least one of these $R$-modules $N_ i \otimes _ R M$ (since $x$ maps to zero in $N \otimes _ R M$). Thus we may assume $N$ is a finite $R$-module.

Assume $N$ is a finite $R$-module. Write $N = R^{\oplus n}/L$ and $K = L'/L$ for some $L \subset L' \subset R^{\oplus n}$. For any $R$-submodule $G \subset R^{\oplus n}$, we have a canonical map $G \otimes _ R M \to M^{\oplus n}$ obtained by composing $G \otimes _ R M \to R^ n \otimes _ R M = M^{\oplus n}$. It suffices to prove that $L \otimes _ R M \to M^{\oplus n}$ and $L' \otimes _ R M \to M^{\oplus n}$ are injective. Namely, if so, then we see that $K \otimes _ R M = L' \otimes _ R M/L \otimes _ R M \to M^{\oplus n}/L \otimes _ R M$ is injective too2.

Thus it suffices to show that $L \otimes _ R M \to M^{\oplus n}$ is injective when $L \subset R^{\oplus n}$ is an $R$-submodule. We do this by induction on $n$. The base case $n = 1$ we handle below. For the induction step assume $n > 1$ and set $L' = L \cap R \oplus 0^{\oplus n - 1}$. Then $L'' = L/L'$ is a submodule of $R^{\oplus n - 1}$. We obtain a diagram

$\xymatrix{ & L' \otimes _ R M \ar[r] \ar[d] & L \otimes _ R M \ar[r] \ar[d] & L'' \otimes _ R M \ar[r] \ar[d] & 0 \\ 0 \ar[r] & M \ar[r] & M^{\oplus n} \ar[r] & M^{\oplus n - 1} \ar[r] & 0 }$

By induction hypothesis and the base case the left and right vertical arrows are injective. The rows are exact. It follows that the middle vertical arrow is injective too.

The base case of the induction above is when $L \subset R$ is an ideal. In other words, we have to show that $I \otimes _ R M \to M$ is injective for any ideal $I$ of $R$. We know this is true when $I$ is finitely generated. However, $I = \bigcup I_\alpha$ is the union of the finitely generated ideals $I_\alpha$ contained in it. In other words, $I = \mathop{\mathrm{colim}}\nolimits I_\alpha$. Since $\otimes$ commutes with colimits we see that $I \otimes _ R M = \mathop{\mathrm{colim}}\nolimits I_\alpha \otimes _ R M$ and since all the morphisms $I_\alpha \otimes _ R M \to M$ are injective by assumption, the same is true for $I \otimes _ R M \to M$. $\square$

[1] Here is the argument in more detail: Assume that we know that the second and fourth arrows are injective. Lemma 10.11.10 (applied to the exact sequence $K \to N_2 \to Q \to 0$) yields that the sequence $K \otimes _ R M \to N_2 \otimes _ R M \to Q \otimes _ R M \to 0$ is exact. Hence, $\mathop{\mathrm{Ker}}\left(N_2 \otimes _ R M \to Q \otimes _ R M\right) = \mathop{\mathrm{Im}}\left(K \otimes _ R M \to N_2 \otimes _ R M\right)$. Since $\mathop{\mathrm{Im}}\left(K \otimes _ R M \to N_2 \otimes _ R M\right) = \mathop{\mathrm{Im}}\left(N_1 \otimes _ R M \to N_2 \otimes _ R M\right)$ (due to the surjectivity of $N_1 \otimes _ R M \to K \otimes _ R M$) and $\mathop{\mathrm{Ker}}\left(N_2 \otimes _ R M \to Q \otimes _ R M\right) = \mathop{\mathrm{Ker}}\left(N_2 \otimes _ R M \to N_3 \otimes _ R M\right)$ (due to the injectivity of $Q \otimes _ R M \to N_3 \otimes _ R M$), this becomes $\mathop{\mathrm{Ker}}\left(N_2 \otimes _ R M \to N_3 \otimes _ R M\right) = \mathop{\mathrm{Im}}\left(N_1 \otimes _ R M \to N_2 \otimes _ R M\right)$, which shows that the functor $- \otimes _ R M$ is exact, whence $M$ is flat.
[2] This becomes obvious if we identify $L' \otimes _ R M$ and $L \otimes _ R M$ with submodules of $M^{\oplus n}$ (which is legitimate since the maps $L \otimes _ R M \to M^{\oplus n}$ and $L' \otimes _ R M \to M^{\oplus n}$ are injective and commute with the obvious map $L' \otimes _ R M \to L \otimes _ R M$).

Comment #396 by Fan on

I don't understand the last sentence of the first paragraph: does it suffice to show that $K_2 \otimes_R M \to N_2 \otimes_R M$ is injective?

Take the chain $0 \to Z \xrightarrow 2 Z$ for example. Tensoring with $Z/2Z$ gives $0 \to Z/2Z \xrightarrow 0 Z/2Z$, which is not exact. However, $K_2 = 0$ and the map $K_2 \otimes_R M \to N_2 \otimes_R M$ is still injective.

Comment #400 by on

OK, yes, that is nonsense! Thanks for this and the other remarks. For the fixes please see here.

Comment #3944 by Lucas Braune on

Possible website issue: I am seeing the four statements of Lemma 10.38.5 (= Lemma 00HD) appear labelled with numbers regarless of whether I choose to display labels with numbers (such as 10.38.5) or with tags (e.g. 00HD). This makes the first sentence of the proof hard for me to read when labels are displayed as tags.

Comment #3945 by Lucas Braune on

Possible website issue: I am seeing the four statements of Lemma 10.38.5 (= Lemma 00HD) appear labelled with numbers regarless of whether I choose to display labels with numbers (such as 10.38.5) or with tags (e.g. 00HD). This makes the first sentence of the proof hard for me to read when labels are displayed as tags.

Comment #3948 by on

Lucas, thanks for noticing! I didn't think of this whilst implementing the framework. I've created an issue on GitHub (see https://github.com/gerby-project/gerby-website/issues/136), and I'll fix it along with some other issues in the next round of bug fixing for Gerby.

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