Testing whether an element of a tensor product of modules is zero

Let $A$ be a commutative ring and let $F$, $E$ be $A$-modules. Then $E$ is the direct limit of its finitely generated submodules. Serge Lang, in his Algebra, p. 604, remarks that a technique to test whether an element of $F \otimes E$ is zero, is to examine its image in $F \otimes N$, as $N$ varies over the finitely generated submodules of $E$. Could somebody please explain why this is a valid technique? What is the image of an element of $F \otimes E$ in $F \otimes N$?

Solution 1:

There is the following (quite unknown ...) criterion when an element of the tensor product $M \otimes_A N$ of two $A$-modules $M,N$ vanishes:

Choose generating sets $E$ of $M$ and $F$ of $N$. Every element of $M \otimes_A N$ can be written as $\sum_{e \in E} e \otimes n(e)$, where $n : E \to N$ is a function with finite support. It vanishes if and only if there is a map $\lambda : E \times F \to A$ with finite support such that

$$\forall e \in E ~: ~ n(e) = \sum_{f \in F} \lambda(e,f) \cdot f$$ $$\forall f \in F~ : ~ 0 = \sum_{e \in E} \lambda(e,f) \cdot e$$

You should think $\lambda$ as a finite matrix which gives a reason why $\sum_{e \in E} e \otimes n(e)$ vanishes, namely why it is optained from $0 \otimes 0$ by applying bilinear relations.

There is a nice proof for this using the general properties of the tensor product, appearing in Pierre Mazet, Caracterisation des epimorphismes par relations et generateurs. There it is also used to give a full characterization of epimorphisms in the category of commutative rings (which, however, is useless in practice; in my opinion only the corollary about the cardinality is interesting).

As for the question about Lang's comment: It is already explained very well, so I don't really know what I should add. Since $M$ is the directed colimit of its finitely generated submodules $M_i$, it follows that $M \otimes_A N$ is the directed colimit of the $M_i \otimes_A N$ (which are, however, not submodules in general!). This means that every element of $M \otimes_A N$ lies in the image of $M_i \otimes_A N \to M \otimes_A N$ for some $i$, and that two elements of $M_i \otimes_A N$ and $M_j \otimes_A N$ become equal in $M \otimes_A N$ iff they become equal in $M_k \otimes_A N$ for some $k \geq i,j$. In particular, an element of $M \otimes N$, which vanishes, already vanishes in some $M_i \otimes N$ for $i$ large enough. This is often used for establishing general results in homological algebra which reduce something concerning tensor products for the finitely generated case. For example, $M$ is flat if and only if $\mathrm{Tor}_{>0}(M,N)=0$ for all finitely generated $N$. And $M$ is flat as soon as every finitely generated submodule of $M$ is flat. For example, this implies that torsionfree $\mathbb{Z}$-modules are flat (try to prove this directly). But note that Lang does not give any criterion when an element in the tensor product precisely vanishes.