Abstract definition of a vector space

Question

Can one define a vector space over a field F as a tensor product of an abelian group (as Z module) with F?

Answer 1

Not quite, but you have a good idea! What is true is that, for any $\mathbb{Z}$-module $A$, $A \otimes_{\mathbb{Z}} F$ has a natural $F$-vector space structure. Moreover, for any $F$-vector space $V$, there is a $\mathbb{Z}$-module $A$ such that $A \otimes_{\mathbb{Z}} F \cong V$. However, the concept of an $F$-vector space is not the same as a $\mathbb{Z}$-module tensored with $F$. You can tell that this is literally true because ${-} \otimes_{\mathbb{Z}} F$ is not an equivalence of the categories in question (more on this later), but I'll try to answer the potential followup question of "why isn't this the definition of vector space instead?"

One reason is that we want to have $F$-vector spaces which are not literally equal to $A \otimes_{\mathbb{Z}} F$ (otherwise, most statements and constructions in linear algebra become extremely tedious, e.g. what does "linearly independent" mean and how do you define the vector space structure on hom-sets? We can certainly answer these questions if we know what an $F$-action is, but then we've already defined vector spaces as normal). It's also much too restrictive in the sense that we only allow ourselves the natural $F$-action to work with on each tensor product.

Another reason is that this definition produces the same vector space in many different ways. To be more precise, what you've really described is a functor $A \mapsto A \otimes_{\mathbb{Z}} F : \mathbb{Z}\,\mathsf{Mod} \to F\,\mathsf{Mod}$ from the category of abelian groups to the category of $F$-vector spaces. This functor is always essentially surjective (as described earlier) but is never fully faithful. For example, if $F$ has characteristic $0$, then $T \otimes_{\mathbb{Z}} F = 0$ for any torsion abelian group $T$. This means we have no hope of uniquely (up to isomorphism) representing $F$-vector spaces as tensor products in this way.

A vector space is truly the data of an abelian group with an $F$-action, and as such the only "good" definition is "a ring homomorphism $F \to \operatorname{End}_{\mathbb{Z}}(A)$ for some abelian group $A$", or something equivalent to that.

Hope this was somewhat helpful! Happy to expand more on these ideas if you'd like.