Let $k$ be a unital commutative ring. Let $k$-Alg denote the category of commutative and unital $k$-algebras. Let Set denote the category of Sets

Fix an arbitrary $k$-module $V.$

Consider the covariant functor $\text{End}_V : k\text{-Alg} \to \text{Set}$ given on objects by $R \mapsto V \otimes_k R,$ and given on morphisms in the following way: If $f: R \to S$ is a map of $k$-algebras and $\varphi : V \otimes_k R \to V \otimes_k R,$ then the map $V \otimes_k S \to V \otimes_k S$ is given by following $V \hookrightarrow V \otimes_k R \to V \otimes_k R \to V \otimes_k S$ and then extending by $S$-linearity, where the second map is $\varphi$ and the last map is $\text{id}_V \otimes_k f.$

By restricting, we may similarly define a covariant functor $\text{GL}_V : k\text{-Alg} \to \text{Set}.$

If $V$ is free of rank $n < +\infty,$ then I know that these two functors are representable by $k[X_{11}, \dots , X_{nn}]$ and by the localization of $k[X_{11}, \dots , X_{nn}]$ at the determinant.

If $V$ is not free, then what can we say about representability? What if $V$ is free of infinite rank? In the latter case, the proof of the representability of these functors in the finite rank case breaks down because for each $j,$ there must exist only finitely many $i$ for which $X_{ij}$ may be mapped to a nonzero element of the $k$-algebra under consideration. And I do not know whether it is possible to express the "finitely many" condition in some sort of algebraic manner.

Thank you very much for reading this. Please let me know if you have any thoughts. Thanks again.

UPDATE: I see that the endomorphism functor is true if $V$ is a finitely generated projective $k$-module, based on Why is $R\mapsto\operatorname{End}_{R-lin}(R\otimes_k V)$ representable by $\operatorname{Sym}(V\otimes_k V^\vee)$?

Are there any other cases in which something can be said?


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