Multilinear and alternating property of $\det(f)$ where $f$ is an endomorphism

Question

Everybody knows the determinant of a matrix $A\in k^{n\times n}$ ($k$ a commutative ring) and everybody knows that the determinant of $A$ is an alternating multilinear map in the columns aswell as in the rows of $A$. Now suppose we do not chose a base and consider the determinant of an endomorphism $f\colon M\longrightarrow M$ where $M$ is some finite free module over $k$. I cannot figure out, what exactly the alternating property says, when translated to a coordinate free situation. Do I need to encode the determinant of $f$ in terms of a base in order to be able to talk about multilinearity or the alternating property? Thanks in advance.

Answer 1

Disclaimer: the question seems too broad so possibly there is not a unique answer to it..... nevertheless here's some thoughts.

In what follows I assume we have fixed a ring $R$.

In general a multilinear map from the modules $M_1,\dots,M_n$ into the module $M$ is just a map $f \colon M_1 \times \dots \times M_n \to M$ which is linear in each component.

The determinant is a multilinear map because you can see the module of the $n\times n$ matrixes, that is $M_n(R)$, as the cartesian product $\prod_{i=1}^n R^n=(R^n)^n$ (identifying a matrix with the $n$-tuple of its columns, or rows), and under this identification it becomes a multilinear map.

In order to define a similar construction over the module of the endomorphisms $End(M)$, for some fixed $R$-module $M$, you would need a decomposition of $End(M)$ as direct product: if $End(M) =\prod_{i \in I} M_i$ for a family of modules $(M_i)_{i \in I}$, then you can see a mapping $f \colon End(M) \to R$ as a mapping $f \colon \prod_i M_i \to R$ and for this multilinearity is meaningful.

Nonetheless there is a problem with this construction: the decomposition of $M_n(R)$ as the product $(R^n)^n$ is a canonical one. There is no garantee that for a generic module $M$ a similiar canonical decomposition of $End(M)$ could be found. So even if you could decompose $End(M)$ as a direct product and define multilinear maps over it, that doesn't mean that these multilinear maps would be meaningful.