I have to solve the following exercise taken from the book "Introduction to Representation Theory" by P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina and S. Gerovitch:
Problem 3.8.4
- Let $V,W$ be finite dimensional representations of an algebra $A$ over a not necessarily closed field $k$. Let $k\hookrightarrow l$ be a field extension and assume that $V\otimes_{k}l\cong W\otimes_{k}l$ as modules over the $l$-algebra $A\otimes_{k}l$. Then $V\cong W$ as $A$-modules.
- (The Noether-Deuring theorem) In the setting of (1), suppose that $W\otimes_{k}l\cong Y\oplus(V\otimes_{k}l)$ for some $A\otimes_{k}l$-module $Y$. Then $V$ is a direct summand in $W$, i.e. $W\cong V\oplus Z$ (I assume: as $A$-modules, what else? - this is not made more explicit).
The exercise is not part of the online version, unless I have overlooked it.
What have I achieved so far? The book gave the following hint on (1):
Reduce to the case of finitely generated, then finite extension, of some degree $n$. Then regard $V\otimes_{k}l$ and $W\otimes_{k}l$ as $A$-modules, and show that they are isomorphic to $V^{n}$ and $W^{n}$ respectively. Deduce that $V^{n}\cong W^{n}$, and use the Krull-Schmidt theorem (valid over any field by Problem 3.8.3) to deduce that $V\cong W$.
I have managed to prove it - I think - for the case $[l:k]=n$ and that if the statement holds for all finitely generated extensions then it holds for all extensions. The first part was proven by giving an explicit isomorphism of $A$-modules between $V^{n}$ and $W^{n}$. This was deduced from: $M\otimes_{k}\bigg(\bigoplus_{i\in I}N_{i}\bigg)\cong\bigoplus_{i\in I}\big(M\otimes_{k}N_{i}\big)\quad\text{as $k$ vector spaces}$ whenever $M,\{N_{i};i\in I\}$ are vector spaces over $k$. For the second part I had to rely on a (in retrospective obvious) hint of my professor: let $\Phi:V\otimes_{k}l\to W\otimes_{k}l$ be the $l$ vector spave isomorphism induced by the module isomorphism assumed, we can pick bases $\{v_{i}\}$ and $\{w_{j}\}$ of $V,W$ respectively to obtain bases of $V\otimes_{k}l$ and $W\otimes_{k}l$ by looking at the simple tensors $\{v_{i}\otimes 1\}$ and $\{w_{j}\otimes 1\}$. Let $M$ be the matrix representation of the isomorphism with respect to these bases and let $m$ be the field extension of $k$ generated by the entries of $M$. Then the restriction of $\Phi$ is an isomorphism $V\otimes_{k}m\to W\otimes_{k}m$ and thus if the statement holds for finitely generated extensions, then it holds for arbitrary extensions. So I end up with the following:
Question 1: Given that $k\hookrightarrow l$ is finitely generated and for all extensions of finite degree the statement (1) holds, how can I deduce that it holds for $k\hookrightarrow l$.
For part (2): here I can tell you only what I have tried. First of all I used the Krull-Schmidt theorem and the fact that tensor products distribute over direct products to write $W\otimes_{k}l$ in two different ways: $W\otimes_{k}l\cong\bigoplus_{i=1}^{s}n_{i}\hat{W_{i}}\text{ and }W\otimes_{k}l\cong\bigoplus_{j=1}^{t}m_{j}(W_{j}\otimes_{k}l)$ where the $\hat{W_{i}}$ were irreducible summands of the $A\otimes_{k}l$-module $W\otimes_{k}l$ and the $W_{j}$ were irreducible representations of the $A$-module $W$. Then I wanted to show that in fact the following holds:
$W$ is a finite dimensional, indecomposable representation of the $l$-algebra $A\otimes_{k}l$ if and only if there exists a finite dimensional, indecomposable representation $V$ of the $k$-algebra $A$ such that $W\cong V\otimes_{k}l$.
From this it would have followed that the summands agree up to permutation. Unfortunately this is not true if $k$ is not algebraically closed. I also considered decomposing $W\otimes_{k}l$ as a finite dimensional representation of $A$ but this, I assume, is not the case as $k\hookrightarrow l$ is not necessarily of finite degree. The next thing to look at was to take the quotient with $Y$ in order to actually end up with an isomorphism as (1). So there is the next question:
Question 2: Let $W,V$ be finite-dimensional $A$-modules, where $A$ is a $k$-algebra and let $k\hookrightarrow l$ be a field extension. Let $\Phi:W\otimes_{k}l\twoheadrightarrow V\otimes_{k}l$ be a module homomorphism. Can we deduce that $\operatorname{ker}\Phi\cong U\otimes_{k}l$ where $U$ is a submodule of $W$?
And finally:
Question 3: Am I on the wrong track? Have I overseen something very simple? Am I a complete moron?
Of course hints are very appreciated.