2
$\begingroup$

Let $A$ be a $C$-algebra, where $C$ is a commutative ring with $1$, and $M$ be a finitely generated left $A$-module.

Question: Is it true that we can always find a positive integer $n$, a $C$-subalgebra $B$ of $M_n(A)$ and an ideal $J$ of $B$ such that $B/J$ is isomorphic to $End(M)\ ?$ If not, what other conditions are needed to make the statement true?

By isomorphism I mean a $C$-algebra isomorphism. $M_n(A)$, as always, is the algebra of $n\times n$ matrices with entries from $A$. By $End(M)$ I mean the algebra of $A$- homomorphisms from $M$ to $M$.

So I'm looking for a homomorphism from some $C$ subalgebra $B$ of $M_n(A)$ onto $End(M)$. Well, I know that there exists a natural surjection $f$ from $A^n$ to $M$ for some positive integer $n$ because $M$ is finitely generated over $A$. One way to define a map $g$ from $M_n(A)$ to $End(M)$ is to define $g(a)(m)=f(ax)$, for all $a \in A$ and $m \in M$, where $x$ is any element of $A^n$ with $f(x) = m$. Ok, this map has obviously the well-definedness issue and that prevents $g$ to be defined on the whole $M_n(A)$. So, we choose B to be the set of those elements $a \in M_n(A)$ such that $f(ax)=0$, for all $x$ from the kernel of $f$. Now $g$ is well-defined on $B$ and $B$ is a $C$-subalgebra of $M_n(A)$. What I'm having trouble with is to show that $g$ is surjective!

PS. I need the above to show that if $S$ is a subalgebra of $R$ and $R$ is finitely generated as $S$-module, then the Gelfand Kirillov dimension of $R$ and $S$ are equal. I didn't know how to prove it directly using the definition. So if anybody knows a direct proof, that would also be great.

Thanks.

$\endgroup$
4
  • 2
    $\begingroup$ This is Lemma 8.2.8 in McConnell and Robson (Noncommmutative Noetherian Rings, AMS Graduate Studies in Mathematics, Vol. 30). They don't give a proof that your $g$ is surjective, but leave it for the reader to check. $\endgroup$
    – PersonX
    Jul 2, 2010 at 11:15
  • $\begingroup$ Dear @Sergiy Kozerenko: You have made a significant number of very minor edits (specifically, seven) to old questions in the past two hours. That is considered too much. Moreover, your edits arguably do not actually improve the readability of the questions. $\endgroup$ Nov 22, 2013 at 12:47
  • $\begingroup$ Finally, and perhaps most importantly, editing questions bumps them to the front page. This reduces the usefulness of the front page, and that is not taken lightly by many users of MathOverflow. Please take the time to read the meta threads meta.mathoverflow.net/questions/599/… and meta.mathoverflow.net/questions/169/… among others. $\endgroup$ Nov 22, 2013 at 17:39
  • $\begingroup$ Dear @Sergiy Kozerenko, in sum, you should not edit more than three old posts a day so as to not flood the front page. By the way, concerning adding TeX/mathjax to old questions (which seems to be most relevant to you), there are also the relevant meta threads meta.mathoverflow.net/questions/478/… and meta.mathoverflow.net/questions/591/… which you may also want to review. Thank you very much for your attention. $\endgroup$ Nov 22, 2013 at 17:42

1 Answer 1

2
$\begingroup$

Let $e_1, \dots, e_n$ be a basis for $A^{\oplus n}$ and let $m_1, \dots, m_n$ be generators for $M$ so that your map $f: A^{\oplus n} \twoheadrightarrow M$ has $f(e_i) = m_i$. Now, suppose $\varphi \in \mathrm{End}_A(M)$.` For each $i$, choose $a_{i,j} \in A$ such that $$\varphi(m_i) = \sum_j a_{i,j} m_j.$$ (Obviously, this choice is not necessarily unique.) Now, define $\tilde{\varphi} \in \mathrm{End}_A(A^{\oplus n})$ by setting $$\tilde{\varphi}(e_i) = \sum_j a_{i,j} e_j.$$ You can show that $f \circ \tilde{\varphi} = \varphi \circ f$, which means that $\tilde{\varphi} \in B$ and $g(\tilde{\varphi}) = \varphi$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.