Let $A$ be a central simple $K-$algebra and $A = M_n(D^{o}), Z(D) = K, [D:K] = m^2$. Let $\psi(a)$ be the reduced trace of an element of $A$. I am not sure if the definition I know(https://ysharifi.wordpress.com/2011/10/11/reduced-norm-and-reduced-trace-1/) is the same as one mentioned in section Schur Indices(12.2) on page 92 of serre's Linear Representation of Finite Groups.
1.In case I am wrong what would be the correct definition?
- If the above definition is the definition of reduced trace meant in the serre's book, then how do I prove the following
Let $\phi(a)$ be the trace of an element $a \in A$ as a $K$ endomorphism induced by left multiplication of $a$ on an vector space of dimension $n$ over $D$. Then I want to see why $\phi(a) = m\psi(a)$ ?.
There is a reference to bourbaki chapter VIII, Algebre section 12.3 But it was of no help to me. I am thankful for any other reference or hints.
The notions are indeed the same (but Serre makes no definition, he just refers to Bourbaki). You should make a change of scalars to an algebraic closure : the trace as a left translation and the reduced trace both are unchanged when tensoring with a field extension.
Then over an algebraically closed field $K$, $A \simeq M_m(K)$, and it is a simple exercise to compute that in a appropriate basis the matrix of the left multiplication by $a\in M_m(K)$ is just $m$ times $a$ as diagonal blocks, hence the formula for the traces.