For any module $P$, we define $\mathrm{tr}(P)=\sum \mathrm{im}(f)$, where $f$ ranges over all elements of $\mathrm{Hom}(P,R)$, and call it trace.

Why does it have such a name? Does it have any relation to the trace of matrix?

  • 4
    $\begingroup$ Why do we call a trace of the matrix trace, btw? $\endgroup$ – Alexei Averchenko Mar 11 '11 at 12:16
  • 2
    $\begingroup$ Like many things in Algebra, the name came from German, "die Spur". It entered English usage [in a 1922 translation of Hermann Weyl's ](jeff560.tripod.com/t.html) Raum, Zeit, Materie. Maybe someone else can provide insight on why the German "Spur" is chosen for the trace of a matrix. $\endgroup$ – Willie Wong Mar 11 '11 at 12:51

I have always viewed the term trace as used to name $$\displaystyle \operatorname{tr}(P)=\sum_{f:P\to R}f(P)$$ as a reference to the fact that that ideal is the part of $R$ which you can reach from $P$... whatever that may mean :) One can check, for example, that $P$ is a generator of the category of modules iff $\operatorname{tr}(P)=R$, and that makes sense (to me!)

A more serious connection is the following. Suppose $V$ is a finite dimensional vector space over a field $k$, and let $\hom_k(V,k)$ be its dual space. Then there is a natural isomorphism $$\phi:V\otimes\hom_k(V,k)\cong\hom(V,V).$$ On the other hand, we have the usual trace map $\operatorname{tr}:\hom(V,V)\to k$, so we can consider the composition $$\operatorname{Tr}=\operatorname{tr}\circ\phi:V\otimes\hom_k(V,k)\to k,$$ which is also sometimes called a trace map. If you now replace $k$ by ring, $V$ by a left $R$-module, then what you wrote $\operatorname{tr}(P)$ is the image of my $\operatorname{Tr}$: it follows that the trace ideal is the image of the trace map.

  • $\begingroup$ Yes,Morita theory! $\endgroup$ – Strongart Mar 14 '11 at 10:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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