2
$\begingroup$

Given a finite dimensional quiver algebra $A$ and a module $M$ with $M^{*}=Hom_A(M,A)$.

Question: Is it possible to obtain the evaluation map $ev_M : M \rightarrow M^{**}$ in the GAP-package QPA for a given module $M$? Here $ev_M(m)=g(m)$ for $g \in M^{*}$.

This might allow one to obtain the $A$-module $Ext_A^1(Tr(M),A)$ (and thus $Ext_A^i(M,A)$ for all $i$ ) as an $A$-module as the kernel of $ev_M$.

$\endgroup$
1
$\begingroup$

With the latest addition to QPA, one can use the function FromIdentityToDoubleStarHomomorphism do the following:

gap> M;
<[ 0, 10, 7 ]>    
gap> f := FromIdentityToDoubleStarHomomorphism( M );
<<[ 0, 10, 7 ]> ---> <[ 3, 16, 13 ]>>
gap> Kernel( f );
<[ 0, 0, 0 ]>
gap> CoKernel( f );
<[ 3, 6, 6 ]>

which computes the natural homomorphism $M\to M^{**}$.

$\endgroup$

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.