# Obtaining the evaluation map with QPA

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 $$ev_M$$.

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 $$M\to M^{**}$$.