Invariants of Linkage of modules

Let $(A,\mathfrak{m})$ be a Gorenstein local ring and let $M,N$ be two Cohen-Macaulay \ $A$-modules with $M$ linked to $N$ via a Gorenstein ideal $\mathfrak{q}$. Let $L$ be another finitely generated $A$-module. We show that $Ext^i_A(L,M) = 0 $forall$i \gg 0$ifandonlyif$Tor^A_i(L,N) = 0$forall$i \gg 0$\mathrm{.}If$D$isCohen-Macaulaythenweshowthat$Ext^i_A(M, D) = 0 $forall$i \gg 0$ifandonlyif$Ext^i_A(D^\dagger, N) = 0$forall$i \gg 0$,where$D^\dagger = Ext^r_A(D,A)$and$r = codim \ D$. As a consequence we get that $Ex{t}_A^i(M,M)=0$ for all $i\gg 0$ if and only if $Ex{t}_A^i(N,N)=0$ for all $i\gg 0$. We also show that $End_A(M)/rad \ End_A(M) \cong (End_A(N)/rad
End_A(N))^{op}$. We also give a negative answer to a question of Martsinkovsky and Strooker.