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 $Ext^i_A(L,M) = 0$ for all $i \gg 0$ if and only if $Tor^A_i(L,N) = 0$ for all $i \gg 0$. If $D$ is Cohen-Macaulay then we show that $Ext^i_A(M, D) = 0$ for all $i \gg 0$ if and only if $Ext^i_A(D^\dagger, N) = 0$ for all $i \gg 0$, where $D^\dagger = Ext^r_A(D,A)$ and $r = codim \ D$. As a consequence we get that $Ext^i_A(M, M) = 0$ for all $i \gg 0$ if and only if $Ext^i_A(N, N) = 0$ 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.

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