Invariants of linkage of modules - Puthenpurakal, Tony J. (MATHEMATICA SCANDINAVICA, 2021)
Let (A,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 q. Let L be another finitely generated A-module. We show that Ext^i_A(L,M) = 0 for all i 0 if and only if Tor^A_i(L,N) = 0 for all i 0. If D is a Cohen-Macaulay module then we show that Ext^i_A(M, D) = 0 for all i 0 if and only if Ext^i_A(D^, N) = 0 for all i 0, where D^= Ext^r_A(D,A) and r = codim(D). As a consequence we get that Ext^i_A(M, M) = 0 for all i 0 if and only if Ext^i_A(N, N) = 0 for all i 0. We also show that End_A(M)/rad End_A(M) (End_A(N)/rad End_A(N))^op . We also give a negative answer to a question of Martsinkovsky and Strooker.