Silting modules over a class of Morita rings

Abstract

Abstract Let Δ = A N B A M A B B \Delta =\left(\begin{array}{cc}A& {}_{A}N_{B}\\ {}_{B}M_{A}& B\end{array}\right) be a Morita ring, where M ⊗ A N = 0 = N ⊗ B M M{\otimes }_{A}N=0=N{\otimes }_{B}M . Let X X be left A A -module and Y Y be left B B -module. We prove that ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) \left(X,M{\otimes }_{A}X,1,0)\oplus \left(N{\otimes }_{B}Y,Y,0,1) is a silting module if and only if X X is a silting A A -module, Y Y is a silting B B -module, M ⊗ A X M{\otimes }_{A}X is generated by Y Y , and N ⊗ B Y N{\otimes }_{B}Y is generated by X X . As a consequence, we obtain that if M A {M}_{A} and N B {N}_{B} are flat, then ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) \left(X,M{\otimes }_{A}X,1,0)\oplus \left(N{\otimes }_{B}Y,Y,0,1) is a tilting Δ \Delta -module if and only if X X is a tilting A A -module, Y Y is a tilting B B -module, M ⊗ A X M{\otimes }_{A}X is generated by Y Y , and N ⊗ B Y N{\otimes }_{B}Y is generated by X X .