Silting modules over a class of Morita rings

Abstract

Let Δ=ANBAMABB\Delta =\left(\begin{array}{cc}A& {}_{A}N_{B}\\ {}_{B}M_{A}& B\end{array}\right) be a Morita ring, where M⊗AN=0=N⊗BMM{\otimes }_{A}N=0=N{\otimes }_{B}M. Let XX be left AA-module and YY be left BB-module. We prove that (X,M⊗AX,1,0)⊕(N⊗BY,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 XX is a silting AA-module, YY is a silting BB-module, M⊗AXM{\otimes }_{A}X is generated by YY, and N⊗BYN{\otimes }_{B}Y is generated by XX. As a consequence, we obtain that if MA{M}_{A} and NB{N}_{B} are flat, then (X,M⊗AX,1,0)⊕(N⊗BY,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 XX is a tilting AA-module, YY is a tilting BB-module, M⊗AXM{\otimes }_{A}X is generated by YY, and N⊗BYN{\otimes }_{B}Y is generated by XX.

Keywords

• (partial)silting modules
• tilting modules
• morita rings
• 16d90
• 16d80