Derived equivalences of triangular matrix rings arising from extensions of tilting modules
Let \(R\) and \(S\) be rings and let \(_RM_S\) be an \(R\)-\(S\)-bimodule. Then one may form a ring as \(2\times 2\)-matrices, where in the upper right corner one has entries in \(M\), the upper diagonal entry are entries in \(R\) and the lower diagonal have entries in \(S\). Denote this ring by \((R,S,M)\). The main result in the paper shows that if \(T\) is a tilting module over \(S\) and if \(M\) has finite projective dimension as \(S\)-module and if \(\text{Ext}^n_S(M,T)=0\) for all \(n>0\), then there is a tilting complex over \((R,S,M)\) with endomorphism ring \((\text{End}_S(T),R,\Hom_S(M,T))\). In particular these two algebras are derived equivalent. The proof uses a construction similar to a recollement construction. The author gives examples showing that the hypotheses are really necessary. Numerous interesting applications are given.
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