Tilting modules, dominant dimension and exactness of duality functors
Let R and S be rings and \({}_ RW_ S\) a bimodule. Denote by \(\Delta_ W\) both the functors \(Hom_ R(\)-,W): R-Mod\(\to Mod\)-S and \(Hom_ S(\)- ,W): Mod-S\(\to R\)-Mod and the composition of the two, in either order, by \(\Delta^ 2_ W\). In the first half of the paper the author studies exactness properties of the functor \(\Delta^ 2_ W\). The case \(W=R\) has been extensively studied in previous papers written by him with K. R. Fuller and the results obtained there are generalized. In the second half of the paper it is shown that a finite dimensional algebra R of positive dominant dimension possesses a canonical pair of tilting left and right modules \({}_ RU\) and \(V_ R\). Let \(W=_ T(V\otimes_ RU)_ S\), where T and S denote the R-endomorphism rings of V and U, respectively. Then the author relates exactness properties of \(\Delta_ U\), \(\Delta_ V\), \(\Delta_ W\) and their squares to dominant dimension and shows that 1) if dom.dim. \(R\geq 2\) then \(\Delta^ 2_ U\) preserves monomorphisms in Mod-S and in R-Mod; 2) if dom.dim. \(R\geq 3\) then \(\Delta^ 2_ U\) is left exact on Mod-S and \(\Delta^ 2_ W\) preserves monomorphisms in Mod-S and in T-Mod. In this case, if \(\Delta_ W: T\)-Mod\(\to Mod\)-S: \(\Delta_ W\) defines a Morita duality, R is QF (and conversely); 3) if dom.dim. \(R\geq 4\) then \(\Delta^ 2_ W\) are left exact on Mod-S and on T-Mod.
scientific article; zbMATH DE number 4099506