Tilting modules, dominant dimension and exactness of duality functors
type: Item
instance of: scholarly article
title: Tilting modules, dominant dimension and exactness of duality functors
zbMATH Open document ID: 0672.16018
published in: Tsukuba Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16L60
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16E10
zbMATH DE Number: 4099506
zbMATH Keywords: duality functors
zbMATH Keywords: finite dimensional algebra
zbMATH Keywords: positive dominant dimension
zbMATH Keywords: tilting left and right modules
zbMATH Keywords: endomorphism rings
zbMATH Keywords: exactness
zbMATH Keywords: Morita duality
author: Robert R. Colby
reviewed by: Yoshiki Kurata
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496160840
OpenAlex ID: W991013765
DOI: 10.21099/TKBJM/1496160840
description: scientific article; zbMATH DE number 4099506