Infinitely generated tilting modules of finite projective dimension (Q2703553)

Tilting theory started about 20 years ago when Brenner and Butler investigated certain functor transformations for the category of finite dimensional modules over a finite dimensional algebra. In the paper under review, the concept of a tilting module of finite projective dimension is introduced for the category of all (not just finitely presented) modules over an arbitrary ring. The proposed definition seems to be appropriate. This is demonstrated by extending a one-to-one correspondence between tilting modules over an Artin algebra and certain contravariantly finite subcategories of the category of finitely presented modules [\textit{M. Auslander} and \textit{I. Reiten}, Adv. Math. 86, No. 1, 111-152 (1991; Zbl 0774.16006)]. The authors use some recent results of Eklof and Trlifaj about the existence of approximations with respect to subcategories of the form \(M^\perp=\{N\mid\text{Ext}^i(M,N)=0\) for all \(i>0\}\). A discussion of cotilting modules is included as well.