Manuel Saorín "A reduction technique for contravariant finiteness of the subcategory of modules with finite projective dimension"

In a recent paper by Green, Psaroudakis and Solberg the authors recover old and show new reduction techniques for the finitistic dimension conjecture. The finitistic dimension fin.dim(A) of a finite dimensional (or, more generally, Artin) algebra A is the supremum of the projective dimensions of the finitely generated A-modules with finite such dimension. The conjecture states that fin.dim(A) is finite, for any such algebra A. Auslander and Reiten showed that a sufficient condition for fin.dim(A) to be finite is that the subcategory 𝒫 ^<∞(mod-A) of the category of finitely generated A-modules mod-A that consists of the modules of finite projective dimension is contravariantly finite in mod-A. In this talk we will discuss reduction techniques for this contravariant finiteness property and show explicit ways of constructing minimal 𝒫 ^<∞-approximation of A-modules out of corresponding approximations over ’smaller’ algebras.