Homological dimensions relative to preresolving subcategories II

Abstract

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒜</m:mi> </m:math> {\mathscr{A}} be an abelian category having enough projective and injective objects, and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒯</m:mi> </m:math> {\mathscr{T}} be an additive subcategory of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒜</m:mi> </m:math> {\mathscr{A}} closed under direct summands. A known assertion is that in a short exact sequence in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒜</m:mi> </m:math> {\mathscr{A}} , the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒯</m:mi> </m:math> {\mathscr{T}} -projective (resp. <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒯</m:mi> </m:math> {\mathscr{T}} -injective) dimensions of any two terms can sometimes induce an upper bound of that of the third term by using the same comparison expressions. We show that if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒯</m:mi> </m:math> {\mathscr{T}} contains all projective (resp. injective) objects of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒜</m:mi> </m:math> {\mathscr{A}} , then the above assertion holds true if and only if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒯</m:mi> </m:math> {\mathscr{T}} is resolving (resp. coresolving). As applications, we get that a left and right Noetherian ring R is n -Gorenstein if and only if the Gorenstein projective (resp. injective, flat) dimension of any left R -module is at most n . In addition, in several cases, for a subcategory <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒞</m:mi> </m:math> {\mathscr{C}} of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒯</m:mi> </m:math> {\mathscr{T}} , we show that the finitistic <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒞</m:mi> </m:math> {\mathscr{C}} -projective and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒯</m:mi> </m:math> {\mathscr{T}} -projective dimensions of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒜</m:mi> </m:math> {\mathscr{A}} are identical.

Topics & Concepts