GORENSTEIN DIMENSIONS OF UNBOUNDED COMPLEXES UNDER BASE CHANGE

Abstract

Abstract. Transfer of homological properties under base change is aclassical field of study. Let R → S be a ring homomorphism. The re-lations of Gorenstein projective (or Gorenstein injective) dimensions ofunbounded complexes between U⊗ LR X (or RHom R (X,U)) and X areconsidered, where X is an R-complex and U is an S-complex. In ad-dition, some sufficient conditions are given under which the equalitiesG-dim S (U⊗ LR X) = G-dim R X + pd S U and Gid S (RHom R (X,U)) =G-dim R X+ id S U hold. IntroductionIt is well known that Gorenstein homological dimensions are refinements ofthe classical homological dimensions. In a different direction, homological di-mensions have been extended to complexes. Avramov and Foxby [2] definedprojective, injective and flat dimensions for unbounded complexes of left mod-ules over associative rings. The concepts of Gorenstein projective, injectiveand flat dimension for homologically bounded complexes were introduced byChristensen [3]. Veliche [10] extended the concept of Gorenstein projectivedimension of homologically bounded complexes to the setting of unboundedcomplexes over associative rings. Dually, Asadollahi and Salarian [1] intro-duced the concept of Gorenstein injective dimension of unbounded complexesover associative rings.Transfer of homological properties along ring homomorphisms is a classi-cal field of study (see, for instance, [9] and its references). Let R and S becommutative Noetherian rings. It was shown in [9] that if ϕ : R → S is aring homomorphism such that every S-module of finite flat dimension is offinite projective dimension over R via ϕ, X is an R-complex and U is an S-complex with finite projective dimension, then one has Gid