On Extensions of Covariantly Finite Subcategories

We prove that in an abelian category with enough projective objects, the extension subcategory of two covariantly finite subcategories is still covariantly finite. This extends a result by Sikko and Smalø. We also prove a triangulated version of the result. As applications, we obtain short proofs to a classical result by Ringel and a recent result by Krause and Solberg. 1. Main Theorems Let C be an additive category. By a subcategory X of C we always mean a full additive subcategory. Let X be a subcategory of C and let M ∈ C. A morphism xM : M −→ XM is called a left X -approximation of M if XM ∈ X and every morphism from M to an object in X factors through xM . The subcategory X is said to be covariantly finite in C, if every object in C has a left X -approximation. The notions of left X -approximation and covariantly finite are also known as X -preenvelop and preenveloping, respectively. For details, see [3, 4] and [5]. To state our main result, let C be an abelian category. Let X and Y be its subcategories. Set X ∗ Y to be the subcategory consisting of objects Z such that there is a short exact sequence 0 −→ X −→ Z −→ Y −→ 0 with X ∈ X and Y ∈ Y, and it is called the extension subcategory of Y by X . Note that the operation “∗” on subcategories is associative. Recall that an abelian category C has enough projective objects, if for each object M there is an epimorphism P −→ M with P projective. The following is our main result, which extends the corresponding results in artin algebras and coherent rings, obtained by Sikko and Smalø via functor categories (see [10, Theorem 2.6] and [11]). Theorem 1.1. Let C be an abelian category with enough projective objects. Assume that both X and Y are covariantly finite subcategories in C. Then the extension subcategory X ∗ Y is covariantly finite. Proof. The argument here resembles the one in the proof of [7, Lemma 1.3]. Assume that M ∈ C is an arbitrary object. Take its left Y-approximation yM : M −→ YM with YM ∈ Y. By assumption the category C has enough projective objects, we may take an epimorphism πM : P −→ YM with P projective. Consider the short exact This project was supported by China Postdoctoral Science Foundation No. 20070420125, and was also partially supported by the National Natural Science Foundation of China (Grant No.s 10501041 and 10601052). The author also gratefully acknowledges the support of K. C. Wong Education Foundation, Hong Kong. E-mail: [email protected]. 1