EXTENSION CLOSURE OF RELATIVE k-TORSIONFREE MODULES
نویسنده
چکیده
Let be a ring. We use mod (resp. mod ) to denote the category of finitely generated left -modules (resp. right -modules). We always assume that and are Artinian algebras and is a faithfully balanced self-orthogonal bimodule, that is, satisfies the following conditions: (1) is in mod and is in mod ; (2) the natural maps → End op and → End are isomorphisms; (3) Ext = 0 and Ext = 0 for any i ≥ 1. We use add (resp. add ) to denote the subcategory of mod (resp. mod ) consisting of all modules isomorphic to direct summands of finite direct sums of copies of (resp. ). Suppose that A ∈ mod (resp. mod ). Then, we call Hom A (resp. Hom A ) the dual module of A with respect to , and denote these modules by A . For a homomorphism f between the -modules (resp. -modules), we put f = Hom f . Let A A → A via A x f = f x , for any x ∈ A and f ∈ A , be the canonical evaluation homomorphism. Then, we call A -torsionless (resp. -reflexive) if A is a monomorphism (resp. an isomorphism). Now let P1 f −→ P0 → A → 0 be a minimal projective resolution of A. Then we have an exact sequence 0 → A → P 0 f −→ P 1 → Coker f → 0. We call Cokerf the transpose (with respect to ) of A, and denote it by Tr A. For a positive integer k, a module A in mod (resp. mod ) is called -k-torsionfree if Ext Tr A (resp. Ext i Tr A = 0 for any 1 ≤ i ≤ k. A is called -k-syzygy if there is an exact sequence 0 → A → X0 → X1 → · · · fk−1 −→ Xk−1 with all Xi in add
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