On the Auslander-Bridger type approximation of modules Tokuji Araya (Tokuyama college of technology) Kei-ichiro Iima (Nara national college of technology)

Throughout this article, let R be a commutative noetherian complete local ring with maximal ideal m and residue field k. All modules considered in this article are assumed to be finitely generated. An R-module C is said to be semidualizing if the natural homomorphism R → HomR (C, C) is an isomorphism and ExtiR (C, C) = 0 for all i > 0. Various homological dimensions with respect to a fixed semidualizing module C such as C-projective dimension are invented and investigated. Here the C-projective dimension of a nonzero R-module M , denoted by C-proj.dimR M , is defined as the infimum of integers n such that there is an exact sequence of the form 0 → C bn → C bn−1 → · · · → C b1 → C b0 → M → 0, where each bi is a positive integer. We denote by mod(R) the category of finitely generated R-modules, by TnC the full subcategory of mod(R) consisting of all modules X such that C TorR i (X, C) = 0 for any 1 ≤ i ≤ n, by Cn the full subcategory of mod(R) consisting of all ∼ modules X such that X = Y ⊗R C for some module Y ∈ TnC . A free module of rank one is a typical example of a semidualizing module. Let M be an R-module. Let ∂

∂

n · · · → Fn → Fn−1 → · · · → F1 →1 F0 → M → 0

be a minimal free resolution of M . We define the C-transpose of M to be the cokernel of the map Hom(∂1 , C) : Hom(F0 , C) → Hom(F1 , C), and denote it by TrC M . We say that M is n-C-torsionfree if the R-modules ExtiR (TrC M, C) equal to zero for all 1 ≤ i ≤ n. If an R-module M belongs to CnC , then there exists an exact sequence ∂n+1

∂

∂

∂

n C bn+1 → C bn → C bn−1 → · · · → C b1 →1 C b0 →0 M → 0.

We define the C-transpose prime of M to be the cokernel of the map Hom(∂1 , C) : Hom(C b0 , C) → Hom(C b1 , C), and denote it by Tr0(C,∂1 ) M . And we define the i-C-syzygy of M to be the image of the map ∂i : C bi → C bi−1 , and denote it by Ωi(C,∂· ) M for each 0 ≤ i ≤ n + 1. The following two theorems is well-known as Auslander-Bridger approximation theorem and Cohen-Macaulay approximation theorem.

Theorem 1 ([1]) The following are equivalent for a finitely generated R-module M : (1) ΩnR M is n-torsionf ree. (2) There exists an exact sequence 0 → Y → X → M → 0 of R-modules such that proj.dimR Y < n, ExtiR (X, R) = 0 (1 ≤ i ≤ n) and X is isomorphic to TrR ΩnR TrR ΩnR M . Theorem 2 ([2]) Let R be a Cohen-Macaulay local ring with the canonical module. Then, for every finitely generated R-module M , there exists an exact sequence 0 → Y → X → M → 0 of R-modules such that inj.dimR Y < n and X is a maximal Cohen-Macaulay R-module. Takahashi unifies above two approximation theorems using a semidualizing R-module. The approximation theorem of Takahashi is stated as follows. Theorem 3 ([5]) Let M and C be finitely generated R-modules. Assume that C is semidualizing. Then the following conditions on M are equivalent: (1) ΩnR M is n-C-torsionf ree. (2) There exists an exact sequence 0 → Y → X → M → 0 of R-modules such that i C-proj.dimR Y < n and ExtR (X, C) = 0 (1 ≤ i ≤ n). Our purpose is to give a middle term of the sequence in Theorem 3 explicitly. To prove our theorem, we establish the following lemma. Lemma 4 Let M and C be finitely generated R-modules. Assume that C is semidualizing, M belongs to C0C and ExtiR (Tr0C M, C) = 0 for 1 ≤ i ≤ n, then M is n-C-torsionf ree. The main result in this article is the following theorem. Theorem 5 Let M be an R-module and let n ≥ 0. If M belongs to CnC and ExtiR (Tr0C ΩnC M, C) = 0 for 1 ≤ i ≤ n, then there exists an exact sequence 0 → T → TrC ΩnR Tr0C ΩnC M → M → 0 of R-modules with C-proj.dimR T < n. We obtain another approximation theorem is the following. Theorem 6 Let M be an R-module and let n ≥ 0. If TrR ΩnR M belongs to TnC and ΩnR M is n-C-torsionfree, then there exists an exact sequence 0 → T → Tr0C ΩnC TrC ΩnR M → M → 0 of R-modules with proj.dimR T < n.

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