L-FUNCTORS AND ALMOST SPLIT SEQUENCES

ABSTRACT

L-functors (Rump, 2001) provide a new tool for the study of Auslander–Reiten quivers associated with an isolated singularity in the sense of M. Auslander. We show that L-functors L, L ⁻:ℳ → ℳ admit an intrinsic definition for an arbitrary additive category ℳ. When they exist, they endow ℳ with a structure closely related to that of a triangulated category. If ℳ is the homotopy category (𝒜) of two-termed complexes over an additive category 𝒜, we establish a one-to-one correspondence between L-functors on (𝒜) and classes of short exact sequences in 𝒜 which make 𝒜 into an exact category with almost split sequences. This applies, in particular, to categories 𝒜 = Λ-CM of Cohen–Macaulay modules over a Cohen–Macaulay R-order Λ for arbitrary dimension of R.

Key Words:

• Almost split sequence
• Exact category
• Ladder functor
• Triangulated category

2000 Mathematics Subject Classification # :

• Primary 16G70
• 18E05
• Secondary 16G30
• 16G50
• 18E30

Notes

^#Communicated by A. Facchini.