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On the Dimension of Modules and Algebras (III): Global Dimension

1955; Cambridge University Press; Volume: 9; Linguagem: Inglês

10.1017/s0027763000023291

2152-6842

Maurice Auslander,

Algebraic structures and combinatorial models

Let Λ be a ring with unit. If A is a left Λ -module, the dimension of A (notation: 1.dim Λ A ) is defined to be the least integer n for which there exists an exact sequence 0 → X n → … → X 0 → A → 0 where the left Λ -modules X 0 , …, X n are projective. If no such sequence exists for any n , then 1. dim A A = ∞. The left global dimension of Λ is 1. gl. dim Λ = sup 1. dim A A where A ranges over all left Λ -modules, The condition 1. dim A A < n is equivalent with ( A, C ) = 0 for all left Λ -modules C . The condition 1.gl. dim Λ < n is equivalent with = 0. Similar definitions and theorems hold for right Λ -modules.