Remarks on Auslander’s depth formula for quasi-projective dimension
Abstract
Abstract For nonzero finitely generated R -modules M and N over a Noetherian local ring R , Auslander’s depth formula is the equality $$ \operatorname {depth} M + \operatorname {depth} N = \operatorname {depth} R + \operatorname {depth}(\operatorname {Tor}_q^R(M,N)) - q, $$ depth M + depth N = depth R + depth ( Tor q R ( M , N ) ) - q , where $$ q:= \sup \{ i \ge 0 \mid \operatorname {Tor}_i^R(M,N) \ne 0 \}$$ q : = sup { i ≥ 0 ∣ Tor i R ( M , N ) ≠ 0 } . Gheibi, Jorgensen, and Takahashi introduced a homological invariant called quasi-projective dimension, which generalizes projective dimension, and proved that Auslander’s depth formula holds when M has finite quasi-projective dimension and $$q=0$$ q = 0 . In this paper, we prove that the formula still holds when M has finite quasi-projective dimension, $$q
Fields of Study & Subjects
Rings
Modules
and Algebras
Polynomial and algebraic computation
Commutative Algebra and Its Applications
Publication details
- Journal / source
- Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas
- Publisher
- Springer Science and Business Media LLC
- Volume
- 120
- Issue
- 3
- Publication date
- 11 June 2026
- Language
- EN