A new Homological Invariant for Modules
Abstract
Let $R$ be a commutative Noetherian local ring with residue field $k$. Using the structure of Vogel cohomology, for any finitely generated module $M$, we introduce a new dimension, called $\zeta$-dimension, denoted by $\zeta-dim_R M$. This dimension is finer than Gorenstein dimension and has nice properties enjoyed by homological dimensions. In particular, it characterizes Gorenstein rings in the sense that: a ring $R$ is Gorenstein if and only if every finitely generated $R$-module has finite $\zeta$-dimension. Our definition of $\zeta$-dimension offer a new homological perspective on the projective dimension, complete intersection dimension of Avramov et al. and $G$-dimension of Auslander and Bridger.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2019
- DOI:
- 10.48550/arXiv.1903.05308
- arXiv:
- arXiv:1903.05308
- Bibcode:
- 2019arXiv190305308I
- Keywords:
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- Mathematics - Commutative Algebra