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Vanishing of relative homology and depth of tensor products
- Publication Year :
- 2016
-
Abstract
- For finitely generated modules $M$ and $N$ over a Gorenstein local ring $R$, one has $depth M + depth N= depth(M\otimes N) +depth R$, i.e., the depth formula holds, if $M$ and $N$ are Tor-independent and Tate homology $\hat{Tor}_{i}^{R}(M,N)$ vanishes for all $i\in\mathbb{Z}$. We establish the same conclusion under weaker hypotheses: if $M$ and $N$ are $\mathcal{G}$-relative Tor-independent, then the vanishing of $\hat{Tor}_{i}^{R}(M,N)$ for all $i\le 0$ is enough for the depth formula to hold. We also analyze the depth of tensor products of modules and obtain a necessary condition for the depth formula in terms of $\mathcal{G}$-relative homology.<br />Comment: To appear in Journal of Algebra
- Subjects :
- Mathematics - Commutative Algebra
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1608.07011
- Document Type :
- Working Paper