Abstract
Consider a non-trivial fiber product R = S ×k T of local rings S, T with common residue field k. Given two finitely generated R-modules M and N, we show that if TorRi (M, N) = 0 = TorRi+1(M, N) for some i ≥ 5, then pdR (M) ≤ 1 or pdR (N) ≤ 1. From this, we deduce several consequences, for instance, that R satisfies the Auslander-Reiten Conjecture.
Original language | English |
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Pages (from-to) | 4661-4674 |
Number of pages | 14 |
Journal | Proceedings of the American Mathematical Society |
Volume | 145 |
Issue number | 11 |
DOIs | |
State | Published - 2017 |
Scopus Subject Areas
- General Mathematics
- Applied Mathematics
Keywords
- Auslander-Reiten
- Ext-index
- Ext-vanishing
- Fiber product
- Injective dimension
- Projective dimension
- Semidualizing complexes
- Tor-vanishing