On the depth of tensor products over Cohen-Macaulay rings

2025/05/01 by Kimura, Kaito, Lyle, Justin, Soto-Levins, Andrew J. · 1 citation
#Commutative Algebra (math.AC) #FOS: Mathematics #Primary 13C15 #Secondary 13D07

paper · doi:10.48550/arxiv.2505.00441

Abstract

Inspired by classical work on the depth formula for tensor products of finitely generated R-modules, we introduce two conditions which we call (ldep) and (rdep) and their derived variations. We show for Cohen-Macaulay local rings that derived (ldep) is equivalent to dim(R) being a uniform Auslander bound for R, and if dim(R)>0 that both are equivalent to (ldep). We introduce an analogous condition we call the uniform Buchweitz condition and provide a corresponding theorem for the (rdep) condition. As a consequence of these results, we show (ldep) implies (rdep) when R is Gorenstein and that the (ldep) and (rdep) conditions behave well under modding out by regular sequences and completion, but we give a concrete example showing they need not localize. Using our methods, we extend work of Jorgensen by calculating the value q_R(M,N):=sup\i | Tor^R_i(M,N) ≠ 0\ under certain conditions.

Citations

• On the vanishing of Ext modules over a local unique factorization domain with an isolated singularity
• Maximal Cohen-Macaulay tensor products and vanishing of Ext modules
• Maximal Cohen-Macaulay complexes and their uses: A partial survey
• Persistence of homology over commutative noetherian rings
• Extremal growth of Betti numbers and trivial vanishing of (co)homology
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• Hom and Ext, Revisited
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• Algebras that satisfy Auslander's condition on vanishing of cohomology
• Nonvanishing cohomology and classes of Gorenstein rings
• Depth and amplitude for unbounded complexes
• Indecomposable canonical modules and connectedness

Cited by

• Finiteness of homological dimensions of Ext modules

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• On the depth of tensor products of modules
• Necessary conditions for the depth formula over Cohen-Macaulay local rings
• Vanishing of relative homology and depth of tensor products
• Quasilifting of hulls and depth of tensor product of modules
• Cohen--Macaulayness of tensor products