On the depth of tensor products over Cohen-Macaulay rings
Kaito Kimura, Justin Lyle, Andrew J. Soto-Levins
TL;DR
This work introduces and studies two depth-inequality conditions, $(oldsymbol{ldep})$ and $(oldsymbol{rdep})$, along with their derived variants, to analyze the depth of tensor products over Cohen–Macaulay local rings. It establishes deep connections between these conditions and classical homological bounds such as the Uniform Auslander Condition $(oldsymbol{UAC})$ and its dual $(oldsymbol{UBC})$, linking them to uniform bounds on Ext/Tor and to maximal Cohen–Macaulay phenomena; notably, derived $(oldsymbol{ldep})$ is tied to a uniform Auslander bound with $b_R=d$, and $(oldsymbol{ldep})$ implies $(oldsymbol{rdep})$ in the Gorenstein case. The paper further analyzes stability under regular sequences and completion, while local behavior can diverge via localization. It also extends Jørgensen’s results by providing principled bounds and formulas for $q_R(M,N)$ under these depth conditions. Overall, the work offers a coherent framework for decomposing and understanding depth formulas in tensor products through homological invariants and dualities, with substantial implications for AB/UBC-type rings and for the structure of MCM modules and complexes.
Abstract
Inspired by classical work on the depth formula for tensor products of finitely generated $R$-modules, we introduce two conditions which we call $(\mathbf{ldep})$ and $(\mathbf{rdep})$ and their derived variations. We show for Cohen-Macaulay local rings that derived $(\mathbf{ldep})$ is equivalent to $\dim(R)$ being a uniform Auslander bound for $R$, and if $\dim(R)>0$ that both are equivalent to $(\mathbf{ldep})$. We introduce an analogous condition we call the \emph{uniform Buchweitz condition} and provide a corresponding theorem for the $(\mathbf{rdep})$ condition. As a consequence of these results, we show $(\mathbf{ldep})$ implies $(\mathbf{rdep})$ when $R$ is Gorenstein and that the $(\mathbf{ldep})$ and $(\mathbf{rdep})$ 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 $q_R(M,N):=\sup\{i \mid \operatorname{Tor}^R_i(M,N) \ne 0\}$ under certain conditions.