Depth-wise Tensor Products (DTP)

• Depth-wise Tensor Products (DTP) are a framework in commutative algebra that capture refined interactions between depth invariants of modules and complexes under tensor operations.
• DTP formalizes conditions like ldep and rdep, establishing inequalities that extend classical depth formulas and control tensor product behavior across various ring types.
• The theory underpins descent of Serre conditions, reflexivity, and rigidity results, impacting practical analysis in Cohen–Macaulay, Gorenstein, and hypersurface rings.

Depth-wise Tensor Products (DTP) are a central object of paper in commutative algebra, capturing refined interactions between the depth invariants of modules and complexes under the tensor product operation. Originating from classical depth formulas over local rings, DTP encompasses a suite of results, metrics, and inequalities unifying and generalizing such phenomena, particularly in the context of Cohen–Macaulay, Gorenstein, hypersurface, and complete intersection rings. The recent formalization of DTP introduces invariants and conditions—ldep, rdep, and their derived forms—tied to uniform bounds, support localization, and reflexivity, extending classical and modern rigidity and depth results.

1. Depth Invariants, Set-up, and Notation

DTP is predicated on the analysis of depth and codimension invariants for modules and complexes over a Noetherian local ring $(R,\mathfrak{m},k)$. For a bounded below $R$-complex $X$ with finitely generated homology, its depth is defined by

$\mathrm{depth}_R(X) = \inf\{\, i \mid \Ext^i_R(k, X) \neq 0\, \} = \inf\{\, i \mid H^i_{\mathfrak{m}}(X) \neq 0 \, \}.$

For a finitely generated $R$-module $M$, $$\mathrm{codepth}_R(M) := \mathrm{depth}(R) - \mathrm{depth}_R(M)$$.

Central invariants for pairs of $R$-modules $M, N$ include:

• $\displaystyle q_R(M,N) := \sup\{\, i \mid \Tor^R_i(M, N) \neq 0 \,\}$
• $\displaystyle b_R(M,N) := \sup\{\, i \mid \Ext^i_R(M, N) \neq 0 \,\}$

The complexity $\mathrm{cx}(M,N)$ measures the polynomial rate of growth in the sequence $\dim_k \Ext^*_R(M,N)$, and CI-dimension $\mathrm{CI\text{-}dim}_R(X)$ denotes the minimal possible deviation from projectivity for a module or complex via complete intersection approximations.

2. Depth Formulas and Main Inequalities

The archetypal result is the depth formula. For $R$ local Cohen–Macaulay or complete intersection, and $M,N$ finitely generated modules with vanishing higher Tor,

$$\mathrm{depth}_R(M) + \mathrm{depth}_R(N) = \mathrm{depth}(R) + \mathrm{depth}_R(M \otimes_R N)$$

(Celikbas et al., 2021, Kimura et al., 1 May 2025, Celikbas et al., 2013).

DTP divides this into two principal one-sided inequalities for the tensor product $M \otimes_R N$ when $q_R(M, N) = 0$:

• ldep: $$\mathrm{depth}_R(M\otimes_R N) + \mathrm{depth}(R) \ge \mathrm{depth}_R(M) + \mathrm{depth}_R(N)$$
• rdep: $$\mathrm{depth}_R(M\otimes_R N) + \mathrm{depth}(R) \le \mathrm{depth}_R(M) + \mathrm{depth}_R(N)$$

Equality (dep) holds if both inequalities are satisfied, yielding $$\mathrm{codepth}_R(M\otimes_R N) = \mathrm{codepth}_R(M) + \mathrm{codepth}_R(N)$$. Derived versions replace modules by bounded complexes with finite homology and Tor-vanishing in high degrees (Kimura et al., 1 May 2025).

Lower and upper bounds for $\mathrm{depth}(M\otimes_R N)$ are refined in several directions. If $M$ is locally free on the punctured spectrum, then either

$$\mathrm{depth}(M\otimes_R N) \geq \mathrm{depth}(M) + \mathrm{depth}(N) - \mathrm{depth}(R)$$

or

$$\mathrm{depth}(M\otimes_R N) \leq \mathrm{cx}(M,N) \leq \min\{\mathrm{cx}(M),\mathrm{cx}(N)\} \leq \mathrm{codim}(R)$$

(Celikbas et al., 2013). In the presence of an isolated singularity, an upper bound is enforced by the dichotomy.

3. Uniform Auslander and Buchweitz Conditions

The Uniform Auslander Condition (UAC) and the Uniform Buchweitz Condition (UBC) are central in DTP and govern the behavior of ldep and rdep, respectively (Kimura et al., 1 May 2025).

• UAC: $R$ has finite Auslander bound $b_R = \sup\{ b_R(A,B) < \infty \}$.
• UBC: For all $M,N$ with $b_R(M,N) < \infty$, $b_R(M,N) \ge \mathrm{codepth}_R(M)$.

The main theorems establish the following equivalences for Cohen–Macaulay $R$ of dimension $d$. Derived ldep holds if and only if $R$ satisfies UAC with bound $b_R = d$ and tensor products of suitably high syzygies of finite-length modules are maximal Cohen–Macaulay:

• If $R$ is Gorenstein, ldep implies rdep and dep holds.
• A dual chain of equivalences holds for rdep governed by UBC.

These conditions are stable under regular sequences and completion, but can fail to localize: there exist rings whose localizations do not retain ldep or rdep (e.g., via fiber product constructions) (Kimura et al., 1 May 2025).

4. Serre Conditions, Reflexivity, and Descent Theorems

DTP provides descent results for Serre conditions under tensor products. For homologically finite complexes $X$, $Y$ over local $R$, a descent theorem (Theorem 3.1) states that if $X$ has finite CI-dimension, $X \otimes_R^L Y$ is bounded and satisfies a Serre condition $(S_n)$, and an auxiliary inequality on the depth holds for primes in the support, then $Y$ also satisfies a weaker Serre condition $(S_{n-m})$ (Celikbas et al., 2021).

Specialization yields corollaries for module reflexivity. For local hypersurfaces, if $M\otimes_R N$ is reflexive and certain depth inequalities hold locally, then $M$ and $N$ are reflexive (Celikbas et al., 2021).

5. Rigidity, Vanishing, and the Role of Auslander's Transpose

Auslander's transpose is pivotal in converting vanishing of Tor into control over Ext and depth (Celikbas et al., 2013). Generalized rigidity theorems state that if $\Tor_i^R(M, N)$ vanishes in a given range, and the module pair has complexity below a threshold, then all higher Tor must vanish. This allows deduction of the lower bound for tensor product depth given initial Tor vanishing and complexity constraints.

The Jørgensen-type formula extends this control: under derived dep conditions, for $q_R(M,N) < \infty$,

$$q_R(M,N) = \sup_{p \in \mathrm{Supp}(M)\cap\mathrm{Supp}(N)}\bigl( \mathrm{depth}(R_p) - \mathrm{depth}_{R_p}(M_p) - \mathrm{depth}_{R_p}(N_p) \bigr)$$

(Kimura et al., 1 May 2025).

6. Behavior Under Localization, Completion, and Regular Sequences

DTP invariants demonstrate stable behavior under completion and passage to quotients by regular sequences:

• Derived ldep (resp. rdep) holds on $R$ if and only if it holds on $R/(x)$ for $x$ a regular element.
• Both properties ascend and descend to the completion $\hat R$ (Kimura et al., 1 May 2025).

However, these conditions may fail to localize: explicit examples show that ldep, rdep, and total reflexivity can fail on certain localizations even when holding globally (Kimura et al., 1 May 2025).

7. Key Examples, Counterexamples, and Applications

• Artinian rings: satisfy derived rdep for modules.
• Golod rings & trivial-vanishing rings: satisfy derived dep.
• Non-AB Artin algebras: provide counterexamples to UAC, hence ldep.
• Hypersurfaces with isolated singularities: for nonfree maximal Cohen–Macaulay modules $M,N$, $\mathrm{depth}(M\otimes_R N)\leq 1$.
• Symbolic power tensor product reflexivity: Over local non-domain hypersurfaces, symbolic powers of nonminimal primes tensor to be reflexive only if both primes are minimal (Celikbas et al., 2021).
• Complexity drop: Examples over $R = k[x, y, z]/(x^2, y^2, z^2)$ illustrate that pairwise complexity can be strictly less than the complexity of either module separately (Celikbas et al., 2013).

Applications of DTP theorems extend to the characterization of tensor product torsion, control over resolution growth, and descent of depth properties in local and derived settings.

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The depth-wise tensor product theory systematizes and extends the classical paper of depth for module and complex tensor products in local algebra, unifying depth formulas, rigidity, reflexivity, and homological invariants within a comprehensive framework (Celikbas et al., 2021, Kimura et al., 1 May 2025, Celikbas et al., 2013).