Tor-pairs: products and approximations

Manuel Cort\'es Izurdiaga

arxiv: 1907.07547 · v1 · pith:HCX6FJALnew · submitted 2019-07-17 · 🧮 math.RA

Tor-pairs: products and approximations

Manuel Cort\'es Izurdiaga This is my paper

Pith reviewed 2026-05-24 19:51 UTC · model grok-4.3

classification 🧮 math.RA

keywords Tor-pairsT-projective dimensionrelative Mittag-Leffler dimensionmodule approximationsdeconstructible classesprecovering classespreenveloping classesproducts of modules

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The pith

For a Tor-pair (T, S), products of modules in T have finite T-projective dimension exactly when modules in S have finite relative T-Mittag-Leffler dimension.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends earlier results on products of flat modules to the setting of general Tor-pairs. It establishes a precise link between the finite T-projective dimension of arbitrary products drawn from T and the relative T-Mittag-Leffler dimension of modules drawn from S. These equivalences are then used to determine when modules in T admit approximations in the module category. Short proofs are supplied showing that every deconstructible class is precovering and that every deconstructible class closed under products is preenveloping.

Core claim

We extend the theory to characterize when products of modules in T have finite T-projective dimension, where T is the left hand class of a Tor-pair (T,S), relating this property with the relative T-Mittag-Leffler dimension of modules in S. We apply these results to study the existence of approximations by modules in T. In order to do this, we give short proofs of the well known results that a deconstructible class is precovering and that a deconstructible class closed under products is preenveloping.

What carries the argument

A Tor-pair (T, S) together with the T-projective dimension on T and the relative T-Mittag-Leffler dimension on S.

If this is right

Products of modules in T have finite T-projective dimension precisely when every module in S has finite relative T-Mittag-Leffler dimension.
The dimension conditions determine whether approximations by modules in T exist.
Every deconstructible class is precovering.
Every deconstructible class that is closed under products is preenveloping.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same dimension link may classify additional rings on which products preserve finite relative dimension beyond the flat case.
The supplied short proofs of the precovering and preenveloping statements could replace longer arguments in the literature on approximations.
The framework invites direct comparison with other relative homological dimensions defined via Tor or Ext.

Load-bearing premise

The assumption that (T, S) forms a Tor-pair and that the relative dimensions are well-defined and behave as stated for the classes in question.

What would settle it

A concrete Tor-pair (T, S) in which some product of modules from T has infinite T-projective dimension while every module in S has finite relative T-Mittag-Leffler dimension.

read the original abstract

Recently the author has studied rings for which products of flat modules have finite flat dimension. In this paper we extend the theory to characterize when products of modules in $\mathcal T$ have finite $\mathcal T$-projective dimension, where $\mathcal T$ is the left hand class of a Tor-pair $(\mathcal T,\mathcal S)$, relating this property with the relative $\mathcal T$-Mittag-Leffler dimension of modules in $\mathcal S$. We apply these results to study the existence of approximations by modules in $\mathcal T$. In order to do this, we give short proofs of the well known results that a deconstructible class is precovering and that a deconstructible class closed under products is preenveloping.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper extends flat module product results to Tor-pairs with a characterization and short proofs of known facts on deconstructible classes.

read the letter

This paper extends the author's prior work on rings where products of flat modules have finite flat dimension to the general case of Tor-pairs (T, S). The central new result characterizes when products of modules in T have finite T-projective dimension by relating it to the relative T-Mittag-Leffler dimension of modules in S. It then applies this to the existence of approximations by modules in T. Along the way it includes short proofs that deconstructible classes are precovering and that deconstructible classes closed under products are preenveloping. What the paper does well is keep the extension direct and provide those short proofs in a clean way. The proofs of the standard facts about deconstructible classes are brief and could serve as handy references. The characterization for general Tor-pairs follows logically from the flat case once the setup is in place, so the logic holds without obvious circularity. The soft spots are mostly about scope. The contribution stays within the established line of research on these pairs and does not add new concrete examples or test cases on specific rings or modules. That makes the advance incremental rather than transformative. The abstract does not indicate any computational verification or independent checks beyond the abstract theory, which is typical for this area but limits broader appeal. This work is aimed at specialists in homological algebra who already deal with relative dimensions, Mittag-Leffler conditions, and approximation theory in module categories. A reader in that niche can extract the characterization and the short proofs for their own use. It is not likely to draw interest from outside that group. The paper shows clear thinking on its own terms and engages honestly with the literature it builds on. I would bring it to a reading group only if the group focuses on relative homological algebra. I would not cite it in my own work in the next year unless I were working directly in this area. It deserves peer review because the extension is well-defined and the proofs are short enough to evaluate without much effort.

Referee Report

0 major / 3 minor

Summary. The paper extends prior results on rings where products of flat modules have finite flat dimension to Tor-pairs (T, S). It characterizes when products of modules in T have finite T-projective dimension in terms of the relative T-Mittag-Leffler dimension of modules in S, applies the results to the existence of T-approximations, and supplies short proofs that deconstructible classes are precovering while product-closed deconstructible classes are preenveloping.

Significance. If the characterizations and applications hold, the work supplies a coherent extension of relative homological algebra from the flat case to Tor-pairs, together with concrete links to approximation theory. The short, self-contained proofs of the two standard facts on deconstructible classes are a clear strength and improve readability.

minor comments (3)

[§1] §1 (Introduction): the definition of a Tor-pair (T, S) is referenced but not restated; a one-sentence reminder of the exact conditions (Tor_1(T, S) = 0 and the pair is maximal) would help readers who have not consulted the cited prior work.
The notation for relative T-Mittag-Leffler dimension is introduced without an explicit comparison to the classical Mittag-Leffler dimension; a short sentence relating the two would clarify the extension.
Theorem statements that characterize finite T-pd of products should explicitly list the standing assumptions on the ring (e.g., whether it is commutative or has finite global dimension) so that the scope is immediately visible.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the clear summary of its contributions, and the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper extends the author's prior results on products of flat modules having finite flat dimension to the setting of Tor-pairs (T,S), providing new characterizations relating finite T-projective dimension of products to relative T-Mittag-Leffler dimension on S, plus applications to T-approximations. It also supplies short independent proofs of two standard facts (deconstructible classes are precovering; product-closed deconstructible classes are preenveloping). These steps rely on the given definition of a Tor-pair and standard module-theoretic notions rather than reducing any claim to a self-citation, fitted parameter, or self-definitional loop. Self-reference to the author's earlier flat-module work is purely motivational and not load-bearing for the new relations or proofs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard definitions and properties from homological algebra without introducing new free parameters or postulated entities.

axioms (2)

domain assumption A Tor-pair (T,S) satisfies Tor_i^R(X,Y)=0 for all i>0, X in T, Y in S
Invoked throughout the abstract as the setting for the extension and characterization.
domain assumption Deconstructible classes are closed under certain operations allowing precovering and preenveloping properties
Used in the short proofs mentioned in the abstract.

pith-pipeline@v0.9.0 · 5641 in / 1296 out tokens · 28162 ms · 2026-05-24T19:51:45.026726+00:00 · methodology

Tor-pairs: products and approximations

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)