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arxiv: 2606.11684 · v3 · pith:GUK2ZBYTnew · submitted 2026-06-10 · 🧮 math.RT · math.RA

τ-tilting modules, depth and delooping level

Mingfei Xu , Xiaojin Zhang This is my paper

Pith reviewed 2026-07-02 22:36 UTC · model grok-4.3

classification 🧮 math.RT math.RA
keywords support au-tilting modulesfinitistic dimensionFac Tdepth relative to Tdelooping levelrepresentation infinite algebrasendomorphism algebras
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The pith

The finitistic dimension of B^op is bounded by the depth of Fac T relative to T and the delooping level of Fac T relative to T.

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

The paper defines the depth relative to T and the delooping level relative to T for the subcategory Fac T of modules generated by a support au-tilting module T. It proves that these two quantities together bound the finitistic dimension of the opposite endomorphism algebra B^op. The bound yields finiteness of the finitistic dimension of B^op when the base algebra A is minimal representation infinite or has finite representation type.

Core claim

Let A be a finite-dimensional basic algebra over an algebraically closed field K, T a finitely generated support au-tilting right A-module and B=End_A T. Denote by Fac T the subcategory of finitely generated right A-modules generated by T. The finitistic dimension of B^op is bounded by the depth of Fac T relative to T and the delooping level of Fac T relative to T. If A is a minimal representation infinite algebra or an algebra of finite representation type, then the finitistic dimension of B^op is finite.

What carries the argument

The depth relative to T and the delooping level relative to T of Fac T, which together bound the finitistic dimension of B^op.

If this is right

  • If A is minimal representation infinite then the finitistic dimension of B^op is finite.
  • If A has finite representation type then the finitistic dimension of B^op is finite.
  • The finitistic dimension conjecture holds for such B^op via the given bound.
  • The relative depth and delooping level control homological dimensions for endomorphism algebras of support au-tilting modules.

Where Pith is reading between the lines

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

  • The bound may allow explicit computation of finitistic dimensions once the relative depth and delooping level can be calculated for concrete algebras.
  • Analogous relative invariants could be defined for other module classes to obtain similar finiteness results.
  • The new measures may relate directly to classical projective dimension or other homological invariants already studied for au-tilting modules.

Load-bearing premise

That T is a finitely generated support au-tilting right A-module over a finite-dimensional basic algebra A over an algebraically closed field, allowing Fac T to admit well-defined finite depth and delooping level relative to T.

What would settle it

An explicit support au-tilting module T for which the finitistic dimension of B^op exceeds the bound given by the depth plus the delooping level of Fac T relative to T.

read the original abstract

Let $A$ be a finite-dimensional basic algebra over an algebraically closed field $K$, $T$ a finitely generated support $\tau$-tilting right $A$-module and $B={\rm End}_A T$. Denote by ${\rm Fac}T$ the subcategory of finitely generated right $A$-modules generated by $T$. We define the depth relative to $T$ and the delooping level relative to $T$ and show that the finitistic dimension of the opposite algebra of $B$ is bounded by the depth of $\textup{Fac}T$ relative to $T$ and the delooping level of $\textup{Fac}T$ relative to $T$. We give applications to the finitistic dimension conjecture. More precisely, we show that if $A$ is a minimal representation infinite algebra or an algebra of finite representation type, then the finitistic dimension of $B^{op}$ is finite.

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.

Referee Report

0 major / 3 minor

Summary. The paper defines the depth and delooping level of the subcategory Fac T relative to a finitely generated support τ-tilting right A-module T, where A is a finite-dimensional basic algebra over an algebraically closed field. It proves that the finitistic dimension of B^op (B = End_A T) is bounded above by these two invariants of Fac T. Applications establish that this finitistic dimension is finite when A is minimal representation-infinite or of finite representation type, yielding new cases of the finitistic dimension conjecture.

Significance. If the bounding inequality holds, the work supplies a direct homological control mechanism inside τ-tilting theory that converts finiteness of the new invariants into finiteness of finitistic dimension for B^op. The applications to minimal representation-infinite algebras and finite-representation-type algebras are concrete and falsifiable, extending existing results on support τ-tilting modules without introducing free parameters or ad-hoc fitting.

minor comments (3)
  1. §2: the recursive definition of depth relative to T is stated only for modules in Fac T; an explicit statement that the invariant is well-defined (finite or infinite) for all objects in the subcategory would clarify the subsequent bounding argument.
  2. The notation 'depth(Fac T / T)' and 'delooping level(Fac T / T)' is introduced without a displayed equation summarizing the two quantities together; adding such an equation before the main theorem would improve readability of the central claim.
  3. The proof that both invariants are finite for minimal representation-infinite A appears in a later section; a forward reference from the statement of the application would help readers trace the logic.

Simulated Author's Rebuttal

0 responses · 0 unresolved

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

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines two new invariants (depth of Fac T relative to T, and delooping level of Fac T relative to T) for a support τ-tilting module T, then proves an upper bound on the finitistic dimension of B^op in terms of these invariants. The bound is a theorem derived from the definitions and standard properties of τ-tilting theory; it does not reduce to the inputs by construction, nor does any step rely on fitted parameters renamed as predictions or on self-citation chains. Applications to finiteness in the minimal representation-infinite and finite-representation-type cases rest on separate verification that the new invariants are finite, which is independent content. The derivation is self-contained against external benchmarks in representation theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard domain assumptions of representation theory of finite-dimensional algebras over algebraically closed fields and the definition of support τ-tilting modules; no free parameters or invented entities beyond the two new invariants (which are definitions rather than postulated objects).

axioms (2)
  • domain assumption A is a finite-dimensional basic algebra over an algebraically closed field K
    Explicit setup in the abstract.
  • domain assumption T is a finitely generated support τ-tilting right A-module
    Required for the definitions of Fac T, depth, and delooping level.

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