Three homological invariants under cleft extensions

Technical Summary: Three Homological Invariants under Cleft Extensions

Problem Statement
This paper investigates the behavior of three specific homological invariants—Igusa-Todorov distance, extension dimension, and Rouquier dimension—under the operation of cleft extensions of abelian categories. While the dimension of triangulated categories (Rouquier dimension) and the extension dimension of abelian categories are well-studied tools for measuring representation-theoretic complexity (e.g., finiteness of representation type or bounds on derived categories), their stability and bounds under specific categorical constructions remain an area of active research. Specifically, the authors aim to establish quantitative bounds for these invariants when an abelian category $\mathcal{A}$ is constructed as a cleft extension of another category $\mathcal{B}$.

Methodology
The authors employ the framework of cleft extensions and cleft coextensions of abelian categories, originally introduced by Beligiannis.

1. Categorical Framework: A cleft extension $(\mathcal{B}, \mathcal{A}, i, e, l)$ involves functors where $e: \mathcal{A} \to \mathcal{B}$ is faithful and exact, admits a left adjoint $l$, and satisfies $el \simeq \text{Id}_{\mathcal{B}}$. This structure induces an endofunctor $F: \mathcal{B} \to \mathcal{B}$. Dually, a cleft coextension involves a right adjoint $r$ and induces an endofunctor $F': \mathcal{B} \to \mathcal{B}$.
2. Homological Conditions: The proofs rely on specific homological properties of these functors, particularly the exactness of $l$ or $r$, the preservation of projective objects by $e$, and the nilpotency of the induced endofunctors ($F^n = 0$ or $F'^n = 0$).
3. Syzygy and Resolution Techniques: The authors utilize syzygy objects ($\Omega^n$) and exact sequences to relate the resolutions of objects in $\mathcal{A}$ to those in $\mathcal{B}$. Key lemmas involve constructing exact sequences in $\mathcal{A}$ by "lifting" sequences from $\mathcal{B}$ via the functor $l$ (or $r$) and analyzing the kernels and cokernels using Schanuel's Lemma and pullback diagrams.
4. Derived Category Analysis: For the Rouquier dimension, the authors utilize the derived functors of the adjoint pairs ($Ll, Lq$) to establish triangle equivalences and bounds within the bounded derived categories $D^b(\text{mod-}\Lambda)$.

Key Contributions and Results

• Igusa-Todorov Distance (Theorem 3.11):
  For a cleft extension $(\mathcal{B}, \mathcal{A}, i, e, l)$ where $l$ is exact, $e$ preserves projectives, and the induced endofunctor $F$ is nilpotent ($F^n=0$) with $\text{Proj}_{\mathcal{B}} = \text{add}(P)$, the Igusa-Todorov distance satisfies:
  $$\text{IT.dist}(\mathcal{B}) \leq \text{IT.dist}(\mathcal{A}) \leq n(\text{IT.dist}(\mathcal{B}) + 1) - 1$$
  This result generalizes previous findings on Artin algebras to the broader context of abelian categories.

• Extension Dimension (Theorem 4.10):
  Addressing the fact that module categories over Noetherian algebras often lack enough injectives, the authors utilize cleft coextensions $(\mathcal{B}, \mathcal{A}, i, e, r)$. If $r$ is exact and the induced endofunctor $F'$ is nilpotent ($F'^n=0$) with $\text{Proj}_{\mathcal{A}} = \text{add}(P)$, then:
  $$\text{ext.dim}(\mathcal{B}) \leq \text{ext.dim}(\mathcal{A}) \leq n(\text{ext.dim}(\mathcal{B}) + 1) - 1$$

• Rouquier Dimension (Theorem 5.8):
  For cleft extensions of module categories over Noetherian algebras where the induced endofunctor $F$ is both left perfect and nilpotent, the Rouquier dimension of the bounded derived categories satisfies:
  $$\text{Rouq.dim} D^b(\text{mod-}\mathcal{B}) \leq \text{Rouq.dim} D^b(\text{mod-}\mathcal{A}) \leq n(\text{Rouq.dim} D^b(\text{mod-}\mathcal{B}) + 1) - 1$$

• Applications to Specific Ring Constructions (Section 6 & 7):
  The theoretical bounds are applied to several important classes of rings:

  • Morita Context Rings: Bounds are established for the invariants of the context ring in terms of the component rings, provided specific bimodule conditions (e.g., projectivity or vanishing Tor) are met.
  • $\theta$-Extensions and Trivial Extensions: The results generalize to rings of the form $\Gamma \ltimes_\theta M$, providing bounds based on the nilpotency index of $M$.
  • Tensor Rings: Similar bounds are derived for tensor rings $T_\Gamma(M)$.
  • Arrow Removals: A significant application is provided for admissible path algebras where specific arrows are removed. Theorem 6.5 demonstrates that if arrows $a_i$ are removed to form $\Gamma$ from $\Lambda$, the invariants satisfy:
    $$\text{IT.dist}(\Gamma) \leq \text{IT.dist}(\Lambda) \leq 2\text{IT.dist}(\Gamma) + 1$$
    (and similarly for extension and Rouquier dimensions). This is shown to be a tight bound in specific examples (e.g., gentle algebras).

Significance and Claims
The paper claims to provide a unified conceptual framework for studying these homological invariants across diverse algebraic constructions. By establishing that these invariants behave predictably under cleft extensions (specifically, that they are bounded linearly by the invariant of the base category and the nilpotency index of the induced functor), the authors offer a tool to:

1. Determine if an algebra is an Igusa-Todorov algebra (and thus satisfies the finitistic dimension conjecture) based on the properties of a simpler sub-structure.
2. Compute or bound the extension dimension and Rouquier dimension for complex rings (like Morita contexts or arrow-removed path algebras) by reducing them to simpler components.
3. Refine existing bounds for Rouquier dimension, as demonstrated in Example 7.3, where the cleft extension bound ($\leq 3$) improves upon the general Loewy length bound ($\leq 5$) for a specific algebra.

The authors emphasize that these results are derived strictly from the homological properties of the cleft extension structure, without requiring the algebras to be of finite representation type, thereby extending the applicability of these invariants to a wider class of Noetherian algebras.