[Paper Review] Ringel modules and homological subcategories

Haishan Chen, Changchang Xi|arXiv (Cornell University)|Jun 4, 2012

Algebraic structures and combinatorial models23 references8 citations

TL;DR

This paper introduces Ringel modules as a unifying framework to study when the kernel of the left-derived functor $T uildrel{ mf L}lacktriangleleft_{B}-$ for an $n$-tilting or $n$-cotilting module $T$ over a ring $A$ is equivalent to the derived category of a ring $C$ via a ring epimorphism $B o C$. It provides general criteria for such realizations and constructs both a positive example (noncommutative) and a counterexample (commutative) for $n \geq 2$, resolving an open question in tilting theory and extending the analysis to cotilting modules with greater complexity.

ABSTRACT

Given a good $n$-tilting module $T$ over a ring $A$, let $B$ be the endomorphism ring of $T$, it is an open question whether the kernel of the left-derived functor $T\otimes^L_B-$ between the derived module categories of $B$ and $A$ could be realized as the derived module category of a ring $C$ via a ring epimorphism $B ightarrow C$ for $n\ge 2$. In this paper, we first provide a uniform way to deal with the above question both for tilting and cotilting modules by considering a new class of modules called Ringel modules, and then give criterions for the kernel of $T\otimes^L_B-$ to be equivalent to the derived module category of a ring $C$ with a ring epimorphism $B ightarrow C$. Using these characterizations, we display both a positive example of $n$-tilting modules from noncommutative algebra, and a counterexample of $n$-tilting modules from commutative algebra to show that, in general, the open question may have a negative answer. As another application of our methods, we consider the dual question for cotilting modules, and get corresponding criterions and counterexamples. The case of cotilting modules, however, is much more complicated than the case of tilting modules.

Motivation & Objective

• To resolve an open question in tilting theory: whether the kernel of $T \otimes^\mathbb{L}_B -$ for an $n$-tilting module $T$ with $n \geq 2$ can be realized as $\mathscr{D}(C)$ via a ring epimorphism $B \to C$.
• To unify the treatment of tilting and cotilting modules by introducing a new class of modules called Ringel modules.
• To provide general criteria for when the kernel of $T \otimes^\mathbb{L}_B -$ is equivalent to the derived category of a ring $C$ with a homological ring epimorphism $B \to C$.
• To construct a positive example of such a realization in noncommutative algebra and a counterexample in commutative algebra for $n$-tilting modules.
• To extend the analysis to cotilting modules, which are shown to be significantly more complex than tilting modules, and provide corresponding criteria and examples.

Proposed method

• Introduce Ringel modules as a unifying framework to analyze the kernel of $T \otimes^\mathbb{L}_B -$ for $n$-tilting and $n$-cotilting modules.
• Establish homological criteria using derived functors, Tor and Ext vanishing, and properties of relative Mittag-Leffler modules.
• Use recollement structures in derived categories to relate $\mathscr{D}(C)$, $\mathscr{D}(B)$, and $\mathscr{D}(A)$ via ring epimorphisms.
• Apply the criteria to construct a positive example from noncommutative algebra and a counterexample from commutative algebra for $n$-tilting modules.
• Analyze the dual case of cotilting modules by adapting the criteria and proving necessary conditions for homological subcategories.
• Use spectral sequences and properties of injective hulls and local cohomology to verify vanishing and non-vanishing of Tor and Ext groups in the counterexample.

Experimental results

Research questions

• RQ1 For $n \geq 2$, does there exist a good $n$-tilting module $T$ over a ring $A$ such that the kernel of $T \otimes^\mathbb{L}_B -$ is equivalent to $\mathscr{D}(C)$ for some ring $C$ via a ring epimorphism $B \to C$?
• RQ2 Is the converse of Corollary 1.2(1) true, i.e., if the kernel of $T \otimes^\mathbb{L}_B -$ is homological, must $T$ be a 1-tilting module?
• RQ3 For a good $n$-cotilting module $U$, does there exist a homological ring epimorphism $\lambda: \mathrm{End}_A(U) \to C$ such that $\mathscr{D}(C)$ realizes the kernel of $U \otimes^\mathbb{L}_B -$?
• RQ4 Can homological subcategories of $\mathscr{D}(A)$ be parameterized, or equivalently, classified via homological ring epimorphisms starting from $A$?
• RQ5 Is the Ringel $R$-module $M$ in Lemma 6.2 always good, and would this imply a generalization of Corollary 6.3?

Key findings

• The paper constructs a positive example of a good $n$-tilting module over a noncommutative ring with $n \geq 2$ such that the kernel of $T \otimes^\mathbb{L}_B -$ is equivalent to $\mathscr{D}(C)$ via a ring epimorphism $B \to C$.
• A counterexample is provided in commutative algebra for $n$-tilting modules with $n \geq 2$, showing that the kernel of $T \otimes^\mathbb{L}_B -$ is not equivalent to $\mathscr{D}(C)$ for any ring $C$ via such an epimorphism.
• For $n$-cotilting modules, the paper proves that the kernel of $U \otimes^\mathbb{L}_B -$ is not homological in $\mathscr{D}(B)$ in general, even for $n=2$, due to non-vanishing $\mathrm{Ext}^n(W, U_n) \neq 0$.
• The criteria for realizing the kernel as $\mathscr{D}(C)$ are shown to be effective for tilting modules but significantly more complex for cotilting modules, which require deeper analysis of Tor and Ext vanishing.
• The paper confirms that for $n$-tilting modules with $n \geq 2$, the existence of such a recollement is not guaranteed, and the answer depends on the ring and module structure.
• In the counterexample, $\mathrm{Tor}_n^A(E(A/\mathfrak{m}), E(A/\mathfrak{m})) \neq 0$ and $\mathrm{Ext}_A^n(W, U_n) \neq 0$, which prevents the kernel from being homological.