Comparing four definitions of cotilting modules

In contrast to the theory of tilting modules, the dual theory lacks a unified definition. Nevertheless, several notions of cotilting modules have been proposed. In this paper, we compare four of the main definitions of cotilting modules that have appeared in the literature. We show that, in the setting of finitely generated right modules, three of these definitions coincide over right Artinian Noetherian algebras, and all four coincide over Artin algebras.

💡 Research Summary

The paper addresses a long‑standing fragmentation in the theory of cotilting modules. While tilting theory enjoys a single, widely accepted definition, its dual—cotilting—has been introduced in several distinct ways. The authors focus on four principal definitions that have appeared in the literature:

1. M‑cotilting (Miyashita) – defined for a right Noetherian ring R with S = End_R C left Noetherian. A module C is M‑cotilting if it is a Wakamatsu tilting module (i.e. gen R ⊆ gen C and cogen C ⊆ cogen R) and both injective dimensions id_R C and id_S C are finite.

2. AR‑cotilting (Auslander–Reiten) – originally formulated for Artin algebras. A finitely generated, self‑orthogonal module C is AR‑cotilting if id_R C ≤ n and there exists an exact sequence 0 → C_n → … → C_0 → D R → 0 with each C_i in add C. Equivalently, R D C is an n‑tilting module.

3. Big cotilting (Angeleri‑Hügel–Coelho) – defined for arbitrary associative rings. A module C is big cotilting if (i) id_R C < ∞, (ii) Ext¹_R(C^I, C) = 0 for every set I, and (iii) there is an injective cogenerator Q and an exact sequence 0 → C_n → … → C_0 → Q → 0 with each C_i in Prod C.

4. AAITY‑cotilting (Aihara–Araya–Iyama–Takahashi–Yoshiwaki) – introduced in 2014 and later extended. A finitely generated, self‑orthogonal C is AAITY‑cotilting if id_R C < ∞ and for every X in K_C (the class of modules N with Ext¹_R(C,N)=0) there exists a short exact sequence 0 → X → C_0 → X_1 → 0 with C_0∈add C and X_1∈K_C. This condition is equivalent to K_C ⊆ cogen C.

The paper first reviews necessary background: self‑orthogonal modules, Wakamatsu tilting, product‑complete modules (those for which Prod C ⊆ Add C), and the homological functors involved. Two technical lemmas (3.6 and 3.7) establish that under coherence assumptions on the endomorphism ring S, an M‑cotilting module is automatically Wakamatsu tilting and that M‑cotilting modules admit Add‑ or add‑resolutions by copies of themselves.

The main results are three theorems:

• Theorem 3.9: (a) If R is right Noetherian, S left Noetherian, and S C is finitely generated, then any big cotilting module is also M‑cotilting. (b) If C is product‑complete, then any M‑cotilting module is big cotilting. The proof uses Lemma 3.6 to turn the Prod‑resolution required for big cotilting into an Add‑resolution required for M‑cotilting, and conversely relies on the equality Prod C = Add C.

• Theorem 3.15: Shows that a product‑complete big cotilting module is AAITY‑cotilting, and conversely any AAITY‑cotilting module is product‑complete big cotilting. The argument hinges on the fact that the class K_C coincides with cogen C when C is product‑complete.

• Theorem 3.18: Establishes the equivalence between M‑cotilting and AAITY‑cotilting under the same coherence hypotheses, essentially by chaining the equivalences from Theorems 3.9 and 3.15.

From these theorems the authors derive two corollaries:

• Corollary 3.20: Over any right Artinian Noetherian algebra, the three notions M‑cotilting, big cotilting, and AAITY‑cotilting coincide for finitely generated right modules.

• Corollary 3.23: Over any Artin algebra, all four notions (including AR‑cotilting) coincide for finitely generated right modules.

The paper also supplies a variety of examples illustrating each definition: hereditary finite‑dimensional algebras yielding M‑cotilting modules, Cohen‑Macaulay orders giving M‑cotilting and AAITY‑cotilting modules, path algebras producing AR‑cotilting modules of arbitrary injective dimension, and injective cogenerators themselves serving as big cotilting modules.

The significance of these results lies in clarifying the landscape of cotilting theory. By pinpointing the exact conditions under which the various definitions agree, the authors provide a unified framework that can be safely used in future work without ambiguity. The product‑complete condition emerges as a pivotal bridge between the “big” and “finite” perspectives, while the coherence of the endomorphism ring ensures the necessary homological control. Consequently, researchers can now select the most convenient definition for a given context, confident that, at least for finitely generated modules over Artin algebras, all definitions describe the same class of objects. This unification is expected to streamline further developments in representation theory, derived equivalences, and the study of t‑structures associated with cotilting modules.