Tilting Right R-Module

• Tilting right R-modules are modules that meet specific axioms (T1-T3), yielding equivalences between module and derived categories through torsion pairs.
• They employ Ext-vanishing and generation conditions to construct tilting classes that induce t-structures and recollement phenomena in homological algebra.
• Applications span finite to infinite-dimensional algebras, with explicit classifications in tame hereditary and Auslander algebras supporting cluster-tilting theory and combinatorial models.

A tilting right $R$-module is a module $T$ over a ring $R$ satisfying specific homological and generation conditions that establish deep connections between module categories, derived categories, and torsion-theoretic data. Tilting modules generalize the notion of progenerators to broader settings, giving rise to equivalences of derived categories, hearts of $t$-structures, and recollement phenomena, and play a pervasive role in the homological paper of both finite and infinite dimensional algebras, as well as in relative and higher homological algebra.

1. Definitions and Fundamental Properties

A (classical) $n$-tilting right $R$-module $T$ is defined by three axioms—see, for example, (Bazzoni, 2016, Bazzoni et al., 2017, Moradifar et al., 2017):

• (T1) Projective dimension: $\operatorname{pd}_R(T) \leq n$.
• (T2) Ext-vanishing: $\operatorname{Ext}_R^i(T, T^{(\lambda)}) = 0$ for all $i > 0$ and any cardinal $\lambda$.
• (T3) Generation: there is an exact sequence

$$0 \longrightarrow R \longrightarrow T_0 \longrightarrow \cdots \longrightarrow T_n \longrightarrow 0,$$

where each $T_i \in \operatorname{Add}(T)$.

The tilting class associated to $T$ is

$$\operatorname{Gen} T = T^{\perp} = \{ M \in \operatorname{Mod}\text{-}R \mid \operatorname{Ext}_R^{>0}(T, M) = 0 \}.$$

This class is a torsion class in $\operatorname{Mod}\text{-}R$, and (under (T1)-(T3)) the torsion pair $(\operatorname{Gen} T, \mathcal F)$ is often called the tilting torsion pair.

Non-classical tilting modules may drop the requirement that $T$ is finitely presented or focus on modules of arbitrary cardinalities, see (Mattiello, 2016, Bazzoni et al., 2017).

2. Tilting Theory and Torsion Pairs

Tilting modules underpin a close relationship with torsion-theoretic data:

• For $T$ a tilting right $R$-module, $(\operatorname{Gen} T,T^{\perp})$ is a torsion pair, and conversely, many tilting torsion pairs arise from such modules (Bazzoni, 2016, Bazzoni et al., 2017).
  • In abelian or Grothendieck categories, tilting objects generalize small projective generators (1011.5345).
  • The heart $\mathcal{H}_t$ of the $t$-structure induced by a tilting torsion pair is often abelian and, under suitable conditions, a Grothendieck category (Bazzoni, 2016, Bazzoni et al., 2017).

The question of when $\mathcal{H}_t$ is equivalent to a module category is governed by the “classical” or “pure projective” nature of $T$:

• If $T$ is tilting-equivalent to a finitely presented module (i.e., is classical), then $\mathcal{H}_t$ is equivalent to $\operatorname{Mod}$–$S$ for some ring $S$ (Bazzoni et al., 2017).
• The heart $\mathcal{H}_t$ is a module category if and only if $T$ is classical (over a Krull–Schmidt or commutative ring, every pure projective 1-tilting module is classical or projective) (Bazzoni et al., 2017).

3. Classification and Structure in Specific Contexts

Tame Hereditary Algebras

Infinite dimensional tilting modules over a tame hereditary algebra $R$ admit a complete classification (1007.4233):

• Large tilting modules are of the form $T = R_U \oplus R_U/R$, where $R_U$ is the universal localization at a union $U$ of tubes and $R_U/R$ decomposes into Pr\"ufer modules.
• The structure of the torsion part of such a tilting module $T(Y, A)$ is uniquely determined by finite-dimensional branch modules $Y$ and a collection of Pr\"ufer modules indexed by a set $A$ of quasi-simple modules.
• Equivalence of tilting modules is detected completely by their torsion part up to specified multiplicities.

Auslander and Radical Square Zero Algebras

The classification of tilting modules over Auslander algebras of radical square zero Nakayama or Dynkin algebras is combinatorial and explicit (Zhang, 2020, Chen et al., 2022):

• For the Auslander algebra of a radical square zero algebra of type $A_m$, the number of tilting right modules is $2^{m}-1$; for types $D_m, E_m$ it is $2^{m-3}\times 14$.
• Each indecomposable summand of a tilting module is either projective or simple.
• Such results connect to cluster-tilting theory and combinatorial models.

4. Connections with Derived and Triangulated Categories

Tilting modules induce $t$-structures on $D(R)$:

• The aisle and co-aisle are determined by vanishing of $D(R)$-Homs from suitable (co)homological shifts of $T$.
• For an $n$-tilting module, the heart of the induced $t$-structure is abelian and is a Grothendieck category if and only if $T$ is pure projective (Bazzoni, 2016).
• The pair $(\mathcal{D},\mathcal{T})$ of the natural $t$-structure and $T$-generated $t$-structure is right filterable (Mattiello, 2016); the heart is derived equivalent to $\operatorname{Mod}$–$R$, thus facilitating translation between module and derived categories.

Extensions to extended module categories and silting theory broaden the scope to higher homological algebra:

• There are bijections between $(m+1)$-term silting complexes, $\tau_{[m]}$-tilting pairs, and functorially finite $s$-torsion pairs in $m$-extended module categories (Zhou, 23 Nov 2024).

5. Variants: Silting, Support $\tau$-Tilting, and Gorenstein Tilting

Silting modules generalize tilting modules by relaxing projective dimension constraints (Gao et al., 2020):

• For a triangular matrix ring $$R = \begin{pmatrix} A & 0 \ M & B \end{pmatrix}$$, a right $R$-module $(X,0)\oplus (Y\otimes_B M,Y)$ is silting if and only if $X$ and $Y$ are silting and $Y\otimes_B M$ is generated by $X$.
• In the finite-dimensional case, support $\tau$-tilting modules obey additional hom-vanishing conditions.

Gorenstein $\pi[T]$-projective modules (relative to a tilting module $T$) are defined via resolutions by $T[T]$-projectives; over $T$-cocoherent rings, all modules are Gorenstein $\pi[T]$-projective if and only if all $T$-injectives are $T[T]$-projective (Amini, 2019).

Support $\tau$-tilting subcategories have descriptions in terms of finendo quasitilting modules, with categorical correspondences to certain cotorsion torsion triples, extending the classical tilting theory (Asadollahi et al., 2022). Every silting module gives rise to a support $\tau$-tilting subcategory.

6. Base Change, Recollement, and Invariance Results

Base Change

Tilting behavior under localization and quotient by a central nonzerodivisor $x$ is tracked by:

• $T$ tilting over $R$ implies $T_x$ tilting over $R_x$ and $T/xT$ tilting over $R/xR$ (Moradifar et al., 2017).
• For classical tilting modules, a strong converse: $T$ is classical tilting if and only if both $T_x$ and $T/xT$ are classical tilting.
• Applications include analyzing tilting over Rees rings and associated graded rings.

Recollement

Recollement of module categories by artin algebras allows ‘gluing’ of tilting modules:

• Given a recollement $$(\operatorname{mod} \Lambda',\operatorname{mod} \Lambda,\operatorname{mod} \Lambda'')$$, if $T'$, $T''$ are tilting over $\Lambda',\Lambda''$, then $T = j_{!}(T'') \oplus M$ is tilting over $\Lambda$ for suitable $M$, ensuring the glued torsion pair is tilting (Ma et al., 2020).

Invariance of Homological Conjectures

For a tilting module $T$ over a Noetherian algebra $R$, $S = \operatorname{End}_R(T)$ inherits the Auslander--Reiten conjecture if $R$ satisfies it (Divaani-Aazar et al., 28 Jul 2025). The dual holds for cotilting modules under co-Noetherianity and artinianity assumptions. The proof relies on dualities between subcategories of $R$- and $S$-modules induced by $T$ and $C$, reducing invariance to preservation of self-orthogonality and projectivity across equivalence.

7. Specialized and Higher Frameworks

Tilting theory has been developed in truncated categories, in highest weight categories, and for quantum groups, with the same core Ext-vanishing and filtration approaches (Bennett et al., 2013, Andersen, 2019). Cellular structures, double centralizer properties, and Schur–Weyl dualities emerge naturally when tilting modules are faithful and their endomorphism algebras recover the original algebra (Hu et al., 2020).

In summary, the tilting right $R$-module unifies and enhances the structure theory of modules via homological and categorical methods. The tilting module framework governs the formation of torsion pairs, hearts of $t$-structures, supports recollements, and guides equivalences between module categories or derived categories, with stability under base changes and connections to deep open conjectures in representation theory. These results provide both concrete classification results and a robust abstract apparatus for further generalizations, such as silting, cosilting, and higher tilting structures.