Pseudo-Dualizing Complexes

• Pseudo-dualizing complexes are bounded complexes of bimodules that generalize dualizing complexes by relaxing finite injective or projective/contraflat dimension requirements.
• They mediate equivalences between intermediate pseudo-coderived and pseudo-contraderived categories, bridging classical derived equivalences and advanced homological contexts.
• They underpin applications in noncommutative and relative duality, enabling effective treatment of duality in singular, torsion, and non-Noetherian settings.

A pseudo-dualizing complex is a bounded complex of bimodules or bicomodules (or, more generally, objects in an appropriate abelian or exact category) that generalizes the notion of a dualizing complex by relaxing the requirement of finite injective (or projective/contraflat) dimension while retaining finiteness and self-homothety properties. Pseudo-dualizing complexes mediate equivalences between “intermediate” (pseudo-coderived and pseudo-contraderived) categories, providing a cohesive framework that interpolates between conventional derived equivalences (Morita/Rickard) and homological correspondences in more exotic contexts (e.g., coderived/contraderived categories). They are central to the paper of relative and noncommutative duality, particularly where the classical dualizing machinery is either unavailable or too restrictive.

1. Definition and Fundamental Properties

Let $A$ and $B$ be associative rings, or more generally, let $\mathcal{C}$ and $\mathcal{D}$ be coassociative coalgebras over a field $k$. A pseudo-dualizing complex $L^\bullet$ for the pair $(A,B)$ (respectively, $(\mathcal{C},\mathcal{D})$) is a bounded complex of $A$–$B$-bimodules (or $\mathcal{C}$–$\mathcal{D}$-bicomodules) satisfying two axioms:

1. Finiteness (Strong Finite Presentation/Copresentation):
   • As a left $A$-module complex, $L^\bullet$ is quasi-isomorphic to a bounded above complex of finitely generated projective (or injective, quasi-finitely cogenerated) $A$-modules (or comodules), respectively. Similarly, on the right $B$-side.
   • In coalgebraic or torsion contexts, this is expressed via quasi-finite generation or copresentation conditions on the terms (e.g., each term is quasi-finitely cogenerated for comodules, or resolves all $J$-torsion modules for a commutative Noetherian $S$ and ideal $J\subset S$ (Positselski, 6 Nov 2025, Positselski, 2019)).
2. Homothety Isomorphism:
   • The canonical graded ring homomorphisms in the derived category

   $A \;\longrightarrow\; \mathbf{R}\Hom_{B}\bigl(L^{\bullet},L^{\bullet}\bigr), \qquad B^{\mathrm{op}} \;\longrightarrow\; \mathbf{R}\Hom_{A}\bigl(L^{\bullet},L^{\bullet}\bigr)$

   are isomorphisms (similarly for bicomodules: $$C^* \rightarrow \operatorname{Hom}_{D^b(\mathrm{comod}-\mathcal{D})}(L^\bullet, L^\bullet[*])$$).

A pseudo-dualizing complex is thus distinguished from a dualizing complex by the absence of any finite injective (or projective/contraflat) dimension hypothesis, but the definition retains both the structural finite generation and the crucial self-homothety condition (Positselski, 2017, Positselski, 2019, Positselski, 6 Nov 2025).

2. Canonical Examples

The following are key instances illustrating the generality and flexibility of the pseudo-dualizing concept:

Setting	Complex $L^\bullet$	Becomes Dualizing When
Commutative Noetherian $(A, m, k)$	$C=A$ (trivial complex)	$A$ Gorenstein (Majadas, 2012)
	Classical dualizing complex $D$	Always dualizing (Majadas, 2012)
	$\omega_A[\dim A]$ ($A$ CM, canonical module)	Always dualizing (Majadas, 2012)
	$R\operatorname{Hom}_Q(A,Q)$ for surj. $Q\to A$	$A$ CM and generically Gorenstein
Associative rings $A,B$	2-sided tilting complex (Rickard)	Projective dim finite both sides (Positselski, 2017)
	Classical dualizing complex	Finite inj. dim. both sides
Coalgebras $\mathcal{C},\mathcal{D}$	Complex of bicomodules strongly quasi-finitely copresented	Finite inj./contraflat dim (Positselski, 2019)
Commutative $S$, ideal $J\subset S$	$K^*_\infty(S, J)$ (infinite dual Koszul complex)	Projective/injective dim finite (Positselski, 6 Nov 2025)

These examples showcase that pseudo-dualizing complexes encompass both the classical theory (dualizing complexes, tilting complexes) and constructions adapted to relative, coalgebraic, and torsion-theoretic frameworks (Positselski, 2017, Positselski, 2019, Positselski, 6 Nov 2025).

3. Corresponding Classes and Pseudo-Derived Categories

For a fixed pseudo-dualizing complex $L^\bullet$ of $A$–$B$-bimodules or $\mathcal{C}$–$\mathcal{D}$-bicomodules, one constructs reflexive subcategories encoding the extent to which $L^\bullet$ mediates derived equivalences. Two fundamental classes are:

• Bass Class $\mathsf{E}_L$ / Maximal Class: The subcategory of $A$-modules (or $\mathcal{C}$-comodules, $J$-torsion $S$-modules) $E$ such that $\Ext^i(L^\bullet,E) = 0$ for $i \gg 0$ and with an identity adjunction morphism $L^\bullet \otimes^\mathbf{L}_{B}\mathbf{R}\Hom_A(L^\bullet,E) \xrightarrow{\sim} E$ (Positselski, 2017, Positselski, 2019, Positselski, 6 Nov 2025).
• Auslander Class $\mathsf{F}_L$ / Maximal Class: The subcategory of $B$-modules (or $\mathcal{D}$-contramodules, $J$-contramodule $S$-modules) $F$ with $\Tor^B_i(L^\bullet, F) = 0$ for $i \gg 0$ and an isomorphism $F \xrightarrow{\sim} \mathbf{R}\Hom_A(L^\bullet, L^\bullet \otimes^\mathbf{L}_{B} F)$.

Derived categories $D(\mathsf{E}_L)$ and $D(\mathsf{F}_L)$, called lower pseudo-coderived and pseudo-contraderived categories, respectively, interpolate between the coderived (resp. contraderived) and conventional derived categories: $$D^{\mathrm{co}}(A\text{-mod}) \rightarrow D(\mathsf{E}_L) \rightarrow D(A\text{-mod})$$

$$D^{\mathrm{ctr}}(B\text{-mod}) \rightarrow D(\mathsf{F}_L) \rightarrow D(B\text{-mod})$$

Minimal classes ($\mathcal{E}^{l_2}, \mathcal{F}^{l_2}$) are similarly defined using inductive generation from projectives/injectives and form the upper pseudo-derived categories (Positselski, 2019, Positselski, 6 Nov 2025).

This structure generalizes to the abelian/exact categories of comodules (with quasi-finite copresentability) and torsion modules (with respect to weakly proregular ideals) (Positselski, 6 Nov 2025).

4. Triangulated Equivalences and t-Structures

The primary theoretical result is that a pseudo-dualizing complex $L^\bullet$ determines an exact equivalence of triangulated categories: $L^\bullet \otimes^\mathbf{L}_B (-) : D(\mathsf{F}_L) \overset{\simeq}{\rightleftarrows} D(\mathsf{E}_L) : \mathbf{R}\Hom_A(L^\bullet, -)$ This equivalence holds for all relevant bounded/unbounded/absolute/coderived variants of derived categories, e.g., $D^*$ for $$* \in \{b, +, -, \varnothing, \mathrm{abs}^{\pm}, \mathrm{co}, \mathrm{ctr}\}$$ (Positselski, 2017, Positselski, 2019, Positselski, 6 Nov 2025).

In the comodule/contramodule context, each pseudo-derived category is equipped with a pair of t-structures, with hearts equivalent to the ambient abelian categories (e.g., $\mathcal{C}$-comod and $\mathcal{D}$-contra), allowing interpretation of classical and "relative" duality in homological terms (Positselski, 2019). In the torsion module context, similar t-structures govern the semi-infinite Matlis–Greenlees–May (MGM) duality (Positselski, 6 Nov 2025).

5. Relation to Dualizing, Semidualizing, and Dedualizing Complexes

Pseudo-dualizing complexes strictly generalize dualizing complexes:

• Dualizing complex: A pseudo-dualizing complex with finite injective dimension on both sides.
• Dedualizing complex: Pseudo-dualizing plus finite projective/contraflat dimension.
• Semidualizing complex: In Noetherian local settings, this is a synonymous term for a pseudo-dualizing complex, characterized by $A \rightarrow \mathrm{RHom}_A(C, C)$ an isomorphism, finite amplitude, but not necessarily finite injective dimension (Majadas, 2012).

The crucial distinction is that the absence of injective/projective dimension constraints allows application in contexts (e.g., certain non-Noetherian, coalgebraic, torsion-theoretic settings) where no dualizing (or dedualizing) complex exists. When the extra dimension condition holds, the corresponding Bass/Auslander classes collapse to injectives/projectives, and pseudo-derived equivalence specializes to known results: comodule–contramodule correspondence, Serre–Grothendieck duality, or Rickard's Morita theory (Positselski, 2017, Positselski, 2019).

6. Applications and Relative Constructions

Pseudo-dualizing complexes provide a powerful framework for “intermediate” homological duality:

• Noncommutative Morita/Relative Duality: For bimodule complexes of rings or coalgebras, $L^\bullet$ mediates between Morita/perfect equivalence (projective case) and Serre–Grothendieck duality (injective case) (Positselski, 2017, Positselski, 2019).
• Torsion/Contramodule MGM Duality: In the torsion theory setting for a commutative ring $S$ and weakly proregular ideal $J\subset S$, a pseudo-dualizing complex of $J$-torsion $S$-modules induces a triangulated equivalence between pseudo-coderived and pseudo-contraderived categories ($D(S_J)$ and $D(S^J)$), underpinning a “semi-infinite” MGM-type duality (Positselski, 6 Nov 2025).
• Base Change and Semi-infinite Derived Categories: For a ring homomorphism $f: (R,I) \rightarrow (S,J)$ under suitable coherence and flatness hypotheses (“quotflatness”), a relative pseudo-dualizing complex $$U^\bullet = D^\bullet \otimes^{\mathbf{L}}_R K_\infty^*(S, J)$$ for dualizing $D^\bullet$ of $R$ yields equivalences of semi/coderived and contraderived categories (Positselski, 6 Nov 2025).
• Gorenstein and Local Duality Criteria: In noetherian local settings, under “$h_2$-vanishing” assumptions on a homomorphism $f:(A,m,k) \rightarrow (B,n,\ell)$ and $C$ semidualizing, if $B$ or its residue field $\ell$ is derived $C$-reflexive* over $A$, then $C$ must be dualizing, providing descent and test criteria for Gorensteinness and the existence of dualizing complexes (Majadas, 2012).

7. Structural Significance and Further Developments

Pseudo-dualizing complexes synthesize and extend several threads of contemporary homological algebra:

• They yield a hierarchy of triangulated categories interpolating between standard, coderived, and contraderived categories, and encode the precise boundary between "enough projectives" and "enough injectives" via the dimension properties of $L^\bullet$ (Positselski, 2017, Positselski, 2019).
• In cohomological studies over non-Noetherian schemes, torsion phenomena, or coalgebraic categories, they provide a practical route to duality and derived equivalence when classical tools fail due to lack of dimension bounds (Positselski, 6 Nov 2025).
• The machinery of corresponding maximal, minimal, and abstract classes underpins the explicit construction of pseudo-derived categories and their categorical t-structures, making them amenable to explicit computation and base change analysis (Positselski, 6 Nov 2025, Positselski, 2019).
• In the context of local rings, semidualizing complexes serve as a diagnostic for Gorenstein properties and the possible upgrade to genuine dualizing complexes via homological tests (e.g., $h_2$-vanishing) (Majadas, 2012).

A plausible implication is that further paper of pseudo-dualizing structures will continue to refine the landscape of relative, noncommutative, and singularity-adapted homological dualities, and will support developments in the paper of derived categories over singular spaces and coalgebraic contexts.