Semidualizing Bimodule in Homological Algebra

• Semidualizing bimodule is a concept in relative homological algebra that generalizes dualizing complexes by providing a framework for studying projective, injective, and Gorenstein dimensions.
• It establishes categorical equivalences via adjoint functors between Auslander and Bass classes, thereby refining homological invariants and duality results.
• Recent generalizations to special semidualizing bimodules extend the theory to higher syzygy categories and (n,d)-injectivity/flatness, enhancing the scope of relative dimension theory.

A semidualizing bimodule is a central concept in relative homological algebra, generalizing both classical dualizing complexes and the identity bimodule by providing a flexible framework for studying projective, injective, and Gorenstein dimensions relative to a particular module. Its axiomatic and categorical characterizations support the construction of Foxby equivalences and a robust relative dimension theory in commutative and noncommutative settings. Recent generalizations to special semidualizing bimodules enable the parametrization of higher syzygy categories and $(n,d)$-injectivity/flattness in both classical and novel contexts.

1. Definition and Axiomatic Properties

Let $R$ and $S$ be (not necessarily commutative) rings, and let ${}_S C_R$ be an $(S,R)$-bimodule. The bimodule $C$ is semidualizing if and only if the following four conditions hold:

1. Projectivity: Both as a left $S$-module and as a right $R$-module, $C$ is degreewise finitely generated and projective (admits a finite projective resolution as a module over each end).
2. Homothety Isomorphisms:
   • The left homothety map $${}_S \gamma: S \xrightarrow{\cong} \operatorname{Hom}_{R^{op}}(C,C)$$ is a ring isomorphism, $s \mapsto (c \mapsto sc)$.
   • The right homothety map $$\gamma_R: R \xrightarrow{\cong} \operatorname{Hom}_S(C,C)$$ is a ring isomorphism, $r \mapsto (c \mapsto cr)$.
3. Vanishing Ext:
   • $\operatorname{Ext}^i_S(C,C)=0$ and $\operatorname{Ext}^i_{R^{op}}(C,C)=0$ for all $i\geq 1$.

In the commutative case, these reduce to White's semidualizing modules; for $C \cong R$ (as bimodules), the semidualizing data is trivial (Zhao et al., 2013, Amini et al., 6 Nov 2025, Salimi et al., 2012).

2. Categorical and Reflexivity Characterization

Semidualizing bimodules admit several categorical formulations:

• The pair of functors $(\operatorname{Hom}_R(C,-), C \otimes_R -)$ induces mutually inverse equivalences between the Auslander class $\mathcal{A}_C(R)$ (modules that behave "projectively" relative to $C$) and the Bass class $\mathcal{B}_C(R)$ (modules that behave "injectively" relative to $C$).
• $C$ is a reflexive generator in the sense that for all finitely generated $R$-modules $M$, the biduality map

$$M \longrightarrow \operatorname{Hom}_{R^{op}}\left( \operatorname{Hom}_R(M,C), C\right)$$

is an isomorphism in the appropriate subcategory; this is equivalent to the vanishing of the higher Ext groups of $C$ against itself (Zhao et al., 2013).

• For bimodules $C$ that are faithfully semidualizing, the above functors reflect and detect zero modules: $\operatorname{Hom}_S(C,N)=0 \implies N=0$ for all $S$-modules and similarly for right-$R$-modules (Amini et al., 6 Nov 2025).

This structure underpins the Foxby equivalence, an essential duality for Gorenstein-type homological theory.

3. Construction and Examples of (Special) Semidualizing Bimodules

For any $d\ge 0$, a $d$-semidualizing bimodule is obtained by taking the $(d-1)$th syzygy module $K_{d-1} = \ker(F_{d-1} \to F_{d-2})$ in any degreewise-superfinite projective resolution

$$0 \longrightarrow F_d \longrightarrow \dotsb \longrightarrow F_0 \longrightarrow C \longrightarrow 0$$

of $C$ (with $F_i$ finitely generated projective), and requiring $K_{d-1}$ to itself be semidualizing. The resulting $K_{d-1}$ is called a special semidualizing bimodule (Amini et al., 6 Nov 2025).

Canonical Examples:

• For $R=S=C$, $C$ is trivially semidualizing and all syzygies are isomorphic to $R$.
• Over a Gorenstein Artin algebra $R$, taking $C$ as a direct sum of indecomposable summands in the minimal injective resolution of $R$ provides nontrivial semidualizing bimodules with significant homological impact (Zhao et al., 2013).
• In the context of $(2,0)$-rings (where every finitely $2$-presented module is projective, but not all are projective for $n=1$), taking $C=R$ yields nontrivial special semidualizing modules and exhibits nontrivial instances of the $(n,d)$-relative theory (Amini et al., 6 Nov 2025).

4. Relative Homological Theory and $(n,d)$-Classes

The presence of a (special) semidualizing bimodule enables the definition and study of homological invariants relative to $C$:

• $C$-projective modules: $$\mathcal{P}_C = \{ C \otimes_R P \mid P \text{ projective} \}$$.
• $C$-injective modules: $$\mathcal{I}_C = \{ \operatorname{Hom}_R(C, I) \mid I \text{ injective} \}$$.
• $C$-flat modules: For commutative $R$, $$\mathcal{F}_C = \{ F \otimes_R C \mid F \text{ flat} \}$$ (Salimi et al., 2012).

Specializations to $(n,d)$-injective and $(n,d)$-flat modules utilize higher syzygy special semidualizing bimodules $K_{d-1}$:

• An $R$-module is $K_{d-1}$-$(n,d)$-injective if it is isomorphic to $\operatorname{Hom}_S(K_{d-1}, I)$ for $(n,d)$-injective $S$-modules $I$ (i.e., $\operatorname{Ext}_S^{d+1}(U,I)=0$ for all finitely $n$-presented $U$).
• Dually, $S$-modules of the form $K_{d-1} \otimes_R F$ with $F$ $(n,d)$-flat (Amini et al., 6 Nov 2025).

This hierarchy generalizes the $C$-injective, $C$-flat, and $C$-$FP_n$ (coherently presented) categories, enabling the refinement of relative dimension-theoretic results.

5. Relative Tor, Global Dimensions, and Foxby Equivalence

Relative homological algebra with respect to $C$ unlocks the construction of proper $C$-projective and $C$-flat resolutions, leading to the definition of relative $\operatorname{Tor}$ bifunctors and the identification: $$\operatorname{Tor}_i^{\mathcal{P}_C M}(M,N) \cong \operatorname{Tor}_i^{\mathcal{F}_C M}(M,N) \cong \operatorname{Tor}_i^R(\operatorname{Hom}_R(C,M), C\otimes_R N)$$ for all $i\ge 0$, which mirrors the classical balance property (Salimi et al., 2012). However, the expected symmetries—e.g., commutativity and compatibility with classical Tor—fail unless $C$ is trivial: $C\cong R$ (Salimi et al., 2012).

Relative Dimensions:

• The global $G_C$-projective dimension of $R$ is defined as $\gldim_{G_C-\mathrm{proj}}(R) = \sup \{ G_C\text{-pd}_R(M) \mid M \textrm{ an }R\text{-module}\}$.
• The global $G_C$-injective dimension $\gldim_{G_C-\mathrm{inj}}(R)$ is defined dually.
• These dimensions coincide: $\gldim_{G_C-\mathrm{proj}}(R) = \gldim_{G_C-\mathrm{inj}}(R)$, and the common value is called the $C$-Gorenstein global dimension, denoted $\mathrm{Ggldim}_C(R)$ (Zhao et al., 2013).

Foxby equivalence, mediated by $(K_{d-1})$, yields a string of mutually inverse equivalences between subcategories of module categories over $R$ and $S$, intertwining $(n,d)$-injective, $(n,d)$-flat, Auslander, and Bass classes: $$K_{d-1} \otimes_R - \dashv \operatorname{Hom}_S(K_{d-1},-) \text{ on suitable subcategories}$$ Dimension-shifting identities persist through this equivalence, directly relating relative injective/flat dimensions over $R$ and $S$ (Amini et al., 6 Nov 2025).

6. Strongly $G_C$-Projective/Injective Modules and Structural Theorems

An $R$-module $M$ is strongly $G_C$-projective if it is a "stable image" in a $\operatorname{Hom}_R(-, \mathcal{P}_C)$-exact acyclic complex built from direct sums of $C \oplus R$ (Zhao et al., 2013). The structure theorem asserts that every $G_C$-projective module is a direct summand of such an object; dually for injectives.

These classes enjoy several closure properties:

• For an $n$-coherent ring $S$ and faithfully special semidualizing bimodule $K_{d-1}$, the categories of $K_{d-1}$-$(n,d)$-injective and $K_{d-1}$-$(n,d)$-flat modules of bounded relative dimension are closed under extensions, kernels of epimorphisms, and cokernels of monomorphisms (Amini et al., 6 Nov 2025).
• Duality pair results ensure these categories are covering and preenveloping, consistent with modern approximative module theory.

The relative dimension theory, periodic resolutions, and two-term exact sequences yield concrete criteria, paralleling classical characterizations of projective and injective modules.

7. Implications, Counterexamples, and Limitations

The presence of a semidualizing bimodule $C$ fundamentally alters the structure of relative homological invariants. While for $C \cong R$ all classical properties recover, nontrivial $C$-relative theory exposes the lack of classical symmetries—such as commutativity and balance for $\operatorname{Tor}$—and the possibility of nontrivial enlargement of relative projective/injective classes over rings of infinite classical global dimension (Salimi et al., 2012, Zhao et al., 2013).

The necessity of the homothety isomorphisms and Ext vanishing is evident in the functional properties of the adjoint pairs $$(K_{d-1} \otimes_R -, \operatorname{Hom}_S(K_{d-1},-))$$ and in the structural theorems underpinning Foxby equivalence and duality results (Amini et al., 6 Nov 2025).

The framework of semidualizing (and special semidualizing) bimodules yields a robust unifying axis along which projective, injective, and Gorenstein relative homological dimensions, as well as categorical equivalences, can be explored in both the commutative and noncommutative settings.