Derived Morita Theory

• Derived Morita theory is defined as an extension of classical Morita theory, characterized by derived equivalences implemented via tilting complexes and derived tensor functors.
• It provides rigorous criteria involving dg-categories and splendid equivalences that connect algebraic geometry, representation theory, and homotopy theory.
• The framework facilitates the study of completions and local-global correspondences in modular representation theory and topological settings.

Morita theory for derived categories concerns the characterization of when two algebras, or more generally two enhanced triangulated categories, have equivalent derived or triangulated categories, extending the scope of classical Morita theory which pertains to module categories. In the derived setting, equivalence is governed not simply by module-level data but by complexes, enhancements, and triangulated structure. The central results of Rickard and subsequent developments provide algebraic, categorical, and geometric criteria for such derived equivalences and have broad implications in algebraic geometry, representation theory, and homotopy theory.

1. Foundations: From Classical to Derived Morita Theory

Classical Morita theory addresses the question: when are the categories of modules over two rings $A$ and $B$ equivalent, $$\mathop{\mathrm{Mod}}(A) \cong \mathop{\mathrm{Mod}}(B)$$? This occurs if and only if there exists a progenerator $P$ for $B$ such that $A \cong \mathrm{End}_B(P)$, with the equivalence realized by the functor $-\otimes_A M$ for an appropriate $A$–$B$ bimodule $M$.

Rickard’s crucial insight was to lift this equivalence to the level of derived categories, specifically, to the (possibly unbounded) derived categories $D(\mathop{\mathrm{Mod}}(A))$ and $D(\mathop{\mathrm{Mod}}(B))$, which are endowed with a triangulated structure reflecting the homological complexity of modules over $A$ and $B$. This shift demands both new criteria for equivalence and new invariants, such as tilting complexes, to witness and construct them (Jasso et al., 8 Sep 2025).

2. Rickard’s Derived Morita Theorem

Rickard’s Derived Morita Theorem establishes the equivalence of the following conditions for two (possibly finite-dimensional) algebras $A$ and $B$ over a commutative ring $k$:

• There is a $k$-linear triangulated equivalence $$\mathcal{F}\colon D(\mathop{\mathrm{Mod}}(A)) \xrightarrow{\sim} D(\mathop{\mathrm{Mod}}(B))$$.
• There exists a complex of $A$–$B$-bimodules $M$ such that the derived tensor product functor $- \otimes_A^{\mathbb{L}} M$ is an equivalence.
• There exists a perfect complex $T \in D(\mathop{\mathrm{Mod}}(B))$ such that:
  • $$A \cong \mathrm{Hom}_{D(\mathop{\mathrm{Mod}}(B))}(T, T)$$ (isomorphism of rings in the derived category),
  • $$\mathrm{Hom}_{D(\mathop{\mathrm{Mod}}(B))}(T, T[n]) = 0$$ for $n \neq 0$,
  • $T$ generates the category of perfect complexes, $K^b(\mathop{\mathrm{proj}}(B))$.

This is encapsulated by the existence of a tilting complex $T$ in $D(\mathop{\mathrm{Mod}}(B))$ such that $$A \cong \mathrm{End}_{D(\mathop{\mathrm{Mod}}(B))}(T)$$ and $T$ has no higher self-extensions, fully recovering the derived category $D(\mathop{\mathrm{Mod}}(A))$ as the category generated by $T$. These results utilize triangulated functoriality, the formalism of total derived functors, and the necessity of constructing suitable projective (or injective) resolutions in the unbounded context (Jasso et al., 8 Sep 2025).

3. Splendid Equivalences and Broué’s Abelian Defect Group Conjecture

In modular representation theory, particularly for finite groups $G$ over a field $k$ of characteristic $p$, blocks of the group algebra $kG$ are basic objects of study. Each block corresponds to a primitive central idempotent and has associated defect groups $D$ measuring its complexity. Broué conjectured that for a block $kG.e$ with abelian defect group $D$, there is a derived equivalence

$$D^b(\mathop{\mathrm{mod}}(kG.e)) \simeq D^b(\mathop{\mathrm{mod}}(kN_G(D).f)),$$

where $N_G(D)$ is the normalizer of $D$ and $f$ is the corresponding Brauer correspondent.

Rickard introduced the notion of splendid equivalence, a special kind of derived equivalence realized by a tilting complex $X$ built from relative permutation modules (i.e., those induced from subgroups like the diagonal $\Delta P$) and thus sensitive to the $p$-local structure. Formally, $X$ is a two-sided tilting complex satisfying $$(A \cong \mathrm{End}_B(X), B \cong \mathrm{End}_A(X))$$ in the homotopy category, with additional structural compatibility. The existence of a splendid tilting complex not only induces a derived equivalence between blocks but guarantees an isotypy between their character groups, via families of perfect isometries that respect scalar products and local subgroups (Jasso et al., 8 Sep 2025).

This framework supplies a structural explanation for Broué’s abelian defect group conjecture: a splendid equivalence between principal blocks of $G$ and $H$ induces an isotypy on ordinary characters, affirming the conjectured connection between local block theory and global character theory.

4. Completion Techniques in Triangulated Categories

Rickard’s methodology permitted alternative proofs and new perspectives via completions of triangulated categories. Given the subcategories:

• $K^b(\mathop{\mathrm{proj}}(A))$: bounded homotopy category of finitely generated projectives,
• $D^b(\mathop{\mathrm{pcoh}}(A))$: bounded derived category of pseudo-coherent modules (those with “nice” finite-type resolutions),

one can obtain $D^b(\mathop{\mathrm{pcoh}}(A))$ as a completion of $K^b(\mathop{\mathrm{proj}}(A))$ using homotopy colimits of Cauchy sequences—sequences in the triangulated category which stabilize appropriately. This perspective, paralleling Milnor’s and Neeman’s developments, allows extension of $k$-linear equivalences of triangulated categories from perfect to pseudo-coherent complexes, and ultimately to the full derived category. Compact objects here are precisely those preserved under coproducts by the Hom functor, reflecting general features of well-generated triangulated categories (Jasso et al., 8 Sep 2025).

5. Connections to dg-Categories and Topological Analogues

The scope of derived Morita theory extends to differential graded (dg) categories, in which the derived Eilenberg–Watts theorem asserts that any equivalence of dg derived categories is realized as a derived tensor functor with a dg bimodule. Explicitly, for small dg-categories $A$ and $B$,

$$\mathop{\mathrm{dgD}}(A \otimes^{\mathbb{L}} B) \simeq \mathop{\mathrm{RHom}}^c(\mathop{\mathrm{dgD}}(A), \mathop{\mathrm{dgD}}(B)),$$

showing that fully faithful exact functors between compactly generated triangulated categories are “of standard type” (via tensoring with a dg-bimodule). This observation underpins the geometric applications, such as Fourier–Mukai transforms in algebraic geometry.

Topological analogues are realized in the work of Schwede and Shipley, who showed that the stable homotopy category of spectra admits a presentation as the derived category of a small spectral category—what may be termed “topological Morita theory”. These results generalize the algebraic version to stable homotopy theory and spectral algebra, demonstrating that the triangulated structure and homotopical data suffice to reconstruct and recognize derived equivalences in highly structured topological settings (Jasso et al., 8 Sep 2025).

6. Impact and Subsequent Developments

Rickard’s derived Morita theory catalyzed a series of advances in the understanding and classification of triangulated categories—algebraic (via Keller’s dg enhancements) and topological (via Schwede and Shipley’s work). The recognition that derived equivalences can be characterized precisely by tilting complexes and that splendid equivalences encode local data supporting conjectures of representation theory provided conceptual advances with ramifications in modular representation theory, algebraic geometry (notably in homological mirror symmetry), and the theory of enhancements of triangulated categories.

Significant subsequent work includes categorical generalizations (toward stable $\infty$-categories, motivic invariants, and noncommutative motives) and the construction of explicit equivalences in examples arising from singularities, algebraic stacks, or geometric representation theory. Completion techniques, as developed by Neeman and others, continue to inform the structure theory of triangulated categories, their compact objects, and the notion of generation in the context of large or “big” derived categories.

Key Concepts	Algebraic Formulation	Significance
Classical Morita theory	$\mathrm{Mod}(A)\cong \mathrm{Mod}(B)$ if $\exists$ progenerator $P$, $A\cong \mathrm{End}_B(P)$	Equivalence at module level
Rickard's derived Morita	$D(\mathrm{Mod}(A))\cong D(\mathrm{Mod}(B))$ via $T$ with $\mathrm{End}_{D(B)}(T)\cong A$, $T$ tilting	Equivalence at derived/triangulated level
Splendid equivalence	Derived equivalence realized by permutation-builted tilting complex	Encodes local data, crucial in modular theory

7. Summary

The theory of Morita equivalence for derived categories, as established by Rickard, provides necessary and sufficient conditions—expressed in terms of tilting complexes and derived tensor functors—for two algebras (and their generalizations to dg or spectral categories) to have equivalent derived (triangulated) categories. The introduction of splendid equivalence and its implications for block theory and local–global conjectures, as well as the formal approaches using completions and compact objects, form the foundational apparatus enabling both abstract classification and concrete computation in modern homological algebra and related areas. These results continue to guide active research in algebra, topology, and geometry (Jasso et al., 8 Sep 2025).