Derived Morita Theorem

• Derived Morita theory is a framework for establishing derived equivalences between categories using tilting complexes and bimodules, preserving invariants like centers and Hochschild cohomology.
• It underpins modular representation theory by using splendid equivalences to reflect p-local structural symmetries and support conjectures such as Broué’s Abelian Defect Group Conjecture.
• Alternative methods employing the completion of triangulated categories extend equivalences from perfect complexes to full derived categories, offering conceptual clarity on derived invariants.

Derived Morita theory provides the framework for understanding equivalences between categories of representations—in particular, derived categories—arising from associative algebras and their bimodules, with profound implications in homological algebra, modular representation theory, and beyond. Three central themes underlie the subject: Rickard’s characterization of derived equivalence via tilting complexes, the role of “splendid equivalence” in modular representation theory and Broué’s conjecture, and alternative categorical methods involving completion of triangulated categories. These developments have shaped subsequent generations of research in algebraic and topological contexts.

1. Rickard’s Derived Morita Theory for Algebras

Rickard’s derived Morita theory generalizes classical Morita theory from the module categories of algebras to their derived categories. For two $k$-algebras $A$ and $B$ (typically flat over $k$), the following are equivalent:

• (a) There exists a $k$-linear triangulated equivalence:

$$\mathrm{D}(\mathrm{Mod}(A)) \simeq \mathrm{D}(\mathrm{Mod}(B)).$$

• (b) There exists a complex $M$ of $A$–$B$ bimodules such that the derived functor

$$- \stackrel{L}{\otimes}_A M : \mathrm{D}(\mathrm{Mod}(A)) \to \mathrm{D}(\mathrm{Mod}(B))$$

is a triangulated equivalence.

• (c) There exists a perfect complex $T$ of $B$-modules (i.e., a bounded complex of finitely generated projectives) satisfying:
  1. The derived endomorphism algebra realizes $A$:

  $$A \cong \operatorname{Hom}_{\mathrm{D}(\mathrm{Mod}(B))}(T, T).$$

2. $T$ is tilting:

$$\operatorname{Hom}_{\mathrm{D}(\mathrm{Mod}(B))}(T, T[n]) = 0 \;\;\; \forall n \neq 0.$$

3. $T$ generates the category of perfect complexes in $\mathrm{D}(\mathrm{Mod}(B))$.

This framework ensures that invariants such as centers, Hochschild cohomology, and Grothendieck groups (modulo subtleties) are preserved under such derived equivalences. The construction that realizes a derived equivalence as a derived tensor product with a tilting complex is central to tilting theory and modern homological algebra.

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

In modular representation theory, group algebras over complete discrete valuation rings decompose into blocks, each associated with a defect group $D$. Broué’s Abelian Defect Group Conjecture asserts that if $D$ is abelian, then the bounded derived category of the block $kGe$ (with primitive central idempotent $e$) is derived equivalent to that of its Brauer correspondent $kN_G(D)f$ in the normalizer of $D$:

$$\mathrm{D}^b(\mathrm{mod}(kGe)) \simeq \mathrm{D}^b(\mathrm{mod}(kN_G(D)f)).$$

Rickard introduced the notion of splendid equivalence: a derived equivalence is called splendid if it is induced by a two-sided tilting complex of bimodules whose terms are direct summands of permutation modules induced from the diagonal subgroup $\Delta P \subseteq G \times H$ for a common Sylow $p$-subgroup $P$. Explicitly, for the derived equivalence to be splendid, its terms must be relatively $\Delta P$-projective.

The structural significance is that such an equivalence not only induces equivalence of derived categories but also respects $p$-local structures, leading to perfect isometries between blocks' character groups and hence “isotypies”. In formulas, for the two-sided tilting complex $X$,

$$\operatorname{Hom}_{\mathrm{D}(kG)}(X, X[n]) = 0 \;\;\; \text{for } n \neq 0, \qquad \operatorname{Hom}_{\mathrm{D}(kG)}(X, X) \cong kH.$$

The existence of a splendid equivalence supports the conjectural philosophy that derived equivalences reflect deep $p$-local and character-theoretic symmetries underlying block theory.

3. Alternative Methods: Completion of Triangulated Categories

An alternative proof of Rickard’s Derived Morita Theorem leverages the completion of triangulated categories. Instead of working directly with derived categories of modules, one considers the subcategory of perfect complexes, $\mathcal{K}^b(\mathrm{proj}\,A)$, which is embedded in the derived category of pseudo-coherent modules, $\mathrm{D}^b(\mathrm{pcoh}\,A)$. The inclusion

$$\mathrm{proj}\,A \hookrightarrow \mathrm{pcoh}\,A \hookrightarrow \mathrm{Mod}(A)$$

is fully faithful. To recover the full derived category, one takes the completion of $\mathcal{K}^b(\mathrm{proj}\,A)$ with respect to Cauchy sequences (those whose Hom-sets stabilize), constructing homotopy colimits of such sequences. This “triangulated-completion” viewpoint, developed by Milnor, Neeman, and Krause, shows that any equivalence at the level of perfect complexes extends uniquely to an equivalence of derived categories, explaining the extension property in Rickard's theorem.

This approach provides conceptual clarity regarding how derived Morita equivalences are governed by generating subcategories and their completions.

4. Invariants Preserved and Broader Significance

Rickard’s results guarantee that derived equivalences preserve a wide array of invariants:

• Centers of algebras
• Hochschild cohomology
• Grothendieck groups (up to base change)

Additional consequences in modular representation theory include:

• Induction of perfect isometries and isotypies between corresponding blocks via splendid equivalences
• Stability of $p$-local structural features relevant to the formulation and paper of conjectures such as Broué’s

Moreover, derived Morita theory lies at the intersection of algebraic geometry, representation theory, and topology, as it underpins the structure of derived invariants, derived categories of coherent sheaves, and modular representation categories.

5. Subsequent Developments and Outlook

Rickard’s categorical and structural perspective catalyzed:

• The development of Morita theory for enhanced triangulated categories, dg-categories, and stable $\infty$-categories (notably by Keller in the algebraic context, and by Schwede and Shipley in topology)
• Enhancements and refinements of tilting theory and its application in algebraic K-theory, noncommutative geometry, and modular representation theory
• Modern categorical techniques for extending and classifying derived equivalences beyond algebra—e.g., in spectral categories, quantales, and enriched contexts

The interplay between splendid equivalence, categorical completions, and invariance results continues to inform current efforts in understanding both structural and finer arithmetic properties of modular representations and their associated categories. This theoretical framework forms the backbone for equivalence-based classification and invariant theory in several mathematical domains.