Questions: Derived Equivalences of Categories

Question 1

Rings R and S are Morita equivalent. What can you immediately conclude about their derived categories?

• A. Their derived categories are NOT equivalent, because Morita equivalence is stronger than derived equivalence and implies ordinary, not derived, structure
• B. Their derived categories are equivalent as triangulated categories, because an ordinary equivalence of module categories induces a derived equivalence
• C. Their derived categories may or may not be equivalent — Morita equivalence and derived equivalence are unrelated notions
• D. Their derived categories are equivalent, but only if R and S are commutative rings

Answer: (B)

Morita equivalence means the module categories Mod-R and Mod-S are equivalent as ordinary categories. An ordinary equivalence of abelian categories induces an equivalence of their derived categories as triangulated categories, because the derived category construction is functorial and preserves quasi-isomorphisms under exact functors. So Morita equivalence implies derived equivalence. The converse fails: two rings can be derived equivalent without being Morita equivalent — derived equivalence is strictly weaker.

Question 2

Two algebras A and B are derived equivalent (but not Morita equivalent). A mathematician computes their Hochschild homology groups. What should she expect?

• A. HH_n(A) ≇ HH_n(B) for most n, because Hochschild homology requires the ordinary categorical structure lost in the derived category
• B. HH_n(A) ≅ HH_n(B) for all n, because Hochschild homology is an invariant of the derived category and is preserved by derived equivalence
• C. HH_0(A) ≅ HH_0(B) but higher Hochschild homology groups may differ
• D. The comparison is undefined unless A and B have the same number of simple modules

Answer: (B)

Hochschild homology is a derived invariant — it can be computed directly from the derived category and is therefore preserved by any derived equivalence. This is one of the key practical consequences of the theory: algebras with different presentations, different numbers of simple modules, and very different ordinary categorical structures can be 'homologically indistinguishable.' The preservation of Hochschild homology (and K-theory, Hochschild cohomology, etc.) makes derived equivalence a useful coarsening — it identifies a meaningful notion of 'same homological type.'

Question 3

If two categories A and B are equivalent as ordinary categories, they are automatically derived equivalent.

• T. True
• F. False

Answer: True

Ordinary equivalence is stronger than derived equivalence — it implies derived equivalence but not vice versa. If F: A → B is an ordinary equivalence of abelian categories, the induced functor on derived categories D(A) → D(B) is a triangulated equivalence. This is because exact functors preserve quasi-isomorphisms, and an ordinary equivalence of abelian categories is exact. So the implication is: ordinary equivalence ⟹ derived equivalence; but derived equivalence ⇏ ordinary equivalence.

Question 4

Two rings can be derived equivalent even if their module categories are not equivalent as ordinary categories.

• T. True
• F. False

Answer: True

This is the key feature that makes derived equivalence genuinely useful beyond Morita theory. A tilting complex T in D^b(Mod-R) can induce an equivalence D^b(Mod-R) ≅ D^b(Mod-S) where S = End(T), even when Mod-R and Mod-S are not equivalent as ordinary categories. In representation theory, this means two algebras with different numbers of simple modules (and hence non-isomorphic module categories) can still be 'homologically the same.' Derived equivalence is strictly weaker than Morita equivalence, capturing a coarser notion of sameness at the homological level.

Question 5

Explain why derived equivalence is described as 'coarser' than ordinary equivalence, and give an example of what is preserved and what may not be preserved when two categories are derived equivalent but not ordinarily equivalent.

Model answer: Derived equivalence identifies more pairs of categories as 'the same' than ordinary equivalence does — it imposes weaker requirements. Two categories are derived equivalent if their derived categories (which retain homological information but lose some fine-grained categorical structure) are equivalent as triangulated categories; they might differ significantly at the level of their ordinary categorical structure. What is preserved: Hochschild homology, Hochschild cohomology, K-theory, and other homological invariants that can be computed from the derived category. What may not be preserved: the number of simple modules (which can differ between derived equivalent algebras), and other structure visible only at the level of individual objects rather than homological complexes.

The coarseness is the whole point: working at the level of derived categories creates equivalences between algebras that look different superficially but share the same deep homological structure. This is useful in representation theory (transfer results between algebras) and algebraic geometry (derived equivalences between coherent sheaf categories on different varieties reveal surprising geometric relationships, as in mirror symmetry).