nLab
Serre functor

Serre functor

Idea

Serre functors were introduced by Aleksei Bondal and Mikhail Kapranov to study admissible subcategories? of triangulated categories. A Serre functor on a triangulated category 𝒜\mathcal{A} is an exact functor such that for any objects AA and BB, Hom(A,B)Hom(B,S(A)) *\Hom(A,B) \simeq \Hom(B, S(A))^*. It does not always exist, but when it does it is unique up to graded natural isomorphism.

The Serre functor is a powerful tool for working with the derived category of coherent sheaves on a variety.

Definition

In the original paper, the following definition was given.

Definition

Let 𝒜\mathcal{A} be a kk-linear triangulated category with finite-dimensional Hom‘s and kk algebraically closed. A Serre functor S:𝒜𝒜S : \mathcal{A} \to \mathcal{A} is an additive equivalence that commutes with the translation functor, with bi-functorial isomorphisms ϕ A,B:Hom 𝒜(A,B)Hom 𝒜(B,S(A)) *\phi_{A,B} : \Hom_\mathcal{A}(A,B) \stackrel{\sim}{\to} \Hom_{\mathcal{A}}(B,S(A))^* for any objects AA and BB, such that the composite

(ϕ S(A),S(B) 1) *ϕ A,B:Hom(A,B)Hom(B,S(A)) *Hom(S(A),S(B)) (\phi^{-1}_{S(A),S(B)})^* \circ \phi_{A,B} : \Hom(A,B) \to \Hom(B,S(A))^* \to \Hom(S(A), S(B))

coincides with the isomorphism induced by SS.

In fact, the last commutativity condition can be deduced from just the bi-functoriality of ϕ A,B\phi_{A,B}, and commutativity with the translation functor also follows from a proposition below. Hence, the following definition is seen in later papers.

Definition

Let 𝒜\mathcal{A} be a kk-linear category with finite-dimensional Hom‘s and kk an arbitrary field. A Serre functor S:𝒜𝒜S : \mathcal{A} \to \mathcal{A} is an additive equivalence with bi-functorial isomorphisms ϕ A,B:Hom 𝒜(A,B)Hom 𝒜(B,S(A)) *\phi_{A,B} : \Hom_\mathcal{A}(A,B) \stackrel{\sim}{\to} \Hom_{\mathcal{A}}(B,S(A))^* for any objects AA and BB.

Of course, formally the definition could be used in categories enriched over a symmetric monoidal category with a sufficiently nice involution.

Examples

Example

In the derived category of finite-dimensional vector spaces over kk, the identity functor is a Serre functor.

Example

In the derived category of coherent sheaves on a smooth projective variety XX, the functor (ω X)[n](\cdot \otimes \omega_X)[n] is a Serre functor, in view of Serre-Grothendieck duality?, where ω X\omega_X is the canonical sheaf and nn is the dimension of XX.

Properties

Proposition

Any autoequivalence F:𝒜𝒜F : \mathcal{A} \to \mathcal{A} commutes with a Serre functor: there is a natural graded isomorphism of functors FSSFF \circ S \stackrel{\sim}{\to} S \circ F.

Proposition

Any Serre functor in a graded category? is graded?.

Proposition

Any Serre functor in a triangulated category is exact? (i.e. distinguished triangles are mapped to distinguished triangles).

Proposition

Any two Serre functors are connected by a canonical graded functorial isomorphism that commutes with the isomorphisms ϕ A,B\phi_{A,B} in the definition of the Serre functor.

References

The original paper and English translation:

The following paper gives the corrected definition and also demonstrates the utility of the Serre functor as a tool for working with the derived category of coherent sheaves on a variety (c.f. Bondal-Orlov reconstruction theorem):

See also: