nLab Serre functor

Idea

A Serre functor on a triangulated category $\mathcal{A}$ is an exact functor such that for any pair of objects there is a natural isomorphism $\Hom(A,B) \simeq \Hom\big(B, S(A)\big)^*$ from their hom-object to the linear dual of their reverse hom-object, “twisted” by the Serre functor.

A Serre functor does not always exist, but when it does then it is unique up to graded natural isomorphism.

The notion of Serre functors was introduced by Bondal & Kapranov 1989 to study admissible subcategories? of triangulated categories.

Serre functors have become 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 $\mathcal{A}$ be a $k$-linear triangulated category with finite-dimensional Hom‘s and $k$ algebraically closed. A Serre functor $S : \mathcal{A} \to \mathcal{A}$ is an additive equivalence that commutes with the translation functor, with natural (in both variables) isomorphisms $\phi_{A,B} \colon \Hom_\mathcal{A}(A,B) \stackrel{\sim}{\to} \Hom_{\mathcal{A}}(B,S(A))^*$ for any objects $A$ and $B$, such that the composite

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

coincides with the isomorphism induced by $S$.

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

Definition

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

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 $k$, the identity functor is a Serre functor.

Example

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

Properties

Proposition

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

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 $\phi_{A,B}$ in the definition of the Serre functor.

References

The original paper and English translation:

• А. И. Бондал, М. М. Капранов, Представимые функторы, функторы Серра и пе­рестройки, Изв. АН СССР. Сер. матем., 53:6 (1989), 1183–1205 pdf

• Alexei I. Bondal, Mikhail M. Kapranov, Representable functors, Serre functors, and mutations, Mathematics of the USSR-Izvestiya 35 3 (1990) 519-541 [doi:10.1070/IM1990v035n03ABEH000716 pdf]

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):

• Aleksei Bondal, Dmitri Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Mathematica 125 (03), 327-344.

See also:

• Oleksandr Manzyk, A-infinity-bimodules and Serre A-infinity-functors, dissertation pdf, djvu; Serre $A_\infty$ functors, talk at Categories in geometry and math. physics, Split 2007, slides, pdf, work with Volodymyr Lyubashenko
• Volodymyr Mazorchuk, Vanessa Miemietz, Serre functors for Lie algebras and superalgebras
• Sefi Ladkani, Categorification of a linear algebra identity and factorization of Serre functors, Mathematische Zeitschrift 285 (2017) 879–896 doi