nLab
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 $𝒜$ is an exact functor such that for any objects $A$ and $B$, $Hom(A,B)\simeq Hom(B,S(A){)}^{*}$. 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 $𝒜$ be a $k$-linear triangulated category with finite-dimensional Hom’s and $k$ algebraically closed. A Serre functor $S:𝒜\to 𝒜$ is an additive equivalence that commutes with the translation functor, with bi-functorial isomorphisms ${\varphi }_{A,B}:{Hom}_{𝒜}(A,B)\stackrel{\sim }{\to }{Hom}_{𝒜}(B,S(A){)}^{*}$ for any objects $A$ and $B$, such that the composite

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

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

Definition

Let $𝒜$ be a $k$-linear category with finite-dimensional Hom’s and $k$ an arbitrary field. A Serre functor $S:𝒜\to 𝒜$ is an additive equivalence with bi-functorial isomorphisms ${\varphi }_{A,B}:{Hom}_{𝒜}(A,B)\stackrel{\sim }{\to }{Hom}_{𝒜}(B,S(A){)}^{*}$ 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 $F:𝒜\to 𝒜$ 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 ${\varphi }_{A,B}$ 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, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1183–1205, 1337.

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 $A_{\mathrm{\infty }}$ 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