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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.
In the original paper, the following definition was given.
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
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.
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.
In the derived category of finite-dimensional vector spaces over $k$, the identity functor is a Serre functor.
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 $X$.
Any autoequivalence $F : \mathcal{A} \to \mathcal{A}$ commutes with a Serre functor: there is a natural graded isomorphism of functors $F \circ S \stackrel{\sim}{\to} S \circ F$.
Any Serre functor in a triangulated category is exact? (i.e. distinguished triangles are mapped to distinguished triangles).
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.
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):
See also:
Last revised on February 14, 2024 at 11:28:04. See the history of this page for a list of all contributions to it.