A Serre functor on a 
-linear
triangulated category with finite-dimensional hom-sets is a
-linear autoequivalence 
with bifunctorial 
-linear isomorphisms
 |
Here 
is a field, and the star denotes the dual
vector space over 
. An autoequivalence is a functor
from the category to itself with an inverse up to natural isomorphism. When it exists, the functor
together with its duality data is unique up to unique natural
isomorphism and commutes with the triangulated shift (The Stacks Project Authors
2026). A Calabi-Yau category requires 
to be isomorphic
to an integer power of the shift as a triangle
functor. An isomorphism only of the underlying
-linear functors
gives the weak Calabi-Yau condition (Keller 2008, corrected author version, p. 8).
