Definition 57.3.2. Let $k$ be a field. Let $\mathcal{T}$ be a $k$-linear triangulated category such that $\dim _ k \mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(X, Y) < \infty $ for all $X, Y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{T})$. We say a Serre functor exists if the equivalent conditions of Lemma 57.3.1 are satisfied. In this case a Serre functor is a $k$-linear equivalence $S : \mathcal{T} \to \mathcal{T}$ endowed with $k$-linear isomorphisms $c_{X, Y} : \mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(X, Y) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(Y, S(X))^\vee $ functorial in $X, Y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{T})$.
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