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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