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57.3 Serre functors

The material in this section is taken from [Bondal-Kapranov].

Lemma 57.3.1. 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}). The following are equivalent

  1. there exists a k-linear equivalence S : \mathcal{T} \to \mathcal{T} and 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}),

  2. for every X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{T}) the functor Y \mapsto \mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(X, Y)^\vee is representable and the functor Y \mapsto \mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(Y, X)^\vee is corepresentable.

Proof. Condition (1) implies (2) since given (S, c) and X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{T}) the object S(X) represents the functor Y \mapsto \mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(X, Y)^\vee and the object S^{-1}(X) corepresents the functor Y \mapsto \mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(Y, X)^\vee .

Assume (2). We will repeatedly use the Yoneda lemma, see Categories, Lemma 4.3.5. For every X denote S(X) the object representing the functor Y \mapsto \mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(X, Y)^\vee . Given \varphi : X \to X', we obtain a unique arrow S(\varphi ) : S(X) \to S(X') determined by the corresponding transformation of functors \mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(X, -)^\vee \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(X', -)^\vee . Thus S is a functor and we obtain the isomorphisms c_{X, Y} by construction. It remains to show that S is an equivalence. For every X denote S'(X) the object corepresenting the functor Y \mapsto \mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(Y, X)^\vee . Arguing as above we find that S' is a functor. We claim that S' is quasi-inverse to S. To see this observe that

\mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(X, Y) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(Y, S(X))^\vee = \mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(S'(S(X)), Y)

bifunctorially, i.e., we find S' \circ S \cong \text{id}_\mathcal {T}. Similarly, we have

\mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(Y, X) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(S'(X), Y)^\vee = \mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(Y, S(S'(X)))

and we find S \circ S' \cong \text{id}_\mathcal {T}. \square

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}).

Lemma 57.3.3. In the situation of Definition 57.3.2. If a Serre functor exists, then it is unique up to unique isomorphism and it is an exact functor of triangulated categories.

Proof. Given a Serre functor S the object S(X) represents the functor Y \mapsto \mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(X, Y)^\vee . Thus the object S(X) together with the functorial identification \mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(X, Y)^\vee = \mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(Y, S(X)) is determined up to unique isomorphism by the Yoneda lemma (Categories, Lemma 4.3.5). Moreover, for \varphi : X \to X', the arrow S(\varphi ) : S(X) \to S(X') is uniquely determined by the corresponding transformation of functors \mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(X, -)^\vee \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {T}(X', -)^\vee .

For objects X, Y of \mathcal{T} we have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits (Y, S(X)[1])^\vee & = \mathop{\mathrm{Hom}}\nolimits (Y[-1], S(X))^\vee \\ & = \mathop{\mathrm{Hom}}\nolimits (X, Y[-1]) \\ & = \mathop{\mathrm{Hom}}\nolimits (X[1], Y) \\ & = \mathop{\mathrm{Hom}}\nolimits (Y, S(X[1]))^\vee \end{align*}

By the Yoneda lemma we conclude that there is a unique isomorphism S(X[1]) \to S(X)[1] inducing the isomorphism from top left to bottom right. Since each of the isomorphisms above is functorial in both X and Y we find that this defines an isomorphism of functors S \circ [1] \to [1] \circ S.

Let (A, B, C, f, g, h) be a distinguished triangle in \mathcal{T}. We have to show that the triangle (S(A), S(B), S(C), S(f), S(g), S(h)) is distinguished. Here we use the canonical isomorphism S(A[1]) \to S(A)[1] constructed above to identify the target S(A[1]) of S(h) with S(A)[1]. We first observe that for any X in \mathcal{T} the triangle (S(A), S(B), S(C), S(f), S(g), S(h)) induces a long exact sequence

\ldots \to \mathop{\mathrm{Hom}}\nolimits (X, S(A)) \to \mathop{\mathrm{Hom}}\nolimits (X, S(B)) \to \mathop{\mathrm{Hom}}\nolimits (X, S(C)) \to \mathop{\mathrm{Hom}}\nolimits (X, S(A)[1]) \to \ldots

of finite dimensional k-vector spaces. Namely, this sequence is k-linear dual of the sequence

\ldots \leftarrow \mathop{\mathrm{Hom}}\nolimits (A, X) \leftarrow \mathop{\mathrm{Hom}}\nolimits (B, X) \leftarrow \mathop{\mathrm{Hom}}\nolimits (C, X) \leftarrow \mathop{\mathrm{Hom}}\nolimits (A[1], X) \leftarrow \ldots

which is exact by Derived Categories, Lemma 13.4.2. Next, we choose a distinguished triangle (S(A), E, S(C), i, p, S(h)) which is possible by axioms TR1 and TR2. We want to construct the dotted arrow making following diagram commute

\xymatrix{ S(C)[-1] \ar[r]_-{S(h[-1])} & S(A) \ar[r]_{S(f)} & S(B) \ar[r]_{S(g)} & S(C) \ar[r]_{S(h)} & S(A)[1] \\ S(C)[-1] \ar[r]^-{S(h[-1])} \ar@{=}[u] & S(A) \ar[r]^ i \ar@{=}[u] & E \ar[r]^ p \ar@{..>}[u]^\varphi & S(C) \ar[r]^{S(h)} \ar@{=}[u] & S(A)[1] \ar@{=}[u] }

Namely, if we have \varphi , then we claim for any X the resulting map \mathop{\mathrm{Hom}}\nolimits (X, E) \to \mathop{\mathrm{Hom}}\nolimits (X, S(B)) will be an isomorphism of k-vector spaces. Namely, we will obtain a commutative diagram

\xymatrix{ \mathop{\mathrm{Hom}}\nolimits (X, S(C)[-1]) \ar[r] & \mathop{\mathrm{Hom}}\nolimits (X, S(A)) \ar[r] & \mathop{\mathrm{Hom}}\nolimits (X, S(B)) \ar[r] & \mathop{\mathrm{Hom}}\nolimits (X, S(C)) \ar[r] & \mathop{\mathrm{Hom}}\nolimits (X, S(A)[1]) \\ \mathop{\mathrm{Hom}}\nolimits (X, S(C)[-1]) \ar[r] \ar@{=}[u] & \mathop{\mathrm{Hom}}\nolimits (X, S(A)) \ar[r] \ar@{=}[u] & \mathop{\mathrm{Hom}}\nolimits (X, E) \ar[r] \ar[u]^\varphi & \mathop{\mathrm{Hom}}\nolimits (X, S(C)) \ar[r] \ar@{=}[u] & \mathop{\mathrm{Hom}}\nolimits (X, S(A)[1]) \ar@{=}[u] }

with exact rows (see above) and we can apply the 5 lemma (Homology, Lemma 12.5.20) to see that the middle arrow is an isomorphism. By the Yoneda lemma we conclude that \varphi is an isomorphism. To find \varphi consider the following diagram

\xymatrix{ \mathop{\mathrm{Hom}}\nolimits (E, S(C)) \ar[r] & \mathop{\mathrm{Hom}}\nolimits (S(A), S(C)) \\ \mathop{\mathrm{Hom}}\nolimits (E, S(B)) \ar[u] \ar[r] & \mathop{\mathrm{Hom}}\nolimits (S(A), S(B)) \ar[u] }

The elements p and S(f) in positions (0, 1) and (1, 0) define a cohomology class \xi in the total complex of this double complex. The existence of \varphi is equivalent to whether \xi is zero. If we take k-linear duals of this and we use the defining property of S we obtain

\xymatrix{ \mathop{\mathrm{Hom}}\nolimits (C, E) \ar[d] & \mathop{\mathrm{Hom}}\nolimits (C, S(A)) \ar[l] \ar[d] \\ \mathop{\mathrm{Hom}}\nolimits (B, E) & \mathop{\mathrm{Hom}}\nolimits (B, S(A)) \ar[l] }

Since both A \to B \to C and S(A) \to E \to S(C) are distinguished triangles, we know by TR3 that given elements \alpha \in \mathop{\mathrm{Hom}}\nolimits (C, E) and \beta \in \mathop{\mathrm{Hom}}\nolimits (B, S(A)) mapping to the same element in \mathop{\mathrm{Hom}}\nolimits (B, E), there exists an element in \mathop{\mathrm{Hom}}\nolimits (C, S(A)) mapping to both \alpha and \beta . In other words, the cohomology of the total complex associated to this double complex is zero in degree 1, i.e., the degree corresponding to \mathop{\mathrm{Hom}}\nolimits (C, E) \oplus \mathop{\mathrm{Hom}}\nolimits (B, S(A)). Taking duals the same must be true for the previous one which concludes the proof. \square


Comments (1)

Comment #9991 by Jonas on

In the last step of the proof of Lemma 57.3.3, how exactly is TR3 used to obtain an element of mapping to both and ?


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