The Stacks project

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

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

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

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

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

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 $, let us form the matress of Remark 57.3.3 from the distinguished triangles $A \to B \to C \to A[1]$ and $S(A) \to E \to S(C) \to S(A)[1]$. By Remark 57.3.4 the $k$-linear dual of this is also a matress. Using that $S$ is a Serre functor, we see as above that this dual matress is the double complex made from the triangles (the second of which we do not yet know to be distinguished) $S(A) \to E \to S(C) \to S(A)[1]$ and $S(A) \to S(B) \to S(C) \to S(A)[1]$. But the matress condition exactly tells us that the maps $S(A) \to S(A)$ and $S(C) \to S(C)$ fit into a morphism of triangles, i.e., that we have a $\varphi $ making the diagram above commute. $\square$