Solution

For a cochain complex $E$, define the shift of a cochain complex byFor a cochain map $a:E\to F$, put $(\Sigma a{)}^i=a^{i+1}$. This is again a cochain map, so the construction is a functor. The sign ensures the standard cone identities; the shifted differential squares to zero. Iteration gives $E[r{]}^i=E^{i+r}$ and differential $(-1{)}^r{d}_E$, and $H^i(E[r])=H^{i+r}(E)$.

For $\varphi :E\to F$, its algebraic mapping cone isApplying this differential twice gives first component $d_F^{2}f+d_F\varphi e-\varphi d_{E}e=0$, since $\varphi$ is a cochain map, and second component $d_E^{2}e=0$. The mapsare cochain maps: $d_{C}u(f)=(d_{F}f,0)=u(d_{F}f)$ and $v(d_C(f,e))=-d_{E}e=d_{\Sigma E}v(f,e)$. They are maps of complexes of sheaves, because every displayed operation is a sheaf morphism in each degree.

The bounded below derived category of sheaves $D(X)=D^{+}(X)$ can be constructed by first quotienting the category of bounded below cochain complexes by cochain homotopy and then formally inverting all quasi-isomorphisms. A morphism may be represented by a roof $E\stackrel{\sim }{\leftarrow }{E}^{\prime }\to F$. Alternatively, take injective resolutions of sheaves and use homotopy classes of maps between the resulting injective complexes. The cohomology sheaf functors descend to the localization, because they already identify homotopic maps and turn quasi-isomorphisms into isomorphisms. If $E\cong 0$ in $D(X)$, applying these functors gives ${\mathcal{H}}^i(E)=0$ for every $i$. Conversely, if all these sheaves vanish, the unique map $E\to 0$ is a quasi-isomorphism, which becomes invertible in $D(X)$. HenceAn exact triangle in a derived category is a triangle isomorphic to a cone triangleIn particular the last arrow is part of the structure; a mere sequence of three objects with successive composite zero does not suffice.

Suppose $0\to E\stackrel{\varphi }{\to }F\stackrel{\psi }{\to }G\to 0$ is degreewise exact. DefineThen $r(d_C(f,e))=\psi (d_{F}f)+\psi \varphi (e)=d_G\psi (f)=d_{G}r(f,e)$, so $r$ is a cochain map. It is natural under maps of short exact sequences. It is degreewise surjective as a sheaf map, and its kernel can be identified with $E^i\oplus E^{i+1}$ using $(a,e)\mapsto (\varphi (a),e)$. In these coordinates the kernel differential isThere is a contracting homotopy $h^i(a,e)=(0,a)$ in degree $i-1$:Thus the kernel has zero cohomology sheaves. The cohomology-sheaf exact sequence is the sequence of the short exact sequence of complexes $0\to \ker r\to C(\varphi )\to G\to 0$, or equivalently its ordinary complex-cohomology sequence at every stalk. It follows that $r$ induces isomorphisms of all cohomology sheaves, provingUnder this isomorphism the cone triangle becomes the exact trianglewhere $\delta =v\circ r^{-1}$ in the derived category.

For the final construction write $U=S^2\setminus S^1$ and set $A=i_{!}{\mathbb{Q}}_U$, $B={\mathbb{Q}}_{S^2}$, $C=j_{\ast }{\mathbb{Q}}_{S^1}$, each initially placed in degree zero. The stated sequence $0\to A\to B\to C\to 0$ is exact on stalks: on $U$ its first map is an isomorphism and its last sheaf is zero; on the equator its first sheaf is zero and the second map is an isomorphism. Take its connecting morphismThe source has only ${\mathcal{H}}^0(C)=C$, whereas the target has only ${\mathcal{H}}^{-1}(A[1])=A$. Therefore every induced map ${\mathcal{H}}^i(\mu )$ is zero: in any degree at least one of its source and target is zero. This is a ghost morphism in a derived category.

To prove the derived morphism itself is nonzero, apply derived global sections, giving the hypercohomology sequence. The extension by zero identifies $H^{\ast }(S^2;A)$ with the compactly supported cohomology of $U$, and closed direct image identifies $H^{\ast }(S^2;C)$ with $H^{\ast }(S^1;\mathbb{Q})$. The complement $U$ is two open disks, each homeomorphic to ${\mathbb{R}}^2$, soThe degree-one part of the resulting exact sequence isThus ${\mu }_{\ast }$ is injective, in particular nonzero. With the hemispheres oriented from $S^2$, the last map is $(a,b)\mapsto a+b$, so an appropriately oriented equator generator maps to $(1,-1)$. A zero morphism in the derived category of sheaves would induce a zero map after derived global sections and taking cohomology. Consequently $\mu \ne 0$, although ${\mathcal{H}}^i(\mu )=0$ for every $i$. This establishes explicitly that the collection of cohomology-sheaf functors detects zero objects but is not faithful on morphisms.