Derived category of sheaves ($D(X)$)

The derived category of sheaves is formed by localizing cochain complexes of sheaves at quasi-isomorphisms, equivalently localizing their homotopy category. The bounded, bounded below and bounded above variants are denoted $D^b(X)$, $D^{+}(X)$ and $D^{-}(X)$. Cohomology sheaves are invariant under quasi-isomorphism and therefore define functors on these categories. Constructible bounded complexes form a natural subcategory on which Verdier duality reverses arrows and, under the usual finiteness hypotheses, is an involution.

A ghost is a morphism inducing zero on every cohomology sheaf, but it need not itself be zero. For the equator $j:S^1\hookrightarrow S^2$ and complement $i:U\hookrightarrow S^2$, the connecting morphism $j_{\ast }{\mathbb{Q}}_{S^1}\to i_{!}{\mathbb{Q}}_U[1]$ is a ghost. Its source and target have cohomology sheaves in different degrees, but its induced map $H^1(S^1)\to H_c^2(U)$ is nonzero, proving that the derived morphism is nonzero.

On bounded constructible complexes over locally compact stratified spaces, Verdier duality is $D_{X}E=R\mathcal{H}om(E,{\omega }_X^{\bullet })$, where ${\omega }_X^{\bullet }$ is the dualizing complex. For an oriented real $N$-manifold, $D_X{\mathbb{Q}}_X\cong {\mathbb{Q}}_X[N]$. Proper direct image commutes with this duality, and $i_x^{!}{D}_{X}E\cong D_{\{x\}}{i}_x^{\ast }E$. These properties exchange stalk support conditions with costalk conditions.

A bounded complex is constructible for a given stratification when each of its cohomology sheaves restricts to a finite-rank local system on every stratum. Constructibility concerns cohomology sheaves, not an arbitrary choice of resolution.

The category $D^{+}(X)$ is obtained from bounded below cochain complexes of sheaves by passing to cochain-homotopy classes of maps and then inverting quasi-isomorphisms. It can equally be modeled by injective resolutions. The cohomology-sheaf functors descend to this localization, and an object is zero exactly when all its cohomology sheaves vanish.

An exact, or distinguished, triangle is one isomorphic to $E\stackrel{f}{\to }F\to \mathrm{Cone}(f)\to E[1]$. For a degreewise short exact sequence $0\to E\to F\to G\to 0$, the map $\mathrm{Cone}(f)\to G$ taking $(v,e)$ to the image of $v$ is a quasi-isomorphism: its kernel is the contractible cone of the identity on $E$. Consequently the short exact sequence defines such a triangle.