Point operations and the open closed constructible line

A sheaf on the point is its vector space of sections, and the point's unique stratification makes constructibility exactly finite-dimensional cohomology by [F1]. For a bounded complex $V$, choose complements $Z^n(V)=B^n(V)\oplus H^n$ and $V^n=Z^n(V)\oplus L^n$ using AC. The differential identifies $L^n$ with $B^{n+1}(V)$, so $V$ is the direct sum of the zero-differential complex $(H^n)$ and a contractible complex of pairs $L^n\stackrel{\sim }{\to }{B}^{n+1}(V)$. If all cohomology is finite-dimensional, $(H^n)$ is a bounded finite-dimensional representative, including the acyclic case where it is zero. Every vector space is injective under AC, because extending a basis splits every vector-space inclusion and extends every linear map from its subspace. Bounded complexes of vector spaces thus compute derived morphisms as homotopy classes by [F2], in both the finite-dimensional and full vector-space categories. For two finite-dimensional complexes these chain maps and homotopies are identical in both categories, proving full faithfulness as well as essential surjectivity. Hence $D_c^b(\mathrm{pt};k)\simeq D^b(k\text{-}{\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}^{\mathrm{f}\mathrm{d}})$.

On the point inverse and direct image of the identity are exact identity functors, and every section has compact support, so the proper-support functor is also the exact identity. Their derived functors are therefore identities. In the exceptional candidate of Derived operations on analytic field sheaves ↗, the fibre opens are only the point and the empty set; compact sections are exact, giving dimension bound $d=0$, and the constant sheaf $k$ is itself $c$-soft, so take $K=k[0]$. For a bounded finite-dimensional complex $B$ every term is injective over $k$, and the candidate has sections ${\mathrm{Hom}}_k^{\bullet }(k,B)=B$ on the point and zero on the empty set. Thus $f^{!}$ is the identity with its ordinary evaluation adjunction, and ${\omega }_{\mathrm{pt}}=a_{\mathrm{pt}}^{}k=k[0]$. The ordinary-operations lemma Derived tensor–Hom and inverse/direct image for field sheaves ↗ identifies tensor and internal Hom with the finite signed complexes: $(A\otimes B{)}^n={⨁}_{p+q=n}{A}^p{\otimes }_k{B}^q$, with $d(a\otimes b)=d_{A}a\otimes b+(-1{)}^{p}a\otimes d_{B}b$, and $\mathcal{H}o{m}^n(A,B)={\prod }_p{\mathrm{Hom}}_k(A^p,B^{p+n})$, with $d\phi =d_B\phi -(-1{)}^n\phi d_A$. Their terms and amplitudes are finite, so they belong to the category identified in step 1.1. Here $f_{\ast }$ and $f_{!}$ in the Example denote their derived operations, which coincide with their exact underived functors in this case.

Apply [F3] to the constant sheaf $\mathcal{F}=k_{\mathbb{C}}$ on $X=\mathbb{C}$, with $U={\mathbb{C}}^{\ast }$ and $Z=\{0\}$: the restriction $k_{\mathbb{C}}{\mathrm{\mid }}_{{\mathbb{C}}^{\ast }}$ is the constant sheaf $k_{{\mathbb{C}}^{\ast }}$, the restriction to the closed point is $k$, and the sequence reads $0\to j_{!}{k}_{{\mathbb{C}}^{\ast }}\to k_{\mathbb{C}}\to i_{\ast }k\to 0$, the terms being computed by [F5] for the extension by zero. Stalkwise the middle sheaf is exact: at $x\mathrm{\ne }0$ the stalk of $j_{!}{k}_{{\mathbb{C}}^{\ast }}$ is $k$ and the stalk of $i_{\ast }k$ is $0$, while at $0$ the first stalk is $0$ and the last is $k$; the quotient of $k_{\mathbb{C}}$ by $j_{!}{k}_{{\mathbb{C}}^{\ast }}$ has no stalks off $\{0\}$, and by [F4] it is canonically the pushforward $i_{\ast }k$ of its restriction to the closed point. All maps are $k$-linear, since they are obtained from the canonical adjunction units of [F5] applied to $k$-sheaves.

In cochain grading, the cone $C$ of the map $g:k_{\mathbb{C}}\to i_{\ast }k$ has $C^{-1}=k_{\mathbb{C}}$, $C^0=i_{\ast }k$ and differential $g$, by reindexing [F6]. Put $A=j_{!}{k}_{{\mathbb{C}}^{\ast }}$. The chain map $u:A[1]\to C$ given by the kernel inclusion in degree $-1$ is a quasi-isomorphism: step 2.2 gives $H^{-1}(C)=A$, $H^0(C)=0$, and $u$ induces the identity on the only possible nonzero cohomology. The cone triangle $k_{\mathbb{C}}\to i_{\ast }k\to C\to k_{\mathbb{C}}[1]$ is distinguished by [F7]. Its signed inverse rotation and the quasi-isomorphism $-u:A[1]\to C$ give the distinguished triangle $A\to k_{\mathbb{C}}\to i_{\ast }k\to A[1]$ with the kernel inclusion as first arrow: the inverse rotation negates the projection $C[-1]\to k_{\mathbb{C}}$, cancelling the sign of $-u$. All three sheaves restrict to constant finite-rank sheaves (including rank zero) on each stratum of $\{{\mathbb{C}}^{\ast },\{0\}\}$, so this localization triangle lies in $D_c^b(\mathbb{C};k)$.