Derived operations on analytic field sheaves

Let $k$ be a field and let $X,Y$ be locally compact Hausdorff spaces (Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not). Write ${\mathrm{Sh}}_k(X)$ for the category of sheaves of $k$-vector spaces (A sheaf on a topological space, Sheaf cohomology as right derived global sections) viewed as modules over the constant ring sheaf $k_X$, and form derived categories in those sheaves (Derived category of an abelian category). For a continuous map $f:X\to Y$, set $f^{\ast }=f^{-1}$ (Inverse image presheaf and inverse image sheaf) and define $f_{!}F$ on an open $V\subseteq Y$ to consist of the sections of $F$ on $f^{-1}(V)$ whose closed support (Support of a section of a sheaf, Compactly supported global sections and their right derived functors) maps properly to $V$, properness being that of Proper continuous maps of topological spaces together with its stability lemma Basic stability of proper continuous maps. Under AC (The Axiom of Choice), $R{f}_{\ast }$ and $R{f}_{!}$ are computed on bounded-below $k$-linear injective replacements. Tensor means the signed total tensor over $k$ (Tensor product of sheaves of modules) and internal derived Hom is the injective construction of Derived tensor–Hom and inverse/direct image for field sheaves (The internal Hom sheaf of two module sheaves).

Suppose the uniform all-fibre-open cohomological dimension bound of Bounded fibrewise-soft resolutions from a uniform all-sheaf dimension bound holds, and choose its finite resolution $k_X\to K$; for an open $U\subseteq X$ put $K_U=j_{U!}{j}_U^{\ast }K$ (Extension by zero for abelian sheaves on an open subspace, c-soft sheaves). For a bounded-below injective complex $I$ on $Y$, define the exceptional inverse-image candidate to be the sheaf complex with open sections ${\mathrm{Hom}}_Y^{\bullet }(f_{!}{K}_U,I)$. Exceptional inverse image as the adjoint of derived proper-support image ↗ proves that this candidate is a sheaf complex and is independent of the chosen resolution. Set ${\omega }_X=a_X^{!}k$ and $D_{X}A=R\mathcal{H}om(A,{\omega }_X)$ when the terminal map satisfies the same finite-dimensional hypotheses. Constructible restriction is a separate geometric theorem; none of these definitions asserts a generic six-functor axiom.

The prescriptions are those of Verdier's exposé: $f^{\ast }$ is the exact inverse image, $f_{!}$ is the sheaf of sections whose closed support is proper over the base — a presheaf by the local nature of properness, whose sheaf property for every sheaf is proved by the exceptional-adjoint theorem recorded in justified_by — $R{f}_{\ast }$ is derived from the direct image Direct image of a sheaf along a continuous map on bounded-below injective replacements, and the terms of the chosen finite resolution are fibrewise $c$-soft, not necessarily $c$-soft on $X$. Their acyclicity for $f_{!}$ follows from the fibre formula in its well-definedness theorem; the restricted fibre sheaves are compact-support acyclic by Compact-support acyclicity of c-soft sheaves. The ordinary-operations lemma also supplies the flasque acyclicity (Flasque sheaf) used to compare resolutions. AC supplies the injective replacements and finite fibrewise-soft resolution; the proper-support sections, extension by zero and the exceptional candidate itself are choice-free constructions. Cohomological shifts obey $H^i(A[r])=H^{i+r}(A)$ in the convention of Derived category of an abelian category.