Coherent six operations on analytic constructible field sheaves

The ordinary and support-locality clauses. Apply [F1] to the actual analytic maps. Its proper-support construction and injective models give the ordinary Cartesian exchange, the projection formula, both support localization triangles, canonical composition, and the units and traces of the adjunctions; the equalities are section-level identities carried through the same resolution comparisons, so no unrelated triangulated-category isomorphisms are selected. Together with [F3] this gives the adjunctions $f^{\ast }⊣R{f}_{\ast }$ and $R{f}_{!}⊣f^{!}$ and the tensor–Hom adjunction on the bounded constructible specialization, their identity and composition comparisons, and the ordinary proper base change for proper maps.

The constructible and extraordinary clauses. Apply [F2] to the actual finite conical and incidence models. It supplies duality, orientation and the extraordinary exchange by dualizing the already canonical ordinary proper base-change map, and identifies dualizing transitivity and $f^{!}\simeq D_X{f}^{\ast }{D}_Y$; the tensor–Hom and inverse/direct-image adjunctions are the signed dg comparisons of [F3]. Each duality comparison is evaluation and transposition of the same support model, so functoriality carries the proved pasting and trace squares of [F1] to their dual squares; this yields the coherent six-operation interface on the bounded constructible subcategory whenever its listed geometric supplier proves preservation of that subcategory, which is thus the exact geometric input rather than an additional axiom.

The product projection. For a locally trivial projection $V\times F\to V$ with finite-type smooth fibre $F$, use the finite product incidence resolution of [F4] on quasi-projective charts: tensor the finite base diagram resolution with the constant-diagram resolution of the fibre. For a general separated finite-type fibre, take a finite affine-open cover and its finite affine intersections; apply this construction to each intersection and totalize the finite ordinary Čech derived augmentation of [F4]. This assembles the same comparison with the finite complex $C^{\ast }(F;k)$, without assuming a single triangulation of a non-quasi-projective fibre. Each realized term is supported on a product of closed cells minus the deleted faces, and its intersection with a small base star times $F$ has contractible factors, so the independently proved constant-sheaf comparison [F4] makes it section-acyclic; applying sections gives the finite double complex $B{\mathrm{\mid }}_{\text{star}}{\otimes }_k{C}^{\ast }(F;k)$. These are natural restriction complexes, and they identify the actual derived image and its adjunction unit with $B{\otimes }_k{C}^{\ast }(F;k)$ and $B\otimes (k\to C^{\ast }(F;k))$. For smooth base change across an open inclusion $j$, a local smooth chart is a product with a contractible ball; the boundary-star section model becomes the previous finite cochain model tensored with the ball cochains, whose augmentation is a quasi-isomorphism by an explicit contraction, so the canonical open-inclusion exchange is a stalkwise quasi-isomorphism. If the supplied map has a factorization $f=p\circ j$ with $p$ proper and $j$ an open inclusion, combining this exchange with the already constructed proper base-change map for $p$ and its compatibility with composition identifies the composite with the canonical exchange for $f$; proper maps already satisfy smooth base change through [F1]. No existence of a compactification is inferred: the clause applies only to the stated supplied factorization.

Coefficient extension. For a field extension $k\to K$ use the geometric supplier's finite local cellular and cochain models [F4]. Scalar extension is exact on vector spaces, commutes with finite sums and cohomology, and commutes with duals of finite-dimensional terms, so ordinary inverse image, finite cochain pushforward, tensor, internal Hom and the support-defined Verdier duality comparisons extend scalars termwise; the exceptional comparison follows from $f^{!}=D_X{f}^{\ast }{D}_Y$ of [F2], and composition and traces extend because their maps are those same finite evaluation and section comparisons. This argument does not assert that infinite products of injective stalks commute with scalar extension.