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Compatible comparisons transport relative objects and support fibers

PDF p. 7: Motivic purity at a smooth closed point

Statement

In a triangulated category, consider two specified distinguished triangles $A \to B \to Q \to A [1]$ and $A^{'} \to B^{'} \to Q^{'} \to A^{'} [1]$ . If isomorphisms $u : A \to A^{'}$ and $v : B \to B^{'}$ commute with the first arrows, there exists an isomorphism $w : Q \to Q^{'}$ completing a morphism of triangles; uniqueness of $w$ is not asserted. In particular any two choices of a cone for the same first arrow are isomorphic compatibly with the first two objects, without making cones a functor on an arbitrary triangulated category.

For a commuting square of cochain maps $C (X) \to C (U)$ and $D (X) \to D (U)$ with vertical quasi-isomorphisms, the induced map on the explicit support fibers is a quasi-isomorphism and respects their maps to the global complexes. Consequently compatible derived comparisons on the ambient scheme and its open complement suffice to transport a purity calculation by this construction. A separately supplied direct compatible comparison of support objects may also suffice; no universal necessity of ambient and open comparisons is asserted. Unrelated isomorphisms of individual cohomology groups do not by themselves supply the commuting derived comparison used here.

Facts & Assumptions

Given: The triangles and commuting first-arrow square; separately the strict commuting square of cochain maps for the explicit-fiber assertion.

Goal: Establish cone transport with its nonuniqueness and compatible support transport.

[F1]

TR3 supplies some third map completing a morphism of distinguished triangles. Triangulated-category axiom TR3

[F2]

Representable Hom gives long exact sequences on such triangles. Long exact Hom sequences of a distinguished triangle

[F3]

The support fiber of $f : A \to B$ has degree $n$ term $A^{n} \oplus B^{n - 1}$ and differential $(a, b) \mapsto (d_{A} a, f (a) - d_{B} b)$ , with projection to $A$ . Support fibers with explicit cohomological indexing

Proof

1.1F1F2given

Apply [F1] to obtain $w$ and a morphism of triangles. For any object $T$ , apply covariant $Hom (T, -)$ and [F2]. In the five-term segment surrounding $Hom (T, Q)$ , all vertical maps except possibly the middle one are isomorphisms because they are induced by $u, v$ or their shifts. Exactness implies the middle map is bijective: surjectivity follows by lifting a target element's image in the next term and correcting the difference by the preceding term; injectivity follows by the same correction for an element with zero image. Thus $Hom (T, w)$ is bijective for every $T$ .

2.1step 1.1algebra

Take $T = Q^{'}$ to lift ${id}_{Q^{'}}$ to $h : Q^{'} \to Q$ with $w h = {id}_{Q^{'}}$ . Then $w (h w - {id}_{Q}) = 0$ ; injectivity for $T = Q$ gives $h w = {id}_{Q}$ . Hence $w$ is an isomorphism. TR3 only gave existence and does not select it uniquely, so no functorial cone assertion follows.

3.1F3givenstep 1.1step 2.1∎

In the cochain-map case, the two vertical maps act coordinatewise on the models in [F3]; the commuting square makes this a cochain map and its projections commute with the ambient vertical map. The short exact sequences $0 \to C (U) [- 1] \to fib (C (X) \to C (U)) \to C (X) \to 0$ and its $D$ counterpart give long exact cohomology sequences. The same five-term argument as in step 1.1, now using the two vertical quasi-isomorphisms and their shifts, proves that the fiber map induces an isomorphism in every degree. This proves support transport under the sufficient compatible-comparison hypotheses stated here.

Depends on

Used by · 1

Gysin purity for a smooth closed pair

Dependency tree · 3 levels

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