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Open excision for a descent theory

PDF p. 7: Excision at a smooth point; PDF p. 13: Excision to the smooth locus

Statement

Let $X$ be a scheme, $T \subseteq X$ a closed subset, $W \subseteq X$ an open subset containing $T$ , $U = X ∖ T$ , and $V = W \cap U = W ∖ T$ . Suppose a contravariant spectrum-valued theory $E$ is defined on these four opens, with coherent restriction maps, and that its restriction square $\begin{array}{ccc} E (X) & ⟶ & E (W) \\ ↓ & ↓ \\ E (U) & ⟶ & E (V) \end{array}$ is homotopy cartesian: the induced map $E (X) \to E (W) \times_{E (V)} E (U)$ is an equivalence, where the product is a homotopy pullback. Define $E_{T} (X) = fib (E (X) \to E (U))$ and $E_{T} (W) = fib (E (W) \to E (V))$ . Restriction induces an equivalence $E_{T} (X) ≃ E_{T} (W)$ . It is natural for compatible maps of such coherent squares, and the square formed by the support-to-global maps and restriction to $W$ commutes coherently.

The same conclusion holds for cochain complexes of abelian groups, with equivalence replaced by quasi-isomorphism, under the following explicit form of two-open descent. Write $A = C (X)$ , $B = C (W)$ , $C = C (U)$ , and $D = C (V)$ , and suppose the restriction square is strictly commutative. If $u : B \to D$ and $v : C \to D$ are its lower restriction maps, require the canonical map $A ⟶ P : = fib (B \oplus C \overset{u - v}{\to} D), a ⟼ (a ∣_{W}, a ∣_{U}, 0),$ to be a quasi-isomorphism, using the explicit fiber complex of Support fibers with explicit cohomological indexing. Then restriction induces a quasi-isomorphism $C_{T} (X) \to C_{T} (W)$ , compatible with the support-to-global maps and natural for maps of such squares. In particular it induces isomorphisms on supported cohomology in every degree. A quasi-isomorphism is a cochain map inducing isomorphisms on all cohomology groups.

These are conditional excision statements: a particular theory must supply the indicated restriction data and descent hypothesis, as well as any quasi-compactness, size, or other conditions required for that hypothesis. No descent property is asserted merely from the existence of a support fiber.

Facts & Assumptions

Given: The opens and either a coherent spectrum square or a strictly commutative cochain-complex square as in the statement, satisfying the corresponding displayed descent hypothesis.

Goal: Prove the restriction equivalence or quasi-isomorphism of support fibers and its stated compatibility and naturality.

[F1]

In the stable spectra interface, fibers and products over a common object are coherent homotopy pullbacks, characterized by their universal properties. Stable spectra and exact triangles

[F2]

For a cochain map $f : M \to N$ , the support-fiber model has degree- $n$ group $M^{n} \oplus N^{n - 1}$ and differential $(m, n) \mapsto (d_{M} m, f (m) - d_{N} n)$ . A commutative square of cochain maps induces a map of these fibers, and if its two vertical maps are quasi-isomorphisms then so is the induced map on fibers. The support-to-global map is projection to the first factor. Support fibers with explicit cohomological indexing

Proof

technique · Paste homotopy pullbacks for spectra; for complexes, compare the explicit descent model with the fiber over the target complex

1.1F1given

In the spectrum case, the descent hypothesis identifies $E (X)$ with $E (W) \times_{E (V)} E (U)$ , compatibly with its two projections. Taking the homotopy pullback of the projection to $E (U)$ along $0 \to E (U)$ gives $E_{T} (X) ≃ (E (W) \times_{E (V)} E (U)) \times_{E (U)} 0 ≃ E (W) \times_{E (V)} 0 = E_{T} (W) .$ The middle equivalence follows by the universal property of the homotopy pullback: a map into either side is a map into $E (W)$ together with a coherent nullhomotopy of its composite to $E (V)$ . The displayed equivalence is the one induced by restriction.

2.1F1step 1.1

The same universal property makes this construction natural for compatible maps of the coherent squares. Its projection to $E (W)$ is the composite of the projection $E_{T} (X) \to E (X)$ with restriction $E (X) \to E (W)$ , with the coherence inherited from the pullback diagram. This proves the support-to-global compatibility.

3.1F2givenstep 2.1

In the complex case, let $q : A \to P$ be the descent quasi-isomorphism and let $p : P \to C$ be projection. The equality $p q = A \to C$ gives a commutative square with vertical maps $q$ and ${id}_{C}$ . By [F2], it induces a quasi-isomorphism $fib (A \to C) \to fib (P \to C)$ .

4.1F2givenconstructstep 3.1

Write an element of $fib (P \to C)^{n}$ as $(b, c, d, t)$ , with $b \in B^{n}$ , $c \in C^{n}$ , $d \in D^{n - 1}$ and $t \in C^{n - 1}$ . Its differential is $(d b, d c, u b - v c - d d, c - d t),$ where each $d$ on the right denotes the differential of the appropriate complex. The map $(b, c, d, t) ⟼ (b, d + v t)$ to $fib (B \to D)$ is a cochain map: the second component of the differential becomes $u b - d_{D} (d + v t)$ . It is split surjective as a cochain map, with section $(b, z) \mapsto (b, 0, z, 0)$ .

5.1step 4.1construct

Its kernel consists of $(0, c, - v t, t)$ and is isomorphic to $fib ({id}_{C})$ , with differential $(c, t) \mapsto (d c, c - d t)$ . This kernel is contractible: the degree-minus-one map $(c, t) \mapsto (t, 0)$ satisfies $d h + h d = id$ . Hence the split surjective map in step 4.1 is a quasi-isomorphism.

6.1F2step 3.1step 5.1

The composite of the maps in steps 3.1 and 4.1 sends $(a, t)$ to $(a ∣_{W}, t ∣_{V})$ , exactly the restriction map between the two support-fiber models. It is therefore a quasi-isomorphism. The formula also shows that it commutes strictly with projection to $A$ and $B$ and is natural for maps of the given commutative squares. Taking cohomology gives the stated supported-cohomology isomorphisms.

7.1step 2.1step 6.1∎

Steps 1.1–2.1 and 3.1–6.1 establish both forms of open excision under their respective descent hypotheses.

Depends on

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