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Clopen reduction of the smooth-point comparison

Statement

Assume the Axiom of Choice and the fixed compatible site and universe size bounds of the motivic and perfect-complex K-theory suppliers. Let $k$ be a perfect field, let $X$ be a quasi-compact quasi-separated $k$ -scheme of pure finite dimension $d \geq 0$ , and let $X = ∐_{a = 1}^{n} X_{a}$ be a finite partition into nonempty clopen subschemes. Each $X_{a}$ has its restricted structure sheaf; clopen means open and closed. Actual restriction induces isomorphisms of subgroups of the same finite tuple group $⨁_{a} K_{0} (X_{a})$ , $F_{m o t}^{d} K_{0} (X) ≅ ⨁_{a} F_{m o t}^{d} K_{0} (X_{a}), F_{s m}^{d} K_{0} (X) ≅ ⨁_{a} F_{s m}^{d} K_{0} (X_{a}) .$ Consequently $F_{m o t}^{d} K_{0} (X) = F_{s m}^{d} K_{0} (X)$ if and only if this equality holds on every $X_{a}$ . In particular, once the finite clopen integral-component decomposition of a normal affine pure threefold has been established, the equality in Theorem 6.7 reduces to its integral components without a connectedness hypothesis.

Here $K_{0} (X) = π_{0} K (X)$ uses nonconnective K-theory of perfect complexes, $F_{m o t}^{d} K_{0} (X)$ is the image of $π_{0}$ of its $d$ th motivic stage in $K_{0} (X)$ , and $F_{s m}^{d} K_{0} (X)$ is the subgroup generated by classes of perfect residue sheaves at smooth closed points. Perfect means locally equivalent to a bounded complex of finite locally free sheaves. Quasi-compact means every open cover has a finite subcover, quasi-separated means intersections of affine opens are quasi-compact, and pure dimension $d$ means every irreducible component has chain dimension $d$ . Perfectness of the field means every algebraic extension is separable. The direct sum over the finite partition is the tuple group with coordinatewise addition.

Facts & Assumptions

Given: The field, pure-dimensional scheme, finite clopen partition, Choice and compatible size bounds in the Statement.

Goal: Decompose the two subgroups under the same actual restriction map and deduce the equivalence of equalities.

[F1]

Finite clopen restrictions split the actual motivic image filtration applies to any finite clopen partition of a qcqs scheme over a field. It gives the actual $K_{0}$ restriction isomorphism, the decomposition of every motivic image group under it, and exact zero extension as the coordinate inclusions. Its Proof 1.1 establishes that every clopen piece is qcqs and constructs zero extension of a sheaf complex on a piece by $(i_{!} E)^{q} (V) = E^{q} (V \cap X_{a})$ , with zero stalks on its open complement. This preserves perfect complexes and identifies their classes with the corresponding coordinate inclusions.

[F2]

Motivic filtration and image groups fixes the actual stage-to-zero image-group convention and its Choice and size bounds. Perfect complexes on schemes and their support fixes the local perfect-complex condition used here. No proper pushforward for motivic stages is required.

[F3]

Subgroup generated by smooth closed points identifies the smooth-point subgroup with finite integer sums of the classes $[i_{x *} κ (x)]$ of perfect residue sheaves at smooth closed points, under the perfect-field, qcqs and pure-dimension hypotheses. It includes the zero subgroup when there are no such points, and fixes the field, dimension and qcqs conventions stated above. Smooth morphism of schemes makes smoothness at a point unchanged on an open neighbourhood in the source. Thus a point in a clopen piece is smooth over $k$ there exactly when it is smooth in $X$ .

[A1]

The Axiom of Choice asserts that every family of nonempty sets has a choice function. It is inherited at [F1]–[F3], together with their fixed compatible size bounds. The finite sums and coordinate projections below require no additional choice principle.

Proof

1.1F1F2F3A1given

By [F1], all pieces are qcqs. Any irreducible subset lies in a single piece: otherwise its intersections with that piece and its clopen complement are a decomposition into two proper closed subsets. The irreducible components of each piece are therefore exactly the components of $X$ contained in it, so each piece is pure of dimension $d$ . A point of a piece is closed in $X$ exactly when it is closed in the piece, since the piece is closed and carries the subspace topology; smoothness agrees by [F3]. Hence the smooth closed points partition by the pieces. Under [A1], invoke [F1] for the actual restriction isomorphism $r : K_{0} (X) \to ⨁_{a} K_{0} (X_{a})$ . Its motivic-subgroup restriction is the first displayed isomorphism, in the convention of [F2]. For a smooth closed point $x \in X_{a}$ , [F3] gives its perfect residue-sheaf class on $X_{a}$ . The explicit zero-extension formula in [F1] gives the same residue sheaf on $X$ : on an open containing $x$ its sections are $κ (x)$ with the residue scalar action, and on an open missing $x$ they are zero. Thus $r$ sends this global point class to its class on $X_{a}$ and zero on every other piece. Every finite integer sum of point classes consequently maps into $⨁_{a} F_{s m}^{d} K_{0} (X_{a})$ . Conversely a tuple of componentwise finite sums lifts to the finite sum of these zero-extended point classes; the number of pieces is finite. Injectivity follows from that of $r$ . This proves the second displayed isomorphism under the very same map, also when a piece has no smooth closed points.

2.1F1F2F3step 1.1∎

If the two subgroups agree on each piece, their finite direct sums agree, so step 1.1 and injectivity of $r$ give equality on $X$ . Conversely equality on $X$ gives equality of those two direct sums under $r$ ; projection to the $a$ th coordinate gives equality on $X_{a}$ . For the stated normal pure-threefold application, use its already established finite clopen integral decomposition and set $d = 3$ . This is a conditional reduction of the equality, not a proof of its integral case.

Depends on

Used by · 1

Bertini curve functional with a prescribed avoidance locus

Dependency tree · 2 levels

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