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Generic localization detects the point subgroup

PDF p. 16: Final paragraph of Theorem 6.7 proof

Statement

Assume the Axiom of Choice and the fixed universe bounds of the completed comparison suppliers. Let $k$ be a perfect field, $U$ a smooth finite-type $k$ -scheme of pure dimension three, and $i : C ↪ U$ an integral closed curve of codimension two, with generic point $η$ . No smoothness or normality of $C$ is assumed. Use the actual Elmanto–Morrow filtered nonconnective perfect-complex $K$ -theory and write $K_{C} (U) = fib (K (U) \to K (U ∖ C))$ and $F_{C}^{j} K (U) = fib (F^{j} K (U) \to F^{j} K (U ∖ C))$ .

The group $π_{0} K_{C} (U)$ is canonically the Grothendieck group of perfect complexes on $U$ supported on $C$ . Under this identification, the arrow in the supported exact sequence $π_{0} F_{C}^{3} K (U) ⟶ π_{0} K_{C} (U) ⟶ Z [C] ≅ Z$ is generic localization: it sends a supported perfect complex $E$ to $\sum_{n} (- 1)^{n} {length}_{O_{U, η}} H^{n} (E)_{η}$ , and the positive cycle $[C]$ corresponds to $1$ . Its kernel, equivalently the image of $π_{0} F_{C}^{3} K (U)$ , is the subgroup generated by the classes of perfect complexes supported at closed points of $C$ . Indeed it is generated by the individual residue-sheaf classes $[i_{x *} κ (x)]$ , where $i_{x} : Spec κ (x) ↪ U$ and $x$ ranges over closed points of $C$ .

If $U$ is an open subscheme of a finite-type $k$ -scheme $X$ , these points are closed smooth points of $X$ . In particular this applies to a curve contained in the smooth locus in the article's algebraically closed-field setting.

Here integral means reduced and irreducible; a curve is one-dimensional. A perfect complex is a complex locally quasi-isomorphic to a bounded complex of finite locally free sheaves. It is supported on a subset when all its cohomology stalks vanish outside that subset. Its Grothendieck group has one generator for each perfect object and the usual distinguished-triangle relations. The length of a finite-length module is the number of simple factors in a composition series. Generic localization is the map induced by restriction to the generic point, expressed through coherent $G$ -theory as generic rank; the alternating-length formula specifies the same map on supported perfect objects.

Facts & Assumptions

Given: AC, the fixed universe bounds, $k$ , $U$ , $C$ and the actual supported filtration as in the Statement. For the final assertion, also let $U \subseteq X$ be an open immersion into a finite-type $k$ -scheme.

Goal: Identify the actual supported sequence and its second arrow, compute its kernel by closed-point classes, and establish the stated smooth-point conclusion.

[F1]

Second stage with integral curve support supplies the actual equivalence $ϵ_{2} : F_{C}^{2} K (U) ≃ K_{C} (U)$ and the exact sequence after transport along $ϵ_{2}$ . Smooth rank, first Chern, determinant and supported positive leading cycle identifies the transported projection with alternating generic length times the positive supported cycle $[C]$ . These suppliers require the stated perfect-field, smoothness, dimension, AC and universe hypotheses.

[F2]

Perfect K theory localization and open excision identifies the category-defined supported nonconnective $K$ -spectrum with the actual restriction fiber for a qcqs scheme and a quasi-compact open complement. Dévissage of K zero with regular ambient support identifies its degree-zero group with $G_{0} (C)$ and sends the coherent class $[F]$ inversely to the class of $i_{*} F [0]$ . Here $G_{0} (C)$ is generated by coherent sheaves with short-exact-sequence relations. The ambient scheme must be regular Noetherian and finite-dimensional.

[F3]

Generic rank and point-supported classes on a curve states, under AC, that for an integral Noetherian one-dimensional scheme the kernel of generic rank on $G_{0}$ is generated by the residue sheaves of closed points. Under the supported comparison it identifies generic rank with alternating generic length; its localization description is the filtered limit over nonempty opens.

[F4]

Perfect residue classes at smooth closed points proves under AC that the residue sheaf of a closed smooth point of a finite-type scheme over a perfect field is perfect and supported at that point, using a local Koszul resolution.

Proof

1.1givenF2

A finite-type scheme over a field is Noetherian, quasi-compact and quasi-separated; $U ∖ C$ is quasi-compact because $U$ is Noetherian. Smoothness over the perfect field makes $U$ regular, and its dimension is finite. Thus the hypotheses of [F2] hold. Its localization equivalence identifies the actual fiber $K_{C} (U)$ with the $K$ -spectrum of the full category of perfect complexes supported on $C$ . The degree-zero comparison and triangle presentation in the supported dévissage identify its $π_{0}$ with the stated Grothendieck group and canonically with $G_{0} (C)$ .

2.1F1F3step 1.1

The pair $C \subset U$ satisfies all hypotheses of [F1]. Transporting the supported graded exact sequence along the actual augmentation $ϵ_{2}$ gives the displayed sequence. By [F1] its second arrow takes $[E]$ to the alternating generic-length coefficient times $[C]$ , with positive generator. By [F3], under the specific $G_{0} (C)$ comparison of step 1.1 this coefficient is generic rank, equivalently localization to the generic point in the stated filtered-limit sense. This identifies the actual arrow, rather than inferring it from an abstract isomorphism of its target with $Z$ .

3.1F2F3F4step 2.1

By [F3], the kernel of generic rank on $G_{0} (C)$ is generated by $[κ (x)]$ for closed points $x$ of $C$ . Since $C$ is closed in $U$ , each such $x$ is closed in $U$ . The inverse in [F2] sends $[κ (x)]$ to the class of the same residue sheaf, viewed as a perfect complex on $U$ in degree zero. It is supported at $x$ ; [F4] independently supplies its local perfectness. Consequently the kernel is generated by these point classes. Conversely every perfect complex supported at a closed point has zero stalk at $η$ , so its alternating generic length is zero. Hence the subgroup generated by all such complexes is exactly the kernel. Exactness in step 2.1 identifies this kernel with the image of $π_{0} F_{C}^{3} K (U)$ .

4.1givenstep 3.1∎

Suppose $U$ is open in a finite-type $k$ -scheme $X$ . A closed point of the finite-type $k$ -scheme $U$ has residue field finite over $k$ . Its image in $X$ is therefore a closed point: on an affine neighbourhood its prime has field quotient finite over $k$ , hence is maximal. Since $U$ is smooth over $k$ , $X$ is smooth over $k$ at each such point by locality of smoothness on the source. Thus all generators in step 3.1 are supported at closed smooth points of $X$ . This proves the final assertion and the lemma.

Depends on

Used by · 1

Smooth points generate the third motivic piece

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.

Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · p. 16, final paragraph of the proof of Theorem 6.7