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Dévissage of K zero with regular ambient support

PDF p. 15: Supported degree-zero dévissage

Statement

Assume the Axiom of Choice and the fixed compatible site and universe bounds of the supplied constructions. Let $U$ be a regular Noetherian finite-dimensional scheme and $i : T ↪ U$ any closed subscheme with defining ideal $I$ . Let $K_{0}^{∣ T ∣} (U) = π_{0} K_{∣ T ∣} (U)$ for the specified nonconnective K-theory of the current supported perfect category. There is a canonical isomorphism $K_{0}^{∣ T ∣} (U) \tilde{\to} G_{0} (T),$ where $G_{0} (T)$ is the Grothendieck group of coherent sheaves on $T$ with short exact relations. For a supported perfect object $E$ it is the coherent-cohomology Euler class followed by ideal-power dévissage, explicitly $[E] ⟼ \sum_{n} (- 1)^{n} \sum_{r \geq 0} [I^{r} H^{n} (E) / I^{r + 1} H^{n} (E)] .$ Each quotient is regarded as a coherent $O_{T}$ -module; both sums are finite after deleting zero terms, and the resulting class is independent of the chosen annihilating powers. The inverse sends $[F]$ to the class of the perfect $U$ -object $i_{*} F [0]$ .

Under this isomorphism the support-to-global map $K_{0}^{∣ T ∣} (U) \to K_{0} (U)$ is coherent pushforward $G_{0} (T) \overset{i_{*}}{\to} G_{0} (U)$ followed by the regular ambient perfect comparison $G_{0} (U) \tilde{\to} K_{0} (U)$ , sending $[M]$ to $[M [0]]$ . No regularity of $T$ , global vector-bundle resolution or resolution-property assumption is imposed. No analogous identification for singular ambient $U$ is asserted.

Statement references

Nonconnective K theory of perfect complexes

Postnikov inverse and the specified nonconnective degree-zero group

Ideal-power dévissage and compatibility with coherent pushforward

Facts & Assumptions

[A1]

The Choice assumption in Given and the stated universes are used through the essentially small perfect-category presentations and the specified nonconnective K-theory model. The Axiom of Choice Given: $U, T, i, I$ as in the Statement, AC and fixed suitable universes. The supported category is the supplied full sheaf-derived ${P e r f}_{∣ T ∣} (U)$ , not complexes internally constructed in a coherent abelian category.

Goal: Prove the stated comparison and support-to-global compatibility in the current nonconnective model.

[F1]

Inherited bounded t structure on supported perfect complexes, Additivity of the supported coherent Euler class and Postnikov inverse and the specified nonconnective degree-zero group give the actual bounded supported t-structure, additive coherent Euler map and Postnikov inverse, together with the general spectral $π_{0}$ premise Nonconnective K theory in degree zero.

[F2]

Finite ideal-power filtrations and coherent sheaves on the closed subscheme and Ideal-power dévissage and compatibility with coherent pushforward give finite ideal-power dévissage, its inverse formula and coherent pushforward compatibility.

[F3]

Perfect complexes on schemes and their support and Nonconnective K theory of perfect complexes supply the full stable idempotent-complete supported categories, their essentially small presentations and the actual support inclusion. Cofiber and triangle presentations of K zero identifies object/cofiber and triangle presentations.

Proof

1.1F1F2F3A1

By [F1], the degree-zero group of the current nonconnective functor identifies with $G_{0} (A_{T})$ by the coherent-cohomology Euler map. By [F2], closed coherent pushforward gives $G_{0} (T) ≃ G_{0} (A_{T})$ . Compose the former map with the inverse of the latter. This is an isomorphism, and its displayed formula is exactly the Euler sum followed by the ideal-power quotient sum. Finiteness and independence follow from [F1]–[F2].

2.1F1F2F3step 1.1

The two inverse constructions take $[F]$ first to $[i_{*} F]$ and then to the class of $i_{*} F [0]$ by [F1]–[F2]. This object is perfect by the imported local regular-perfect theorem applied to its coherent degree-zero cohomology. Thus the inverse has the asserted actual-object interpretation without any global locally free resolution.

3.1F1F2F3step 1.1step 2.1∎

The Euler/Postnikov comparisons commute with supported-to-global inclusion by [F1]. Under abelian dévissage the corresponding map is precisely coherent pushforward by [F2]. Their commutative squares compose to the stated support-to-global formula, using the same ambient comparison with $T = U$ . This establishes the full proposition in the specified nonconnective model.

Depends on

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