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External supported slice projection with positive generic coefficient

PDF p. 16: Supported leading weight two

Statement

Let $k$ be perfect, let $V$ be a smooth quasi-projective finite-type $k$ -scheme, and let $W \subset V$ be integral closed support of codimension $q$ . In classical slice K-theory the actual stage augmentation $ϵ_{q} : (f_{q} K)_{W} (V) \to K_{W} (V)$ is an equivalence. Set $ψ_{W} = π_{0} ((f_{q} K)_{W} (V) \to (s_{q} K)_{W} (V)) \circ (π_{0} ϵ_{q})^{- 1} .$ The canonical slice-to-cycle comparison identifies the generic coefficient of $ψ_{W}$ with $K_{0}^{W} (V) ⟶ G_{0} (W) ⟶ K_{0} (k (W)) \overset{\dim}{\to} Z .$ The first arrow is the coherent Euler/devissage comparison, the second is actual generic restriction, and dimension sends $[k (W)]$ to $+ 1$ . This statement includes support: the q-th supported slice restricts equivalently to its codimension-q generic-point coefficient. It does not identify a supported class with its possibly zero image in the ambient Chow group.

Source contract

Given: Exactly the perfect-field, smooth quasi-projective ambient and integral codimension-q support hypotheses above.

Goal: Import the exact actual supported projection and its positive cycle normalization.

[F1] Kahn–Levine Proposition 2.2.2 gives the compatible coniveau/slice towers; Lemmas 2.3.1–2.3.2 give the supported augmentation equivalence and generic detection. Definition 3.1.5 defines precisely $ψ_{W}$ .

[F2] Their Definitions 3.1.9–3.1.15, Theorem 3.2.4 and Lemma 6.4.2 identify the slice-cycle map with the support-pair K-to-G map and generic restriction constructed explicitly in Lemma 6.2.1.

[F3] Section 5.9 identifies $K_{0} (F)$ with $Z$ by dimension, compatible with the Serre face convention. In the ordinary case no reduced-norm scaling or negative generator is inserted.

External input

Import the restricted package [F1]–[F3], assembled from the displayed source constructions and commuting diagrams with exactly their hypotheses. It supplies a map and its coefficient, not an inference from equality of groups. Its application to a smooth ambient that is not quasi-projective is made only after restricting to an affine neighborhood of the generic point and using the local detection lemma; quasi-projectivity is not added as a global hypothesis. The full coniveau, purity and moving-lemma constructions are substantial external inputs and are not locally rebuilt.

The S1 coniveau/slice construction is connected to the P1-stable KGL tower by the actual augmentation-adjoint comparison of Levine section8.3/Theorem9.0.3, as specified in the dedicated reference; this stabilization is an imported map-level input, not an identification inferred from the equality of cohomology groups.

Depends on

Used by · 2

Dependency tree · 1 level

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.

External dependencies

Bruno Kahn and Marc Levine, Motives of Azumaya algebras, JIMJ 9 (2010), 481–599, published final PDF · Notation p.487; Proposition 2.2.2 p.496 and compatible tower diagram p.498; Lemmas 2.3.1–2.3.2 pp.499–500; Definition 3.1.5 p.503.

Over a perfect field, coniveau and slice tower comparisons are isomorphisms of towers natural for smooth maps. For closed support W of codimension at least q, the actual augmentation (f_q E)_W(Y)→E_W(Y) is an equivalence. Definition 3.1.5 defines the supported leading map as its inverse on pi0 followed by the slice projection to (s_q E)_W(Y). The q-th supported slice is detected by its codimension-q generic points.

Hypotheses: Part I works over a perfect field; Sm/k in this published source means smooth quasi-projective finite-type k-schemes. W is any closed subset, not required smooth. For the augmentation codim W≥q; generic detection in weight q uses the generic points of W of codimension exactly q. In the application only affine open neighborhoods of the generic point of C are used.

Reading limits: Read actual published final PDF pp.487,494–500 and complete Definition 3.1.5 on p.503, including displayed proof context and augmentation diagrams. The coniveau moving-lemma and slice constructions referenced there are imported, not reconstructed; no full-paper reading is claimed.

Kahn–Levine, published construction of the cycle map and its K/G comparison · Definitions 3.1.9–3.1.15 pp.505–507; Theorem 3.2.4 and its complete proof pp.508–510; Lemma 6.2.1 p.535; Lemmas 6.4.1–6.4.3 pp.539–540.

The source constructs supported cycle maps by restriction to the codimension-q generic points. Its K-to-G natural transformation is defined on support pairs and commutes with these cycle maps. For ordinary K-theory (Azumaya algebra A=k), the slice-cycle map is an equivalence and its degree-zero coefficient is the restriction from G0(W) to G0(k(w)). This identifies the actual inverse-augmentation slice projection, not merely its abstract target group.

Hypotheses: Specialize the source Azumaya algebra to A=k. The ambient scheme is smooth quasi-projective over the perfect field, W is closed support and may be singular. q≥0. The comparison is through the actual K_W→G_W morphisms on the category of support pairs.

Reading limits: Read the complete displayed definitions, the commutative diagrams and proof of Theorem 3.2.4, and the complete statements/displayed proofs of Lemmas 6.2.1 and 6.4.1–6.4.3. The referenced well-connectedness and moving-lemma foundations are not locally reconstructed. The current perfect-spectrum pi0 comparison is supplied by the already authored devissage/model contracts, not invented as a new abstract isomorphism here.

Kahn–Levine, positive cycle coefficient and face convention · Section 5.9, printed pp.531–532, before Lemma 5.9.1; for orientation context Theorem 6.5.5 and its proof pp.542–543.

For A=k, K0(F) is canonically Z by the dimension function, so a one-dimensional F-vector space represents the positive cycle generator. Face pullbacks agree with Serre intersection multiplicities. The slice identification is built from these actual cycle maps, not an independent generator choice.

Hypotheses: Ordinary K-theory A=k; residue fields F; admissible cycles for the face statement. The slice theorem uses a perfect base field. No normality of the supported subvariety is imposed.

Reading limits: Read the complete indicated coefficient/face passage and Theorem 6.5.5 statement and displayed proof. Its general Azumaya and motive-theoretic applications are not imported. No invalid old rank-trivialization display is reproduced or used.

Marc Levine, The homotopy coniveau tower, arXiv math/0510334v1 · Section8.3 pp.46–47, equation(8.3.1); Theorem9.0.3 and Remark9.0.4 p.47; model conventions pp.44–45

The stable homotopy coniveau tower is built termwise from the S1 coniveau towers with coherent P1-loop bonding maps. Its actual augmentation-adjoint map phi_p E→f_p E is an isomorphism over a perfect field. For Bott-periodic K this transports the supported coniveau/slice projection to the KGL slice tower used by the canonical EM bridge; applying support fibers preserves the actual augmentation and projection.

Hypotheses: Perfect field, E in SH(k); the construction uses a fibrant P1-Omega-spectrum. K is its usual Bott-periodic KGL realization, whose termwise K-space convention is the classical slice K convention in the EM comparison. Remark9.0.4 explicitly handles finite perfect fields through axiomA3; no infinite-field strengthening is inserted.

Reading limits: Read actual section8.3, Theorem9.0.3 statement and accompanying Remark9.0.4 plus pp.44–45 model conventions. The proof of Theorem9.0.3 in section9.1 is imported, not reconstructed. The separate v1 Corollary4.2.4 proof-display error is retained as a finding, not used as a locally valid proof step. No claim to have read its 2008 published version.

Sources

Bruno Kahn and Marc Levine, Motives of Azumaya algebras, JIMJ 9 (2010), 481–599, published final PDF · Notation p.487; Proposition 2.2.2 p.496 and compatible tower diagram p.498; Lemmas 2.3.1–2.3.2 pp.499–500; Definition 3.1.5 p.503.
Kahn–Levine, published construction of the cycle map and its K/G comparison · Definitions 3.1.9–3.1.15 pp.505–507; Theorem 3.2.4 and its complete proof pp.508–510; Lemma 6.2.1 p.535; Lemmas 6.4.1–6.4.3 pp.539–540.
Kahn–Levine, positive cycle coefficient and face convention · Section 5.9, printed pp.531–532, before Lemma 5.9.1; for orientation context Theorem 6.5.5 and its proof pp.542–543.
Marc Levine, The homotopy coniveau tower, arXiv math/0510334v1 · Section8.3 pp.46–47, equation(8.3.1); Theorem9.0.3 and Remark9.0.4 p.47; model conventions pp.44–45