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The actual EM Chern operation on the standard point

PDF p. 17: Corollary6.9 supported factor calculation

Statement

Assume the Axiom of Choice and the compatible nested universe and site bounds of the supplied actual EM coefficient and perfect K-theory constructions. Over any algebraically closed field $k$ , let $P = P_{k}^{3}$ and $p = [1 : 0 : 0 : 0]$ . Use the actual supported Chern operation, the residue-object generator $u_{p} = [O_{p}]$ and the positive cycle generator $δ_{p}$ selected by the supplied comparison $Ψ$ . Then, for every integer $n$ , $c_{3, p} (n u_{p}) = 2 n δ_{p} .$ On the standard affine chart $D_{+} (X_{0}) = A_{k}^{3}$ this gives the same formula at the origin, with the same positive generators under open excision. There $u_{0}$ is represented by the ordinary coordinate Koszul resolution of $k$ over $k [t_{1}, t_{2}, t_{3}]$ .

Facts & Assumptions

[A1]

Assume The Axiom of Choice and the compatible nested universes containing $ω$ , the fields, schemes, small sites and enhanced categories required by the supplied constructions. Their existence is a separate size assumption. These are exactly the construction assumptions retained in [F1]–[F3].

[F1]

An integral P³ test fixes the supported RR coefficient as positive two, Facts F0, F2–F5 and Proof 1.1–2.1, provides the independent projective test calculation for the actual EM operations. Its F0 checks that this $P$ is a smooth pure three-dimensional Noetherian finite-type qcqs scheme. With $h = c_{1} (O (1))$ , its F2 gives $H_{m o t}^{6, 3} (P, Z) = Z h^{3}$ . Its coordinate Koszul resolution gives $[O_{p}] = 1 - 3 [O (- 1)] + 3 [O (- 2)] - [O (- 3)]$ in actual perfect $K_{0} (P)$ . Its actual total-Cartan calculation, including negative integer multiples, gives $c_{3} (n [O_{p}]) = 2 n h^{3}$ . Only those subarguments are used; the later RR application in that supplier's Proof 3.1 is not used.

[F2]

Positive supported coordinates at a smooth threefold point gives $K_{0, p} (P) = Z u_{p}$ and $H_{p}^{6, 3} (P, Z) = Z δ_{p}$ , with their actual excision maps and their residue/cycle normalizations. The coordinate-flag normalization in [F1]'s F2 gives $ι_{H} (δ_{p}) = h^{3}$ : its three smooth Cartier-divisor pushforwards give the positive global hyperplane cube. The specified refined Gysin, coefficient and forget-support comparisons used in that flag argument are supplied by The field comparison preserves refined smooth Gysin and the point class. This uses a Gysin comparison of coefficient theories, not an identification of their independently constructed K-space Chern operations.

[F3]

Actual supported Chern operation and its smooth comparison gives the actual Chern support-to-global square and its open-excision compatibility. In particular $ι_{H} c_{3, p} (z) = c_{3} (ι_{K} z)$ . Its qcqs-complement hypothesis holds on $P$ and its affine chart by [F1].

Proof

Given: The field, projective test pair, actual operations, Choice and size assumptions above.

Goal: Prove $c_{3, p} (n u_{p}) = 2 n δ_{p}$ for every integer $n$ , and the stated affine-chart formula and Koszul representative.

1.1A1F1

Use the Koszul and actual Cartan subarguments of [F1]. Explicitly, the total series of the displayed residue-object class is $(1 - 2 h t)^{3} / ((1 - h t)^{3} (1 - 3 h t))$ . Through degree three its numerator is $1 - 6 h t + 12 h^{2} t^{2} - 8 h^{3} t^{3}$ and its denominator is $1 - 6 h t + 12 h^{2} t^{2} - 10 h^{3} t^{3}$ , so its quotient is $1 + 2 h^{3} t^{3}$ modulo $t^{4}$ . Here congruence means equality of coefficients through degree three. The actual total-series homomorphism in [F1] makes the series of $n [O_{p}]$ its $n$ th power. Modulo $t^{4}$ this changes the degree-three coefficient to $2 n h^{3}$ , also for negative $n$ , since the inverse of $1 + a t^{3}$ is $1 - a t^{3}$ in that quotient. No integer is inverted. Thus $c_{3} (n [O_{p}]) = 2 n h^{3}$ in the original coefficient group.

2.1F1F2F3step 1.1∎

By [F2], the support-forgetting map from $Z δ_{p}$ to $Z h^{3}$ sends its specified generator to its specified generator and is injective. By [F3] and step 1.1, both $c_{3, p} (n u_{p})$ and $2 n δ_{p}$ map to $2 n h^{3}$ ; injectivity gives the asserted supported equality. Restrict to the standard affine chart. The excision comparisons in [F2]–[F3] preserve the actual operation and both positive generators, so the origin has the same formula. The restriction of the coordinate Koszul resolution in [F1] is precisely the coordinate Koszul resolution on that chart. This argument uses no independently constructed RR Chern operation.

Depends on

Used by · 1

The actual EM point calculation through an étale chart

Dependency tree · 2 levels

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