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An integral P³ test fixes the supported RR coefficient as positive two

PDF p. 17: Corollary6.9 supported factor calculation

Statement

Assume the Axiom of Choice and the compatible site-size and universe bounds of the supplied motivic and perfect K-theory constructions. For the RR polynomial of codimension $d = 3$ and class index $q = 3$ , $P_{3}^{3} (n) = 2 n$ for every integer rank $n$ . In particular the supported third Chern class of the positive point K-generator is twice the positive motivic point generator. The coefficient is integral and the same in every characteristic.

Facts & Assumptions

[A1]

Assume The Axiom of Choice and fix compatible universes and small scheme sites as in Scheme sites and sheafification. These are the foundational assumptions of the projective-bundle, smooth-comparison, point-purity, virtual-Chern and actual perfect K-space suppliers in [F1]–[F3], used in steps 1.1–3.1. The finite Koszul resolution and integral formal-series arithmetic require no additional choice.

[F1]

External actual supported Riemann–Roch in codimension three gives the actual supported formula, and Actual supported Chern operation and its smooth comparison identifies the actual supported Chern operation and its compatibility with forgetting support through the smooth comparison. When $q = d = 3$ , every positive Chern/normal variable disappears and only rank remains.

[F2]

By Smooth motivic comparison, $H^{2 j, j} (k) = C H^{j} (k) = 0$ for $j > 0$ and $H^{0, 0} (k) = Z$ . The projective-space formula of Motivic projective bundle formula gives the diagonal ring on $P_{k}^{3}$ with hyperplane $h = c_{1} (O (1))$ and $H^{6, 3} = Z h^{3}$ . The triple coordinate-hyperplane point has positive fundamental class $h^{3}$ by normalized divisor Gysin and its functoriality, included in [F1]. Motivic purity and lower-weight vanishing at a smooth closed point gives $H_{x}^{6, 3} (P^{3}) = Z δ_{x}$ and forgetting support sends $δ_{x}$ to $h^{3}$ , hence is injective.

[F3]

The actual perfect K-space operations satisfy total Cartan supplies the actual perfect total-series homomorphism on every noetherian finite-dimensional scheme, hence on the nonaffine scheme $P_{k}^{3}$ . Its bundle values are the original projective-relation classes. Thus the virtual bundle rules of Whitney identities and virtual Chern arithmetic and Total Chern series for actual and virtual bundles compute these actual perfect operations on the explicit Koszul class and on all its integer multiples. No global bundle-representation assertion for arbitrary qcqs schemes is used.

Proof

Given: [A1], any algebraically closed $k$ , a coordinate point $x \in P_{k}^{3}$ , and integer $n$ .

Goal: Determine the universal integer polynomial value without a sign convention guess or division.

1.1A1F2F3construct

The three coordinate linear sections cut out $x$ regularly. Their Koszul resolution gives $[k (x)] = 1 - 3 [O (- 1)] + 3 [O (- 2)] - [O (- 3)]$ in actual perfect $K_{0} (P^{3})$ . The actual perfect total Chern operation on this explicit Koszul class is computed by [F3] from the line-bundle values in [F2], giving $(1 - 2 h t)^{3} / ((1 - h t)^{3} (1 - 3 h t))$ .

2.1A1F2F3step 1.1

Expand only through degree3 in the integral formal-series ring. The numerator is $1 - 6 h t + 12 h^{2} t^{2} - 8 h^{3} t^{3}$ ; the denominator is $1 - 6 h t + 12 h^{2} t^{2} - 10 h^{3} t^{3}$ modulo $t^{4}$ . Thus their quotient is $1 + 2 h^{3} t^{3}$ modulo $t^{4}$ , with first two coefficients zero. For $n$ times this perfect class, the actual total Cartan identity of [F3] gives the $n$ th power of this series, whose degree-three coefficient is $2 n h^{3}$ , including negative $n$ by formal inversion.

3.1A1F1F2F3step 2.1∎

Apply supported RR [F1] to $a = n \in K_{0} (k) = Z$ and forget support. Its right side is $P_{3}^{3} (n) h^{3}$ , while step 2.1 gives $2 n h^{3}$ . Since $h^{3}$ freely generates $H^{6, 3} (P^{3})$ , their equality gives $P_{3}^{3} (n) = 2 n$ . The injective support-forgetting map in [F2] also fixes its supported sign. This test determines the universal polynomial value for every regular codimension-three point pair.

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Sources

Fasel–Yakerson, Vector bundles on singular surfaces and threefolds · Printed p. 17, Corollary 6.9 and proof
Alberto Navarro, On the higher Riemann–Roch without denominators, v1 · Example 1.1; sections 2.1–2.7; Theorem 3.1 and proof, printed pp. 3–8