On the Adams–Riemann–Roch theorem in positive characteristic

Abstract

We give a new proof of the Adams-Riemann-Roch theorem for a smooth projective morphism X → Y , in the situation where Y is a regular scheme, which is quasi-projective over F p . We also partially answer a question of B. Köck.

The Deligne pairing and a functorial Riemann Roch theorem in positive characteristic

“…It can be showed that τ (E) satisfies the defining properties of the p-th Bott element. In other words, we have the following proposition (see [PR12], Prop. 2.6).…”

“…The all ingredient will be used in Section 4. On a quasi-compact scheme of characteristic p > 0, Pink and Rössler constructed an explicit representative of the p-th Bott element (see [PR12], Sect. 2).…”

“…Let Ω f be the sheaf of relative differentials of f . Also, they proved the following key proposition which can be used to prove the p-th Adams-Riemann-Roch theorem in positive characteristic (see [PR12], Prop. 3.2).…”

“…Remark 3.10. In [PR12], there is an inverse for a vector bundle E in K(X) Q of a regular scheme X because the Grothendieck-γ filtration is finer than the topological filtration (see Chap.III [FS85]) so that for any vector bundle…”

“…In [PR12], in the setting of the positive characteristic, R. Pink and D. Rössler proved the following version of the Adams-Riemann-Roch theorem: Let f : X → Y be projective and smooth of relative dimension r, where Y is a quasi-compact scheme of characteristic p > 0 and carries an ample invertible sheaf. Then the following equality…”