Cyclic Adams Operations

TL;DR

This paper introduces cyclic Adams operations on the Grothendieck group of complexes over a Noetherian ring, satisfying key axioms and providing a new proof of Serre's Vanishing Conjecture, while aligning with existing Adams operations in specific cases.

Contribution

It develops cyclic Adams operations on $K_0^Z(Q)$ that satisfy Gillet and Soulé's axioms, offering a shorter proof of Serre's Vanishing Conjecture and establishing agreement with prior Adams operations.

Findings

Cyclic Adams operations satisfy Gillet and Soulé's axioms.

Provided a shorter proof of Serre's Vanishing Conjecture.

Showed agreement with existing Adams operations in certain cases.

Abstract

Let $Q$ be a commutative, Noetherian ring and $Z \subseteq \operatorname{Spec}(Q)$ a closed subset. Define $K_0^Z(Q)$ to be the Grothendieck group of those bounded complexes of finitely generated projective $Q$-modules that have homology supported on $Z$. We develop "cyclic" Adams operations on $K_0^Z(Q)$ and we prove these operations satisfy the four axioms used by Gillet and Soul\'e in their paper "Intersection Theory Using Adams Operations". From this we recover a shorter proof of Serre's Vanishing Conjecture. We also show our cyclic Adams operations agree with the Adams operations defined by Gillet and Soul\'e in certain cases.