Lambda-rings and the field with one element

TL;DR

This paper develops Lambda-algebraic geometry, an extension of classical algebraic geometry over the integers, which models properties expected of geometry over the hypothetical field with one element, connecting to arithmetic algebraic geometry.

Contribution

It introduces Lambda-algebraic geometry as a robust framework that generalizes algebraic geometry over Z and aligns with properties predicted for the field with one element.

Findings

Lambda-algebraic geometry over Z exhibits properties of geometry over the field with one element.

The framework is formally robust and connected to current research in arithmetic algebraic geometry.

It extends classical algebraic geometry to a deeper algebraic setting.

Abstract

The theory of Lambda-rings, in the sense of Grothendieck's Riemann-Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual algebraic geometry over the ring Z of integers to produce Lambda-algebraic geometry. We show that Lambda-algebraic geometry is in a precise sense an algebraic geometry over a deeper base than Z and that it has many properties predicted for algebraic geometry over the mythical field with one element. Moreover, it does this is a way that is both formally robust and closely related to active areas in arithmetic algebraic geometry.