Lambda-rings and the field with one element

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.

Preliminary version released in advance of the Frobenius Lifts workshop. All comments are welcome!

References in corpus (3)

• New Approach to Arakelov Geometry
• Les variétés sur le corps à un élément
• Cyclotomy and analytic geometry over F_1

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