Lambda-rings and the field with one element

Details Lambda-rings and the field with one element Lambda-rings and the field with one element

2009-06-17

arxiv.org/abs/0906.3146v1

Mathematics

Number Theory

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

Scientific paper

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.

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