Algebraic geometry over the residue field of the infinite place

Abstract

Nikolai Durov introduced the theory of generalized rings and schemes to study Arakelov geometry in an alternative algebraic framework, and introduced the residue field at the infinite place, F∞. We show an elementary algebraic approach to modules and algebras over this object, define prime congruences, show that the polynomial ring of n variables is of Krull dimension n, and derive a prime decomposition theorem for these primes.Proof. These are all trivial consequences of theorems in universal algebra.Proposition 2.11. Hom(M 1 , M 2 ) has a natural F-structure.Proof. The pointwise sum and scalar multiple of two homomorphisms is a homomorphism.Definition 2.12. Given two F-modules M 1 and M 2 , the tensor product M 1 ⊗ M 2 is a module with a natural bilinear map M 1 ×M 2 → M 1 ⊗M 2 where M 1 ×M 2 is the set of pairs, such that for any bilinear map M 1 × M 2 → N for some other module N , there is exactly one map M 1 ⊗ M 2 → N that makes the diagramProposition 2.13. Given two F-modules M 1 and M 2 , M 1 ⊗ M 2 exists and is unique. It is generated by elements of the form m 1 ⊗ m 2 with m 1 ∈ M 1 and m 2 ∈ M 2 . Furthermore · ⊗ M is a covariant functor.

Non-Noetherian representation categories of generalized fields

“…In this section, we introduce our motivating example, the category of finitely generated F∞-modules. Our main reference for this section is [8].…”

“…In the theory of Durov, the generalized field, F∞ is the residue field at infinity, while the Boolean semi-ring B is treated as a semi-field of characteristic one [5]. Even though basic algebro-geometric properties of the generalized fields, B and F∞ resemble the properties of usual finite fields [6,10,8], the homological properties are far from ordinary [5,15].…”