This article describes an entirely algebraic construction for developing conformal geometries, which provide models for, among others, the Euclidean, spherical and hyperbolic geometries. On one hand, their relationship is usually shown analytically, through a framework comparing the measurement of distances and angles in Cayley-Klein geometries, including Lorentzian geometries, as done by F. Bachmann and later R. Struve. On the other hand, such a relationship may also be expressed in a purely linear algebraic manner, as explained by D. Hestens, H. Li and A. Rockwood. The model described in this article unifies these approaches via a generalization of Lie sphere geometry. Like the work of N. Wildberger, it is a purely algebraic construction, and as such it works over any field of odd characteristic. It is shown that measurement of distances and angles is an inherent property of the model that is easy to identify, and the possible models are classified over the real, complex and finite fields, and partially in characteristic 2, revealing a striking analogy between the real and finite geometries.
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.
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