By Durov N.

The important objective of this paintings is to supply an alternate algebraic framework for Arakelov geometry, and to illustrate its usefulness via offering a number of basic purposes.

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5. e. a classical Λ-algebra. In other words, we cannot obtain any “new” generalized algebras over a classical commutative ring. 18. ) Let’s list some generalized rings. g. Z, Q, R, C,. . • Classical commutative semirings: N0 , R≥0 , T,. . g. Z∞ ⊂ R, Z(∞) ⊂ Q, or AN := Z(∞) ∩ Z[1/N] ⊂ Q. • Archimedian valuation rings V ⊂ K, defined as follows. Let K be a classical field, equipped with an archimedian valuation |·|. Then V (n) = {(λ1 , . . 1) For example, Z(∞) ⊂ Q and Z∞ ⊂ R are special cases of this construction.

Under composition and base change. We have a minimal localization theory, namely, the unary localization theory T u , consisting only of unary localizations A → Af . We have a maximal localization theory, called the total localization theory T t , which contains 40 Overview all open pseudolocalizations. Any other localization theory is somewhere in between: T u ⊂ T ? ⊂ T t . 6. ) Once we fix a localization theory T ? , we can construct the T ? -spectrum Spec? A of a generalized ring A as follows.

At first glance commutativity of two operations of arity ≥ 2 seems to be not too useful. We are going to demonstrate that in fact such commutativity relations are very powerful, and that they actually imply classical associativity, commutativity and distributivity laws, thus providing a common generalization of all such laws. 13. ) Let’s start with some simple cases. Here Σ is some fixed algebraic monad. 5. Commutativity 33 • Two constants c, c′ ∈ Σ(0) commute iff c = c′ . In particular, a generalized ring contains at most one constant.