Abstract
We give a systematic study of notions of real algebra analogous to those of local rings, valuation rings, localizations and structural sheaf on the Zariski spectrum studied in commutat ive algebra. This study provides a simpler presentation of the theory in [9] : the properties of real strict localizations and of the structural sheaf on the real spectrum. In the geometric case real strict localizations are rings of germs of Nash functions and the structural sheaf is the sheaf of Nash functions. Except Theorem 3.3.3 there are few new results in this paper. The proofs we present here are different and we hope simpler than those in [9]. Logical considerations are replaced by local algebraic arguments. Global results on the structural sheaf are obtained from properties of the real strict localizations by using the compactness of the constructible topology in the real spectrum.
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