This book studies the universal constructions and properties in categories of commutative algebras, bringing out the specific properties that make commutative algebra and algebraic geometry work. Two universal constructions are presented and used here for the first time. The author shows that the concepts and constructions arising in commutative algebra and algebraic geometry are not bound so tightly to the absolute universe of rings, but possess a universality that is independent of them and can be interpreted in various categories of discourse. This brings new flexibility to classical commutative algebra and affords the possibility of extending the domain of validity and the application of the vast number of results obtained in classical commutative algebra. This innovative and original work will interest mathematicians in a range of specialities, including algebraists, categoricians, and algebraic geometers.
'If you prefer your Zariski categories to be cosliced and your prelocal morphisms to be co-universal, or equivalently, interminable, then this is the book for you. If not, then you probably won't appreciate the elegance of the exposition.'
Mathematika, 39 (1992)
'reduced schemes correspond to schemes on the Zariski category RedCRing of reduced commutative rings, etc., thus making the notion of a scheme even more natural in the present context. The book formalizes this point of view in a very elegant way, providing a wide variety of well-chosen examples to make it attractive to a broad, mixed audience. The last chapter provides methods to construct new Zariski categories from known ones, proving, once
again, the universality and wide applicability of the techniques covered in this book.'
A. Verschoren, Mathematics Abstracts, 772/93
Introduction; Zariski categories; Classical objects; Spectra; Schemes; Jacobson ultraschemes; Algebraic varieties; Zariski toposes; Neat objects and morphisms; Flatness properties; Etale objects and morphisms; Terminators; Some constructions of Zariski categories.