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Categories of commutative algebras. (English) Zbl 0772.18001
Oxford Science Publications. Oxford: Clarendon Press. ix, 271 p. (1992).
This book, which could as well be named “Zariski categories”, starts from the point of view that the essence of commutative algebra, as we understand it nowadays, is not to be found in its calculus of individual rings, but in the overall, universal calculus in the category of commutative rings.
If one wishes to study the universal properties in categories of commutative algebras, which make commutative algebra “work”, then one is automatically lead to the notion of Zariski categories, as these are exactly the categories with such a structure on which basic commutative algebra and algebraic geometry can be developed. Moreover, as these are defined by universal properties, they enrich traditional commutative algebra by an increased input from concepts, techniques and tools stemming from category theory, thus adding new flexibility to many concepts of “classical” commutative algebra. As an example, schemes over an affine scheme $$\underline{\text{Spec}}(A)$$ correspond to schemes on the Zariski category $$\underline{\text{CAlg}}(A)$$ of commutative algebras over $$A$$, reduced schemes correspond to schemes on the Zariski category $$\underline{\text{RedCRing}}$$ 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 first two chapters present the basic machinery and could be viewed as the “commutative algebra” of Zariski categories. Chapters 3 to 6 treat spectra, schemes, Jacobson schemes and algebraic varieties within the context of Zariski categories and could be viewed as the basic “algebraic geometry” of Zariski categories. The last chapters are more specialized and treat topics like Zariski toposes, neatness, flatness, étale morphisms and Chevalley’s theorem. The topics covered in these chapters should convince the reader of the efficiency of the present set-up as well as stimulate him towards further research. 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.

##### MSC:
 18-02 Research exposition (monographs, survey articles) pertaining to category theory 18B25 Topoi 13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra 18F05 Local categories and functors 18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) 18E35 Localization of categories, calculus of fractions