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The Zariski category of graded commutative rings. (English) Zbl 0798.18008
Seely, R. A. G. (ed.), Category theory 1991. Proceedings of an international summer category theory meeting, held in Montréal, Québec, Canada, June 23-30, 1991. Providence, RI: American Mathematical Society. CMS Conf. Proc. 13, 171-181 (1992).
The main object of study in this paper is the category GradCRng of \(\mathbb{Z}\)-graded commutative rings with units. Using properties of the forgetful functor form GradCRng to the category of commutative rings with units, and a right adjoint of this functor, it is shown that GradCRng is a Zariski category, as defined in the author’s book [“Categories of commutative algebras” (Oxford 1992; Zbl 0772.18001)]. Simple and algebraically closed simple objects of GradCRng are identified as certain graded fields, and they are used to formulate and prove a version of the Nullstellensatz for GradCRng. The author also examines graded prime spectra and graded schemes, i.e., schemes on GradCRng. In particular, the restriction to the 0th homogeneous component yields a morphism of Zariski categories which extends to a morphism of the corresponding categories of schemes mapping \(\text{Proj}_{gr}(A)\) to \(\text{Proj}(A)\).
For the entire collection see [Zbl 0771.00047].

18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
13A02 Graded rings
14A05 Relevant commutative algebra
18B99 Special categories