Finitely coordinated and finitely copresentable affine algebraic sets (Q1575026)

This paper is in connection with the paper ``Lokal präsentierbare Kategorien'' of \textit{P. Gabriel} and \textit{F. Ulmer} [Lect. Notes Math. 221 (1971; Zbl 0225.18004)]. The notions of finitely presentable and finitely generated objects in an arbitrary category, introduced by Gabriel and Ulmer [op. cit.] work perfectly well in any algebraic category in the sense of \textit{F. W. Lawvere} [Proc. Natl. Acad. Sci. USA 50, 869-872 (1963; Zbl 0119.25901)]. But neither these notions nor their dual ones work as well in geometrical categories. In oder to describe the finite character of geometrical objects, the author introduces the notions of finitely coordinated and finitely copresentable objects, and then it is shown that these notions work well in geometrical categories of affine algebraic sets. Their connections with the finitely generated and finitely presentable objects of Gabriel-Ulmer are established. The notherian case, where finitely coordinated objects are identical to finitely copresentable ones, is studied.