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Proceedings of the American Mathematical Society

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A functor to ringed spaces

Author: Gail L. Carns
Journal: Proc. Amer. Math. Soc. 29 (1971), 222-228
MSC: Primary 18.20
MathSciNet review: 0276304
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Abstract: With the set of orders $ \mathcal{O}$ on a field and the Harrison topology induced from the set of all primes as a base space we define a ringed space $ (\mathcal{O},\mathcal{F})$. For each field homomorphism we find an associated ringed space morphism producing a contravariant functor from the category of fields to the category of ringed spaces. An equivalence relation $ \sim $ is defined on the set of orders and again a ringed space $ (\mathcal{O}/\sim, \bar{\mathcal{F}})$ and a contravariant functor from fields to ringed spaces is obtained along with a natural transformation from the first to the second functor. Finally, we obtain a ringed space morphism $ (\mathcal{O}/\sim, \bar{\mathcal{F}}) \to (Y, \mathcal{O}_Y)$ where Y is the spectrum of the ring of bounded elements and $ {\mathcal{O}_Y}$ is the structure sheaf.

References [Enhancements On Off] (What's this?)

  • [1] Glen E. Bredon, Sheaf theory, McGraw-Hill, New York, 1967. MR 36 #4552. MR 0221500 (36:4552)
  • [2] D. W. Dubois, Infinite primes and ordered fields, Dissertationes Math. 69 (1970). MR 0257051 (41:1705)
  • [3] Samuel Eilenberg and Norman Steenrod, Foundations of algebraic topology, Princeton Univ. Press, Princeton, N. J., 1952. MR 14, 398. MR 0050886 (14:398b)
  • [4] Roger Godement, Théorie des faisceaux, Publ. Inst. Math. Univ. Strasbourg, Hermann, Paris, 1958. MR 0102797 (21:1583)
  • [5] A. Grothendieck, Éléments de géométrie algébrique, Inst. Hautes Études Sci. Publ. Math. No. 4 (1960).
  • [6] D. K. Harrison, Finite and infinite primes for rings and fields, Mem. Amer. Math. Soc. No. 68 (1966). MR 34 #7550. MR 0207735 (34:7550)
  • [7] John L. Kelly, General topology, Van Nostrand, Princeton, N. J., 1955. MR 16, 1136. MR 0070144 (16:1136c)
  • [8] I. G. MacDonald, Algebraic geometry. Introduction to schemes, Benjamin, New York, 1968. MR 39 #205. MR 0238845 (39:205)

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Keywords: Field, sheaf, ringed space, order, structure sheaf
Article copyright: © Copyright 1971 American Mathematical Society

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