Abstract
In this short chapter, we summarize geometric properties of a variety and a tower of varieties we need in the book. One reason for adding this chapter is to make the book logically complete, and another is to give the foundation of the theory of towers of varieties in the language of proschemes, since the Shimura variety is a tower of varieties fundamental to the number-theoretic study of automorphic forms. If the reader is familiar with the subject, he or she can take a brief look at the content of this chapter and go directly to Chap.6.
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References
N. Bourbaki, Algèbre Commutative (Hermann, Paris, 1961–1998)
H. Matsumura, Commutative Ring Theory. Cambridge Studies in Advanced Mathematics, vol. 8 (Cambridge University Press, New York, 1986)
A. Grothendieck, J. Dieudonné, Eléments de Géométrie Algébrique. Publications IHES, vol. 4 (1960), vol. 8 (1961), vol. 11 (1961), vol. 17 (1963), vol. 20 (1964), vol. 24 (1965), vol. 28 (1966), vol. 32 (1967)
M. Nagata, Theory of Commutative Fields (American Mathematical Society, Providence, RI, 1993)
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Hida, H. (2013). Geometry of Variety. In: Elliptic Curves and Arithmetic Invariants. Springer Monographs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6657-4_5
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DOI: https://doi.org/10.1007/978-1-4614-6657-4_5
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