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Elements of Algebraic Geometry and Commutative Algebra

  • H. E. A. Eddy CampbellEmail author
  • David L. Wehlau
Chapter
  • 1.2k Downloads
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 139)

Abstract

In this chapter we summarize the basic elements of algebraic geometry and commutative algebra that are useful in the study of (modular) invariant theory. Normally, these techniques are most useful in questions about the structure of rings of invariants, and, as well, they seem to be most useful in proving theorems that hold true for all groups, modular or not. It is worth noting that a large fraction (as much as one third — see the paper of Fisher (1966, Page 146)) of the papers in mathematics in the latter stages of the 19th century were studies of invariant theory. It is worth noting as well that commutative algebra was invented, discovered if you prefer, by Hilbert, in order to clarify and understand invariant theory more fully, see Fisher (1966). There are several excellent references, including Atiyah and Macdonald (1969), Dummit and Foote (2004), Eisenbud (1995), Lang (1984), Matsumura (1980), and a forthcoming book by Kemper, (2010).

Keywords

Prime Ideal Homogeneous System Polynomial Ring Commutative Algebra Hilbert Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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    M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 39:4129 zbMATHGoogle Scholar
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    David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. MR 97a:13001 zbMATHGoogle Scholar
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    Charles S. Fisher, The death of a mathematical theory: a study in the sociology of knowledge, Arch. History Exact Sci. 3 (1966), no. 2, 137–159. MR 0202546 zbMATHCrossRefMathSciNetGoogle Scholar
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    Hideyuki Matsumura, Commutative algebra, second ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344 (82i:13003) zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Sir Howard Douglas Hall, Dept. MathematicsUniversity of New BrunswickFrederictonCanada
  2. 2.Dept. Mathematics & Computer ScienceRoyal Military College of CanadaKingstonCanada

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