Elements of Algebraic Geometry and Commutative Algebra

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).