ABSTRACT
Let G be a finite group acting on the finite dimensional vector space V over F . Then W[V]G is finitely generated as an algebra over F by 2.3.1. We may therefore write W[V]G = F [ / i , . . . , fd\/J where J is the ideal of re lations for this presentation of F [F ]G as a quotient of a polynomial alge bra. The ideal J in F [ / i , . . . , fd] is finitely generated since F [ / i , . . . , fd] is a Noetherian ring. Thus there is a finitely generated free F [ / i , . . . , /^-module F' and an epimorphism d : Ff — > J. The kernel J' of the map d is a finitely generated F [ / i , . . . , /^-module, and so there exists a finitely generated free IF[/i, . . . , /^-module F" and an epimorphism df : F" — > J'. Proceeding in this way we obtain short exact sequences
which may be spliced together to form a free resolution (see [136] or [173])
of F [P ]g as an F [ / i , . . . , /J-module. The successive modules J^ n\ which appear first here and then there, are called the syzygies (that is about what the word ‘syzygy’ means with reference to the phases of the moon) of W[V]G. Of course they depend on the choice of the generators / i , . •., fd £ F[1/]G. It is natural to ask if there is some choice of generators for which the process stops, i.e. for which the free resolution T has finite length. Moreover is there anything invariant to be gotten out of the syzygies, that is, something independent of the choice of the generators. This is the subject of Hilbert’s Syzygy Theorem and one of the main sources of the origins of homological algebra.
