The topics treated in the following pages were largely covered in two seminars, both given at Sheffield University, one during the session 1976/7 and the other during 1978/9. I had noted sometime earlier that M. Hochster and J. A. Eagon had established a connection between Determinantal Ideals and Invariant Theory. However in order to understand what was involved I had first to acquaint myself with the relevant parts of the theory of Algebraic Groups. With this in mind, I began to read J. Fogarty's book on Invariant Theory.

Almost at once my interest broadened. It had been my experience to see Commutative Algebra develop out of attempts to provide classical Algebraic Geometry with proper foundations, but it had been a matter of regret that the algebraic machinery created for this purpose tended to conceal the origins of the subject. Fogarty's book helped me to see how one could look at Geometry from a readily accessible modern standpoint that was still not too far r~moved from the kind of Coordinate Geometry which now belongs to the classical period of the subject.

When my own ideas had reached a sufficiently advanced stage I decided to try and develop them further by committing myself to giving a seminar. With a subject such as this, and in circumstances where the time available was very limited, it was necessary to assume a certain amount of background knowledge. Indeed most accounts of aspects of the theory of Algebraic Groups assume a very great deal in the way of prerequisites. In my case the audience could be assumed to be knowledgeable about Commutative Noetherian rings and I planned the lectures with this in mind.