Computational Commutative Algebra 2

Hofstadter's Law: It always takes longer than you think it will take, even if you take into account Hofstadter's Law. (Douglas R. Hofstadter) Dear Reader, why did we begin the foreword of this second volume with the same quote as the ?rst? There we wrote that it took three years of intense work just to ?ll three centimeters of your bookshelf. The completion of this volume took four years and it is about four centimeters thick. Thus we have a con?rmed invariant which governs our writing: our velocity is one centimeter per year, after all e?ects due to Hofstadter's Law have been taken into account. When westartedthisprojectinthelastmillennium,weplannedabookforlearning, teaching, reading and, most of all, enjoying the topic at hand. Surely there is no law which says that a mathematical book has to be dull, boring, dry, or tedious. But how do you make it enjoyable? Our approach has been to ?ll it with amusing quotes, varied jokes, funny word games, ?owery metaphors and occasional literarye?orts. There are two possible drawbacks of this method. Firstly, not everyone has the same sense of humour and not every metaphor works as intended. For instance, it is easy to joke about certain politicians, but what happens if they read this book? And when we wrote of a small boat sailing slowly into the Brazilian sunset, it was pointed out to us that this entails a geographical problem. Secondly, it is very di?cult to write humorously in a foreign language.