The goal of these notes is to introduce the concept of uniformity in commutative algebra. Rather than giving a precise definition of what uniformity means, we will try to convey the idea of uniformity through a series of examples. As we’ll soon see, uniformity is ubiquitous in commutative algebra: it may refer to absolute or effective bounds for certain natural invariants (ideal generators, regularity, projective dimension), or uniform annihilation of (co)homology functors (Tor, Ext, local cohomology). We will try to convince the reader that the simple exercise of thinking from a uniform perspective almost always leads to significant, interesting, and fundamental questions and theories. This theme has also been discussed by Schoutens [2000], who shows how uniform bounds can be useful in numerous contexts that we do not consider in this paper.

The first section of this paper, based on the first lecture in the workshop, is more elementary and introduces many basic concepts. The next three sections target specific topics and require more background in general, though an effort has been made to minimize the knowledge needed to read them. Each section has some exercises which the reader might solve to gain further understanding. The first section in particular has a great many exercises.