Regular Local Rings

Abstract

As mentioned before, local rings serve for the study of the local behavior of a global object, such as an affine variety. In particular, notions of local “niceness” can be defined as properties of local rings. There is a range of much-studied properties of local rings. This includes the Cohen–Macaulay property, the Gorenstein property, normality, and regularity. In this book only normality and regularity are dealt with at some length, and one exercise, 13.3, is devoted to the Cohen–Macaulay property. It turns out that regularity is the nicest of these properties, meaning that it implies all others. After defining the notion of regularity of a Noetherian local ring R, we will see that this is equivalent to the condition that the associated graded ring gr(R) is isomorphic to a polynomial ring. If R is the coordinate ring of an affine variety, localized at a point, gr(R) can be interpreted as the coordinate ring of the tangent cone at that point (see Exercise 12.5). So in this situation regularity means that the tangent cone is (isomorphic to) affine n-space.