Part IV Local Rings
13.1 Basic Properties of Regular Local Rings
Chapter 13
Regular Local Rings
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) affinen-space.
How can we determine the pointsxof an affine varietyXwhere the local- ized coordinate ringK[X]xis regular? This is the topic of the second section of this chapter, where we prove the Jacobian criterion. A consequence is that an affine variety is “nice” at “most” of its points.
(a) M is generated by m1, . . . , mn as an R-module.
(b) M/mM is generated by m1+mM, . . . , mn+mM as aK-vector space.
In particular, all minimal generating systems of M have the same size, namely dimK(M/mM).
Proof. It is clear that (a) implies (b). Conversely, assume (b) and setN :=
(m1, . . . , mn)⊆M. Then (b) impliesM ⊆N +mM, soM/N ⊆m·M/N.
By Nakayama’s Lemma7.3, this impliesM/N={0}, soM =N. ut Applying Lemma 13.1 to M = m shows that every minimal generating systems of m hat the size dimK m/m2
. Since we know from the principal ideal theorem (more precisely, from Corollary7.6) thatmcannot be generated by fewer than dim(R) (the Krull dimension) elements, we obtain
dimK m/m2
≥dim(R). (13.1)
This inequality prompts the definition of regularity.
Definition 13.2.(a) Ris called regularif dimK m/m2
= dim(R).
(Heredim(R)signifies the Krull dimension.) SoR is regular if and only ifm can be generated bydim(R)elements, which in turn is equivalent to the condition thatRhas a system of parameters which generatesm. Such a system of parameters is called aregular system of parameters.
(b) Let S be a Noetherian ring and X := Spec(S). An element P ∈ X is called anon-singular pointif the localizationSP is regular. Otherwise, P is called a singular point. S is called a regular ring if X has no singular points.
(c) A pointx∈X of an affine variety is callednon-singularif the localiza- tionK[X]x of the coordinate ring atxis regular. Otherwise, xis called singular.X is callednon-singularif every point is non-singular.
Remark. The above definition of a regular ring raises the following question:
Is a regularlocalringRalso a regular ring in the sense of Definition13.2(b)?
In other words, is RP regular for every P ∈ Spec(R)? This is indeed true, but not at all easy to prove (see Matsumura [37, Corollary 18.G]). / Example 13.3. By Definition13.2(a), a zero-dimensional local ring is regular if and only if it is a field (equivalently, if and only if it is reduced). / Before treating more examples, it is useful to establish the following regu- larity criterion in terms of the associated graded ring. It is a consequence of Theorem12.8.
Theorem 13.4(Associated graded ring and regularity). Ris regular if and only if the associated graded ring gr(R) is isomorphic to a polynomial ring overK.
13.1 Basic Properties of Regular Local Rings 193 Proof. WriteA:= gr(R). By Theorem12.8we have dim(A) = dim(R) =:n.
First assume thatR is regular, so the maximal idealmis generated by n elements. By the discussion preceding (12.5) on page182, it follows that A is generated by nelements as aK-algebra, so by Theorem5.9 and Proposi- tion5.10these elements must be algebraically independent. It follows thatA is isomorphic to a polynomial ring.
Conversely, assume that Ais isomorphic to a polynomial ring. By Corol- lary 5.7 it follows thatA is generated by nelements b1, . . . , bn. By the dis- cussion preceding (12.3) on page181 we have a gradingA=L
d∈N0Ad with A0∼=KandA1∼=m/m2. We may assume that the homogeneous component of degree 0 of everybiis 0. Letπ:A→A1be the projection on the component of degree 1. Everya ∈A can be written as a polynomial over K in the bi, and it follows thatπ(a) is aK-linear combination of theπ(bi). ThereforeA1 is generated byπ(b1), . . . , π(bn), and we get dimK m/m2
= dimK(A1)≤n.
With (13.1) this implies thatR is regular. ut
To get a geometric interpretation of regularity, consider the case where R =K[X]x is the coordinate ring of an affine variety, localized at a point x∈X. Exercise 12.5gives a presentation gr(R)∼=K[x1, . . . , xn]/J of gr(R) in this case, where the varietyVKn(J) can be interpreted as the tangent cone at x, i.e., the best approximation of X by an affine variety made up of lines passing through x. So roughly speaking, Theorem 13.4 tells us that x is a non-singular point if and only if the tangent cone at xis an affine n-space.
Geometrically, this makes a lot of sense since non-singularity should mean that the variety looks “nice” locally. However, there is a catch. Even if the tangent cone is some affinen-space, gr(R) need not necessarily be isomorphic to a polynomial ring, since J need not be a radical ideal. This happens, for example, ifxis a cusp ofX. In such a case, the geometric interpretation may be saved by viewing the affine variety of a non-radical ideal J has having
“double components” or “hidden embedded components”.
Example 13.5. (1) The formal power series ring R := K[[x1, . . . , xn]] in n indeterminates over a field is a regular local ring. This can be seen by doing Exercise12.6(a) (the result is gr(R)∼=K[x1, . . . , xn]) and applying Theorem 13.4, or by observing that the maximal ideal is generated by x1, . . . , xn and using Exercise7.10to conclude that dim(R) =n.
(2) LetX1, X2, X3 be the cubic curves from Exercise12.6(b), shown in Fig- ure12.1 on page189. LetRi be the localization of the coordinate ring ofXi at the pointx:= (0,0). In Exercise12.6(b) the associated graded rings gr(Ri) are determined, and the result is isomorphic to a polynomial ring only forR3. Soxis a singular point ofX1andX2, but not ofX3.X1
is particularly interesting, since it has a cusp atx. Here the tangent cone may be viewed as a “double line”. By changing coordinates one can also determine the associated graded ring of the localization at other points.
The result is that all points other than the origin are non-singular. This is what one expects from looking at Figure12.1. /
This may be a good place for a short digression on completions. By con- sidering two elements fromRas “near” if their difference lies in a high power ofm, we get a concept of convergence. Krull’s Intersection Theorem12.9guar- antees that with this concept, the limit of a convergent series is unique. Ex- ercise13.4gives more background on this. We also have a concept of Cauchy sequences.R is calledcompleteif every Cauchy sequence has a limit in R.
Most of the local rings we have seen in this book are not complete. However, one can construct an extensionRbofRwhich is a complete local ring.Rb also is Noetherian, and it has the property that all of its elements are limits of R-valued sequences.Rb is called the completion of R. In a sense, the con- struction of completion mimics the passage from the rational numbers to the real numbers. From the construction ofRb it can be shown that that R and Rb have the same associated graded ring. So the associated graded ring may serve to transport properties from R to Rb and vice versa. For example, it follows from Theorem12.8that dim(R) = dim(R), and it follows from The-b orem 13.4that Rb is regular if and only ifR is regular. A nice example of a complete local ring is the formal power series ringK[[x1, . . . , xn]] inninde- terminates over a field. In fact, it is the completion ofK[x1, . . . , xn](x1,...,xn) (see Exercise 13.5). Another well-known example of a complete ring is the ringZpofp-adic integers (withpa prime number), which plays an important role in algebraic number theory and computer algebra. Zp is the completion ofZ(p), the ring of rational numbers with denominator not divisible byp.
Completion is an important tool in commutative algebra. Philosophically, the idea is that localization is not “local enough”, but completion describes the behavior of a variety on a smaller scale. For example, the local ring at a point of an irreducible affine varietyX contains all the information of the va- riety that is invariant under birational equivalence, since the field of fractions of the local ring is the function field K(X) = Quot(K[X]), and birational equivalence is just defined as isomorphicness of the function fields. So the local ring still contains some sort of global information, even if it is regular.
However, if we assume that the local ring is regular, it turns out that its completion is isomorphic to the formal power series ringK[[x1, . . . , xn]] with n = dim(X) (see Matsumura [37, Corollary 2 to Theorem 60, p. 206]). So in this case completion eliminates global information. Another illustration of this philosophy is contained in Exercise 13.6. For more on completion, we refer to Eisenbud [17, Chapter 7].
By putting together Theorem 13.4, Proposition 8.8 and Theorem 12.10, we obtain
Corollary 13.6.(a) Every regular local ring is an integral domain.
(b) Every regular local ring is normal.
In fact, a bit more is true: Every regular local ring is factorial. This is clear for zero-dimensional rings by Example13.3, and will be proved for one- dimensional rings on page207. In dimension>1, the result is much harder
13.2 The Jacobian Criterion 195