Lecture Notes on Local Rings

Blowing up finitely supported complete ideals in a regular local ring

A commutative Noetherian local ring of finite global dimension is a regular local ring and the structure of such a ring is well described 1n many texts (see §1.9). It is perhaps surprising that very little seems to be known about the corresponding non-commutat1ve rings and it is our aim 1n this thesis to examine some of the ways in which the commutative theory can, and cannot be extended to a right Noetherian local ring of finite right global dimension. Chapter 1 contains basic definitions and, in Chapter 2, results are obtained on the projective and Injective dimensions of modules over right Noetherian local rings. Commutative regular local rings are domains and we begin Chapter 3 by considering the question of when a right Noetherian local ring R of finite right global dimension is a prime ring. An example of Stafford's shows that R is not necessarily prime; however, by examining the Nilpotent radical of R, we are able to extend results of Ramras [57] and Walker [75] and show that R is indeed prime when certain prime ideals are localisable. In Chapter 4 we consider the lattice of prime ideals of R when R is an AR-r1ng and provide theorems which generalise some of the basic results from the theory of commutative regular local rings. Section 4.3 contains examples which not only illustrate points arising in Chapters 3 and 4 but also show that some of the techniques which are mainstays of the commutative theory, fall dramatically in a non-commutat1ve setting. In Chapter 5, we generalise the concepts of regular sequences and Cohen Macaulay rings enabling us to prove, in §5.2 and chapter 6, that a right Noetherian local ring of finite right global dimension, which 1s integral over its centre, is a prime ring and exhibits many properties similar to those enjoyed by commutative regular local rings. Examples are provided which show that some alternative approaches are not applicable to the rings considered in these chapters. Regular local rings are Gorensteln rings and, as such, are the subject of an elegant structure theorem due to Bass [5]. In Chapter 7, we generalise his theorem to the situation of rings Integral over their centres. Each chapter begins with a summary of the results contained in that chapter.

Let V be a complete regular local (Noetherian) ring and let f ∈ V be a nonunit. We find necessary and sufficient conditions for to be the completion with respect to the maximal ideal of an integral domain of the form where S is a regular local ring whose completion with respect to its maximal ideal is V. In addition, if contains the rationals, we give necessary and sufficient conditions for to be the completion of an excellent local domain of the form , where f ∈ S, and S is a regular local ring whose completion is V.

The object of this paper is to discuss the conjecture, which will be abbreviated (E), that every complete local ring of dimension n possesses a finitely generated module of depth n. It is noted that several conjectures which have been open for some time follow from (E), and the connection of (E) with Serre's conjecture on multiplicities over regular local rings is discussed. In fact Serre's conjecture is proved for dimension ≤ 4 using the ideas under consideration.

Let (X, x O) be a normal complex analytic space andA⋐X a connected Stein compact set, i.e. a compact subset ofX which has a basis of open neighborhoods which are Stein spaces. We restrict attention to thoseA such thatR=H 0 O) is Noetherian. In Section I various exact sequences involving the divisor class group ofR, denotedC(R), are developed. (IfA is a point, one of these sequences is well-known [24], [39].) LetB be a connected compact Stein set on a normal varietyY such thatS=H o(B, Y O) andT=H o(A×B, X×Y O are Noetherian. In II we give a Kunneth-type formula which relatesC(R), C(S) andC(T). In III we show certain analytic local rings are unique factorization domains, study the divisor class groups of local rings on the quotient of an analytic space by a finite group, and prove a simple result on the topology of germs of complex analytic sets. We give a function-theoretic proof that complete intersections of dimensions greater than three which have isolated singularities have local rings which are unique factorization domains.

Let $R$ be a Noetherian local ring with the maximal ideal $\mathfrak{m}$. Assume that $R$ contains ideals $I$ and $J$ satisfying the conditions (1) $I\subseteq J$, (2) $I : \mathfrak{m}\nsubseteq J$, and (3) $J$ is $\mathfrak{m}$-full, that is $\mathfrak{m}J : x = J$ for some $x \in \mathfrak{m}$. Then the theorem says that $R$ is a regular local ring, if the projective dimension $\mathrm{pd}_{R}I$ of $I$ is finite. Let $\mathfrak{q} = (a_{1}, a_{2}, \ldots , a_{t})R$ be an ideal in a Noetherian local ring $R$ generated by a maximal $R$-regular sequence $a_{1}, a_{2}, \ldots , a_{t}$ and let $\bar{\mathfrak{q}}$ denote the integral closure of $\mathfrak{q}$. Then, thanks to the theorem applied to the ideals $I = \mathfrak{q} : \mathfrak{m}$ and $J = \mathfrak{\bar{q}}$, it follows that $I^{2} = \mathfrak{q}I$, unless $R$ is a regular local ring. Consequences are discussed.

Applications of dualizing complexes to Buchsbaum rings

Cartan maps, clean rings, and unique factorization

Let \((S, \mathfrak {n})\) be a regular local ring and let \(I \subseteq \mathfrak {n}^{2} \) be a perfect ideal of S. Sharp upper bounds on the minimal number of generators of I are known in terms of the Hilbert function of R = S/I. Starting from information on the ideal I, for instance the minimal number of generators, a difficult task is to determine good bounds on the minimal number of generators of the leading ideal I∗ which defines the tangent cone of R or to give information on its graded structure. Motivated by papers of S. C. Kothari and S. Goto et al. concerning the leading ideal of a complete intersection I = (f, g) in a regular local ring, we present results provided ht (I) = 2. If I is a complete intersection, we prove that the Hilbert function of R determines the graded Betti numbers of the leading ideal and, as a consequence, we recover most of the results of the previously quoted papers. The description is more complicated if ν(I) > 2 and a careful investigation can be provided when ν(I) = 3. Several examples illustrating our results are given.

Extended plus closure in complete local rings

A criterion is established, in terms of the Massey products structure carried by the homology of partial resolutions, for the Yoneda cohomology algebra ExtA(k, k) to be a free module over the universal envelope of a free graded Lie subalgebra.It is shown that several conjectures on the (co)homology of local rings, in particular on the asymptotic behaviour of the Betti numbers, follow from such a structure.For all rings with edim A -depths * 3, and for Gorenstein rings with edim A -depth A = 4, the following dichotomy is proved: Either A is a complete intersection, or xiA(k, k) contains a nonabelian free graded Lie subalgebra.0. Ten conjectures on the homology of local rings (in place of Introduction).In this paper (A, rrt, k) will denote a noetherian local ring A with maximal ideal m and residue field k.A well-known homological invariant of A is its Poincar series 00 ; = 0 where b (= b(A)) is the /'th Betti number of A, i.e. the rank of the /'th free module in the unique, up to noncanonical isomorphism, yl-free resolution X of k.Much of the work on the homology of local rings has been motivated by the following famous conjecture:C0. PA(t) represents a rational function.Recently this has been settled in the negative by D. Anick,1 who constructed for each k a A>algebra A requiring 5 generators for m and having m3 = 0, with transcendental Poincar series.On the other hand, more than 20 years of research have produced different kinds of conditions, which ensure the rationality of PA(t).Before collecting for further reference several of them in a theorem, we introduce some notation, to be kept for the rest of the paper.edim A = embedding dimension of A := dimt m/m2 (= bx(A)); dim A -Krull dimension of A ;depth A = maximal length of an ^-regular sequence in m; d(A) = complete intersection defect of A '-dim^ a/no.-(dim R -dim A), where (R, n) is a regular local ring such that R/a is isomorphic to the m-adic

This paper describes the implementation of recursive algorithms in unique factorization domains, namely multivariate polynomial greatest common divisors (gcd) and factorization into irreducible parts in the Java computer algebra library (JAS). The implementation of gcds, resultants and factorization is part of the essential building blocks for any computation in algebraic geometry, in particular in automated deduction in geometry. There are various implementations of these algorithms in procedural programming languages. Our aim is an implementation in a modern object oriented programming language with generic data types, as it is provided by Java programming language. We exemplify that the type design and implementation of JAS is suitable for the implementation of several greatest common divisor algorithms and factorization of multivariate polynomials. Due to the design we can employ this package in very general settings not commonly seen in other computer algebra systems. As for example, in the coefficient arithmetic for advanced Grobner basis computations like in polynomial rings over rational function fields or (finite, commutative) regular rings. The new package provides factory methods for the selection of one of the several implementations for non experts. Further we introduce a parallel proxy for gcd implementations which runs different implementations concurrently.

Gorenstein injective, Gorenstein flat modules and the section functor

1. Let L be a noetherian domain that is integrally closed in its quotient field F. To each ideal A of L is assigned an ideal A,a' the integral closure or completion of A, that consists of all elements x E L for which an equation of the form, xn + a1xn + * + an = 0, ai eAr, holds. If Q is the set of all valuations v of F such that the associated valuation ring R, contains L, then Aa coincides with the ideal Ab = {x; v(x) > v(A), V v E Q}, [2; 3]. An ideal A is said to be complete if A = Aa, and the set of all complete ideals is denoted by F(L). If A and B belong to F(L), the product AB may not belong to F(L), but the completion of the product (AB)a does. Hence a binary composition x is defined on F(L) by the condition, A x B = (AB)a. Under this composition F(L) is a commutative semigroup with an identity in which the cancellation law holds [2]. In case L is a two dimensional regular local ring, Zariski has shown [4, Appendix 5] that (1(L), x) is a Gaussian semigroup, and that the composition x is ordinary product. In this paper we study the case in which L is a two dimensional normal local domain which is subject to conditions less stringent than regularity. (See ?2 below.) It is shown that modulo a simple equivalence relation the semigroup (FO(L), x) is Gaussian, where Fo(L) is the subset of F(L) that consists of primary ideals belonging to the maximal ideal of L. However (FO(L), x) is not Gaussian in an absolute sense for in simple examples it is seen that the maximal ideal M of L is an irreducible element of (F(L), x) that is not prime. (Here we are using the semigroup terminology of Jacobson [1, Chapter IV].) Our methods are direct extensions of those of Zariski. In case L is regular, the form ring associated with L and the sequence of powers of the maximal ideal M is a polynomial ring over a field and Zariski's arguments are based in part on the fact that such a ring is a unique factorization domain. In our case the form ring is an integrally closed noetherian domain, and we obtain results analogous to Zariski's by using the Artin theory of factorization in the sense of quasi-equality that is valid in such domains.

Completions of Local Rings with an Isolated Singularity