Showing papers on "Global dimension published in 1976"

On finite weak and injective dimension

Abstract: An extension is given of the theorem relating projective dimen- sion to depth for a finitely generated module of finite projective dimension over a commutative Noetherian local ring. This extension is dualized to relate injective dimension to a concept of codepth when the injective dimension is known to be finite. 1. Introduction and notation. A classical result on the projective dimension of modules over a commutative Noetherian local ring says that if the module is finitely generated and of finite projective dimension, then that dimension is the difference between the depth of the ring and that of the module (1, Theorem 3.7). We extend this result by changing the context to weak dimension. We then dualize to injective dimension, and the result we get seems to be known only in the special case of finitely generated modules. For the duration of the paper, all rings are commutative Noetherian with unit, and all modules are unitary. If M is an R-module, then we denote the weak, projective, and injective dimensions of M over R by w.d.RM, p.d.RM, and i.d.RM respectively, and we use ER(M) to denote the injective envelope of M over R. If R is local with residue field k, then we use Mv to stand for the Matlis dual HomR(M, ER(k)) of M. If M is any R-module, and I is an ideal in R, then we define the depth of M over I as the lowest degree in which ExtR (R/I, M) is nonzero, and the codepth of M over I as the lowest degree in which TorR(R/I, M) is nonzero; we denote these by depth, M and codepth, M respectively. If ExtR(R/I, M) is 0 in all degrees, we say depth, M = ox, and likewise for codepth. We adopt the convention that if we ask for the largest dimension in which a positively graded object is nonzero, and that object is 0 in all degrees, then the requested dimension is - ox. Thus, for example, the projective dimension of 0 over R will be - oo, since it is the supremum over all R-modules A of the largest dimension in which ExtR (0, A) is nonzero. The author wishes to thank Professor Everett L. Lady for referring him to the article (5) which appears to be the origin of the spectral sequence needed here, and Professor Melvin Hochster for pointing out nonhomological proofs of two of the following corollaries.

Maximal ideal transforms of Noetherian rings

Abstract: Let J? be a commutative Noetherian ring with unit. Let T be the set of all elements of the total quotient ring of R whose conductor to R contains a power of a finite product of maximal ideals of R. If A is any ring such that R C A C T, then A/xA is a finite R module for any non-zerodivisor x in R. It follows that if, in addition, R has no nonzero nilpotent elements, then any ring A such that R C A C T is Noetherian. Let fi be a commutative Noetherian domain with unit of Krull dimension one. The Krull-Akizuki theorem [6, Theorem 33.2] states that if Pis an integral domain containing R and contained in a finite algebraic extension of the quotient field of R then P is Noetherian. By adjoining a finite number of elements to R and letting this new ring be called R, one proves the theorem by proving the following: Any ring A between a Noetherian domain R of Krull dimension one and its quotient field is Noetherian. This is equivalent to showing that if x is any nonzero element of R then A/xA is a finite R module. We shall restate this reduction of the Krull-Akizuki theorem in such a way that the final statement is true for any Noetherian ring of any Krull dimension. In order to do this we first characterize the relationship between a one dimensional Noetherian domain R and its quotient field P. If y belongs to P, then its conductor to R, or its denominator ideal contains some powered product of a finite number of maximal ideals of R. In the special case that R has precisely one maximal ideal M, T is the set of all elements whose conductor to R contains a power of M. If B is any commutative Noetherian ring with unit and / is any ideal of B that contains a non-zero-divisor, the Ptransform of B is defined to be the set of all elements of the total quotient ring whose conductor to B contains a power of I. The Ptransform is a ring between B and its total quotient ring. If B is a Noetherian domain of Krull dimension one and B has one maximal ideal M, then the quotient field is the A/-transform of B. Now if R is any commutative Noetherian ring with unit, we call P the global transform of R if P is the set of all elements of the total quotient ring whose conductor to R contains a power of a finite product of maximal ideals of R. If M is any maximal ideal of R, TM = P ®R RM, where RM is the localization of R at M, is the A/fi -transform of RM. Also P contains the M-transform of R for M any maximal ideal of R. If R is a domain of Krull dimension one, we Received by the editors May 16, 1974 and, in revised form, November 4, 1974. AMS (MOS) subject classifications (1970). Primary 13E05, 13E10; Secondary 13B99.

Generalized uniserial rings

Abstract: A description is given of two-sided Noetherian rings over which all finitely generated modules are semichained. Bibliography: 31 titles.

On the Global Dimension of Some Filtered Algebras

Abstract: Let k be a commutative field of characteristic 0. Let g be a Lie algebra over k. Let / be a A> valued 2-cocycle on the \" standard complex\" for g. We set 9(/) = T(Q)/Uf(Q), where T(g) denotes the tensor algebra of the vector space g and Uf(o) the two-sided ideal of T(g) generated by all elements of the form x®y-y® x-[x,y]-f(x,y) for x,ye$. It is known [15] that g(/) is a filtered ^-algebra whose associated graded algebra is isomorphic to a polynomial algebra over k and that every filtered A>algebra with this property is isomorphic to one such. In this paper we determine (§2, Theorem 2.6) the global dimension of g(/), where g is a finite-dimensional nilpotent Lie algebra over k, and deduce some interesting corollaries. In §1 we prove some results which are used in the proof of the main theorem.

On non-commutative regular local rings

Abstract: Let R be a ring (with identity). We shall call R a local ring if R is aright noetherian ring such that the Jacobson radical M is a maximal ideal (and so is the only maximal ideal), and R/M is a simple artinian ring. A local ring R with maximal ideal M is called regular if there exists a chain of ideals M i of such that M i–1 / M i is generated by a central regular element of R/M i (1 ≦ i ≦ n). For such a ring R , Walker [6, Theorem 2. 7] proved that R is prime and n is the right global dimension of R , the Krull dimension of R , the homological dimension of the R -module R/M and the supremum of the lengths of chains of prime ideals of R . Such regular local rings will be called n-dimensional . The aim of this note is to give examples of regular local rings. These arise as localizations of universal enveloping algebras of nilpotent Lie algebras over fields and localizations of group algebras of certain finitely generated finite-by-nilpotent groups.

The Global Dimensions of Ore Extensions and Weyl Algebras

Abstract: This chapter discusses the global dimensions of ore extensions and Weyl algebras. If R is a ring and D is a derivation of R , and if S = R[t, D] is the ore extension of R with respect to D , that is, S is additively the group of polynomials in an indeterminate t with multiplication subject to tr = rt + D(r) for all r in R , then an extension of D to a derivation of S by setting D(t) = 0 for all elements s of S leads to the result ts = st + D(s). If R is a commutative noetherian ring, and if M → 0 is a left A 1 R -module that is finitely generated as a left R -module, then M is an abelian torsion group.

Homological dimension of abelian groups over their endomorphism rings

Abstract: The projective dimension of an abelian group with torsion reduced part, as a module over its endomorphism ring, is determined. In particular, a group of projective dimension 2 is exhibited. Any abelian group G may be considered as a module over its endomorphism ring E. It then is natural to study the properties of G as an P-module. In particular, you can try to compute the various homological dimensions of G. For example, in [6] we showed that a /J-group G is flat over E (i.e. has weak dimension zero) unless its reduced part is bounded and its divisible part is nonzero. In [7] we characterized those G which are injective over P. Douglas and Farahat [2] showed that if G is torsion, then dimEG < 1, where dim£G denotes the projective dimension of G over P. Later [3] they showed that if G is divisible, then dim£C < I. In both of these papers they raised the question as to whether dim£C7 < 1 for all abelian groups G. The purpose of this paper is to determine the projective dimension of groups whose reduced part is torsion. This combines and generalizes the results in the two Douglas and Farahat papers and exhibits groups of projective dimension 2. We begin with an observation of Douglas and Farahat [2, 1], a simplified version of a theorem in Northcott [5, Lemma 2, p. 139], and a (very slightly) generalized version of Douglas and Farahafs theorem about torsion groups, which we prove mainly for the convenience of the reader. Lemma 1. Let G be an abelian group, E its endomorphism ring, and A a summand of G. Then Hom(4, G) is a projective E-module. Proof. Let e G P be a projection on A. Then Wom(A, G) = Ee is a summand of E. Lemma 2. Let A0 C Ax C A2 C • • • be R-modules such that dimRA0 < n and dimRA-/Ai_l < n for i = 1,2, ■ • • . Then dimR (J At < n. Proof. See Northcott [5, Lemma 2, p. 139] or fill in the details in the following sketch. If n = 0, then U A, s A0 ® Ax/A0 ® A2/Ax ® ■ ■ ■ so the lemma is true. If n > O.map projectives P, onto the Ai/Aj_x and lift to A-r Then ©P; maps onto U Aand the kernel has dimension less than n by induction. Received by the editors January 27, 1975. AMS (MOS) subject classifications (1970). Primary 18G20, 20K40. 1 This research was supported in part by NSF Grant GP-28379. F> American Mathematical Society 1976 65 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 66 FRED RICHMAN AND E. A. WALKER In particular, if A0 C Ax C A2 C ■ ■ ■ are all projective, then dim U At < 1. This is easily proved directly by mapping @A, onto (J Ai and observing that the kernel is isomorphic to ®Ar Lemma 3 (Douglas and Farahat). Let G be an abelian group, E its endomorphism ring, and 77 a pure, fully invariant, torsion subgroup of G. Then dim£77 < 1. Proof. Since 77 is the direct sum of its primary subgroups, which are invariant under A, we may assume that 77 is/?-primary. Let N = [n: H has a cyclic summand of order n}. Note that a cyclic summand of 77 is also a summand of G. If N is infinite then 77= U H[n], neN and since H[n] = Hom(C, G) where C is a cyclic summand of order n, we have that 77 is the union of a countable chain of A-projectives, by Lemma 1, so dim£77 < 1 by Lemma 2. If N is finite, let n be its largest element. Let A-t = H[np']. Now A0 = H[n] is A-projective as before. Since 77 = A © D where nB = 0 and D is divisible, the A-module Ai/Aj_l is generated by any element x in D of order «/?'. So Ai/Aj_x at A/7 where 7 = {e £ A: ex £ A,_x }. Write G = Gx © G2 where G, is a rank-one summand containing x. Then 7 = Ee0 where e0 is the identity map on G2 and multiplication by p on G{. Since Ee0 = E is projective, we have dimEAjAt__x < 1 and so dim£77 < 1 by Lemma 2. The following lemma is a slight generalization of the characterization in [6] of /?-groups that are projective over their endomorphism rings. Lemma 4. Let G be an abelian group, E its endomorphism ring, and 77 a pure, fully invariant, torsion subgroup of G. Then H is E-projective if and only if every p-primary subgroup of 77 is bounded. Proof. We may assume that 77 is /?-primary. If 77 is bounded, then 77 is isomorphic to Hom(C, G) where C is a cyclic summand of 77 of maximum order. So 77 is A-projective by Lemma 1. Suppose 77 is A-projective. Consider the map 4>: Hom£(77, A) £77 ^ Hom£(77, 77) given by <£(/ <8> A)(x) = /(x)A. By the dual basis theorem [1, Proposition 3.1, p. 132], the image of , G)^ H given by , G) which is isomorphic to Hom(Q/Z, G) which is isomorphic to Hom(^, G) where A is a divisible summand of G containing one copy of each Z(px) that occurs in G. Thus, by Lemma 1, we have dim£r7 < 1. Theorem. Let G be an abelian group, E its endomorphism ring, and H = T ® D a pure, invariant subgroup of G, with T torsion reduced and D divisible. Then (1) dimEH < 2. (2) dimEH < 1 if and only if D is torsion or Hom(T,D) is bounded. (3) dimEH = 0 if and only if D is torsion free and Tp (the p-primary part of T) is bounded for all p. Proof. If H is torsion, then (1) and (2) follow from Lemma 3, while (3) is Lemma 4. So we may assume that H is not torsion and, hence, contains a summand of G isomorphic to Q. Let N = {n: H has a cyclic summand of order n) and P = {p: H has a cyclic summand of type Z(/>°°)}. The "if" half of (3) follows from Lemma 4 and the observation that if D is torsion free, then D = Wom(Q, G) is P-projective. The "if" half of (2) follows from Lemmas 3 and 5 since H is the direct sum of the invariant subgroups "S.p^PTp and D + "2pEPTp. To complete the proof we consider the projective resolution O^K^ © H[n] ®Hom(Q,G)^H->0 where ©a, + /maps to 2a, + /(l). Let M = K n Hom(£>, G) = {/ G Hom(Q, G):f(l) = 0}. Then M = HomC2p(EPZ(p'x'), G) is algebraically compact, torsion free, reduced (see [4, Proposition 44.3]), and P-projective. The torsion subgroup of AT is

The integral closure of a Noetherian ring

Abstract: Let R be a commutative ring with identity and let R' denote the integral closure of R in its total quotient ring. The basic question that this paper is concerned with is: What finiteness conditions does the integral closure of a Noetherian ring R possess? Unlike the integral domain case, it is possible to construct a Noetherian ring R of any positive Krull dimension such that R' is non-Noetherian. It is shown that if dim R S 2, then every regular ideal of R' is finitely generated. This generalizes the situation that occurs in the integral domain case. In particular, it generalizes Nagata's Theorem for two-dimensional Noetherian domains.

Inert extensions of Krull domains

Abstract: Let A C B be integral domains with B an inert extension of a Krull domain A. Let 9(A) be the set of height one primes of A, and let T = H e9,(/))fi A . When each Bp = B ® Ap is a UFD, a necessary and sufficient condition for T to be a Krull domain is obtained. If T is a Krull domain and each B is a UFD, then the divisor class groups of A and T are isomorphic under the natural mapping. These results are applied 10 A Q B when B is a symmetric algebra over A and when B is locally a polynomial ring over A. Let A C B be integral domains. Following Cohn [3] we say that B is an inert extension of A if for any nonzero x, y E B, xy E A implies x E A, y G A. If B is an inert extension of A and a Krull domain, then /. is easily seen to be a Krull domain. In this paper we give a partial converse to this result, and we compare the class groups of A and B, when both are Krull domains. This paper originated as a study of extensions A Q B with A a Krull domain and B a symmetric algebra over A. Under these hypotheses it is known that when A is a UFD, fi is a UFD if and only if B is a factorial Amodule [4], [5]. An A -module M over a UFD A is factorial if it is torsion-free and every element x E M may be written uniquely x = ax*, with a E A and x an element of M divisible only by units of A. Elements of M divisible only by units are called primitive. We generalize the idea of a factorial module as follows. Let A be a Krull domain, M an A -module, and 9iA) the set of height one primes of A. We say that M is a Krull A -module if M is torsion-free, Mn = M ®. A„ is a factorial p TM p A -module for eachp E 9iA), M = f]peq>tA)Mp, and each nonzero x in M is primitive in all but a finite number of M (i.e. the intersection is "locally finite"). Our main result, Theorem 1.6, shows that if A C B is an inert extension with A a Krull domain and B a UFD for each p E 9iA), then T = ne^\i) is a Krull domain if and only if it is a Krull /.-module, in which case Cl iA) = Cl (T). In the second section we show that in the case when B is a symmetric algebra over A, the condition that B be a UFD for each p E 9{A) is superfluous. We conclude with some examples. We owe a debt of originality to §10 of [7], as our main results are generalizations of Theorem 10.11 of [7] and our arguments, for the most part, Received by the editors February 13, 1976. AMS (MOS) subject classifications (1970). Primary 13F05; Secondary 13B99.

Global Dimension of Ore Extensions

Abstract: This chapter discusses the global dimension of ore extensions If R is a left and right noetherian and of a finite left global dimension, a necessary condition for equality to hold is the existence of a left S -module M that is finitely generated as an R -module and with ldim R M = lgldim R If R is a ring with derivation D , then S = R [ t ] is the ore extension of R with respect to D , that is, S is additively the group of polynomials in an indeterminate t with multiplication subject to tr = rt + D(r) for all r in R The chapter presents a situation in which R is commutative

The unity in rings with Gabriel and Krull dimension

Abstract: The main result is that all rings with right Krull dimension and divisible torsion free additive group have a right identity. Furthermore it will be proved that a simple ring with characteristic 0, right Gabriel dimension < 2 and finite right uniform dimension has a unity. This is false for higher Gabriel dimensions, as demonstrated by a counterexample. A similar construction gives an example for a ring with unity and Gabriel dimension, but without Krull dimension, all factor rings having finite uniform dimension.

A remark on the first neighbourhood ring of a Noetherian Cohen-Macaulay local ring of dimension one

Abstract: There is an isomorphism between the first neighbourhood ring of a noetherian Cohen-Macaulay local ring A of dimension one and the ring of endomorphisms of a large power of its maximal ideal. Let A be a noetherian Cohen-Macaulay local ring of dimension one and m be its maximal ideal. An element a of m' is superficial of degree t if, for every large integer n, m"a = mn + l. The following results are well known: every superficial element is regular; for every large integer t, there exists a superficial element of degree t [4]. The first neighbourhood ring R of A is the subring {(x/y)\x G m', y superficial of degree t) of the total quotient ring K of A. For every large integer n, the product Rm" = m" [2, 12.1]. Let v be the least such n. Let End^m") denote the algebra of A -endomorphisms of m". There is a sequence (1) A g End^(m) c • • • C End^m") c • • • . (1) Theorem. 1. The integer v is the least integer n such that xm" = m"+' for every superficial element x, where t is the degree of x. 2. For every integer n > v, the ring End/1(m") = End/4(m") and there exists an isomorphism F of A-algebras of End/4(m") onto R such that F(FlomA(m",m"+1)) is the ideal Rm of R. Proof. 1. Let x be a superficial element of degree t. Then length(m"/*nT") = te where e is the multiplicity of A [2, 12.5]. If xm" = m" + ', then t-\ length^/jcm") = 2 length(m"+7m',+'+1) = te. i = 0 Received by the editors December 13, 1974. AMS (MOS) subject classifications (1970). Primary 13H10.

On Arnold’s formula for the dimension of a polynomial ring

Abstract: If R is a commutative integral domain with quotient field K and x, ..., x,, are indeterminates, then there exist 1, 6A in K such that dim R[xl, . . . ,xjI = n + dim R[1, * * , n] If R is a commutative ring, the Krull dimension of R is the maximum of the lengths of all chains of prime ideals in R. If R = (C [V] is the coordinate ring of an affine variety V over the complex numbers, then increasing chains of primes in R correspond to decreasing chains of irreducible subvarieties. In this "geometric case" the Krull dimension corresponds to our intuitive notion of (complex) topological dimension. Moreover, since R[X] corresponds to V x e (the product of V and an affine line), intuition would lead us to suspect (*) dim R[X] = dim R + 1. In [7], W. Krull established (*) for any noetherian ring. Seidenberg [9], [101 investigated the validity of (*) for arbitrary commutative rings and observed that it does not hold in general. He observes that one always has dim R + 1 < dim R[X] < 2 dim R + 1, and he provides examples to show that within these bounds anything can happen. Jaffard [6] made an extensive study of the dimension theory in polynomial rings. He introduced the notion of valuative dimension of a domain R. This is just the maximum of the ranks of the valuation overrings of R. Jaffard showed that when (*) fails, the valuative dimension of R must exceed the dimension of R. In addition, he studied the asymptotic behavior of the function f (n) = dim R[X1, . .. , X" ] and showed that if R is a domain of finite valuative dimension, then for all suitably large n one has f (n + 1) = f (n) + 1. In [4] Gilmer and Bastida call the sequence {f (i )},0 the dimension sequence of the ring R, and they investigate which sequences are dimension sequences of a certain class of rings. In [2] Arnold and Gilmer determine all sequences which are the dimension sequence of a commutative ring. Both [2] and [4] depend upon a result of Arnold [1, Theorem 5, p. 323] which we refer to as Arnold's formula. We state the result as follows: Received by the editors September 12, 1974 and, in revised form, November 21, 1974. AMS (MOS) subject classifications (1970). Primary 13A15, 13B25, 13C15, 13G05.