On modules of finite projective dimension

Generalized tilting functors and Ringel–Hall algebras

Let A be a noetherian commutative ring which is a homomorphic image of a Gorenstein ring.The first result in this note is that A is a Cohen-Macaulay ring if and only if every finitely generated /I-module can be embedded in a finitely generated Amodule T which is pointwise of finite injective dimension (i.e.all localizations of 7"are of finite injective dimension).The second result is that if A is Cohen-Macaulay, then every finitely generated Amodule can be embedded in a finitely generated /(-module of finite projective dimension if and only if A is Gorenstein.Let A denote a commutative noetherian ring.As a running hypothesis we will assume that A is a homomorphic image of a Gorenstein ring (a commutative noetherian ring R is Gorenstein if Rv has finite injective dimension as an /?p-module, for every prime ideal p of R; see [2] for other definitions equivalent to this one).However, this class of noetherian rings is rather extensive, since it contains most of the rings occurring in algebraic geometry, and, in particular, it contains the complete local rings, by Cohen's structure theorem [5].All modules will be finitely generated (f.g.), and for a f.g.A-module M and p g Spec A, depth Mp denotes (as usual) the maximal length of a Mpregular sequence in pAp, while grade M is the maximal length of an /(-regular sequence in Ann M. Furthermore, recall that a f.g.nonzero Amodule Q is said to be a Gorenstein module if Í2 pointwise is of finite injective dimension (i.e.inj dim Op< co for all p e Supp O) and depth Í2p= depth Ap for all p e Supp O. Since A is a homomorphic image of a Gorenstein ring, by assumption, there exists a Gorenstein /1-module which is of rank one (i.e.Hom(i2, D.) = A), if A is a Cohen-Macaulay ring (cf.Sharp [II], and see [10], [6] and [7] for further properties of Gorenstein modules).Theorem I.A is a Cohen-Macaulay ring if and only if every f.g.Amodule can be embedded in a f.g.A-module which pointwise is of finite injective dimension.

Let [Formula: see text] be a local commutative Noetherian ring and [Formula: see text] an ideal of [Formula: see text] Assume that [Formula: see text] is a finitely generated [Formula: see text]-module of finite projective dimension [Formula: see text] and [Formula: see text] is a finitely generated [Formula: see text]-module of dimension [Formula: see text] We prove that [Formula: see text] is Artinian. Moreover, if [Formula: see text] is finitely generated, then [Formula: see text] is an [Formula: see text]-cofinite [Formula: see text]-module.

Let M M be a finitely generated module over a local complete intersection R R of characteristic p &gt; 0 p&gt;0 . The property that M M has finite projective dimension can be characterized by the vanishing of ext R i ⁡ ( f n R , M ) \operatorname {ext}_R^i({}^{f^n}\!\! R,M) for some i &gt; 0 i&gt;0 and for some n &gt; 0 n&gt;0 .

Tilting Modules Associated with a Series of Idempotent Ideals

Let ${\mit\Lambda}$ be an artinian ring and let ${\mathfrak r}$ denote its Jacobson radical. We show that a simple module of finite projective dimension has no self-extensions when ${\mit\Lambda}$ is graded by its radical, with at most two simple modules

Central elements and deformations of local rings

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 A be a commutative ring and let E be an A-module with a finite free resolution (see below for definition). Extending results known previously for noetherian rings, it is shown that ann ( E ) = annihilator {\text {ann}}(E)=\text {annihilator} of E is trivial if and only if the Euler characteristic of E = χ ( E ) &gt; 0 E = \chi (E) &gt; 0 ; in addition, if χ ( E ) = 0 , ann ( E ) \chi (E) = 0,{\text {ann}}(E) is dense (i.e. ann ( ann ( E ) ) = 0 {\text {ann}}({\text {ann}}(E)) = 0 ). Also, a local ring is constructed with its maximal ideal with a finite free resolution but consisting exclusively of zero-divisors and thus, contrary to the noetherian case, one does not always have a nonzero divisor in ann ( E ) {\text {ann}}(E) if χ ( E ) = 0 \chi (E) = 0 . Finally, if E has a finite resolution by (f.g.) projective modules it turns out that ann ( ann ( E ) ) {\text {ann}}({\text {ann}}(E)) is generated by an idempotent element.

Approximating Modules by Modules of Finite Projective Dimension

Syzygy Theorems via comparison of order ideals on a hypersurface

In this paper I shall demonstrate certain algebraic properties of ideals of finite homological dimension in local rings.In the first section, I show that no non-zero ideal of finite homological dimension in a local ring can be of zero grade (this is stated by Auslander and Buchsbaum in the appendix to (l), but I cannot find a proof in the literature). Using this result, together with the complex defined by a matrix, which is described by Eagon and Northcott in (2), I prove that an ideal of homological dimension one in a local ring Q may always, for some integer n, be described as the set of determinants of matrices obtained by adjoining to a certain (n–l)× n matrix with elements in the maximal ideal of Q another row with elements arbitrarily chosen in Q. (This result was established in (3), under the additional condition that Q should be a domain.)

Let M be a finitely generated module of rank r and finite projective dimension over a commutative Noetherian local ring S. We show that M has a free direct summand of rank r if and only if its Buchsbaum–Eisenbud invariant J(M) is a principal ideal. As an application, we consider the problem of finding rigid triples. Recall that the module M is said to be rigid if for each i ≥ 0 and each finitely generated S-module N the condition implies for every j ≥ i. A triple of positive integers b = (b 0, b 1, b 2) is said to be rigid if for every commutative Noetherian local ring S one has that every S-module with minimal free resolution of the form 0 → S b 2 → S b 1 → S b 0 → 0 is rigid. We use our criterion for free direct summands of maximal rank to derive a necessary and sufficient condition for a triple b to be rigid in terms of the rigidity of the universal modules of type b over complete regular local rings. In particular, we exhibit the rigidity of any triple of the form (k, m + 1,m), thus providing a new class of rigid modules.

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.

In this chapter we prove a localization theorem of Quillen for singular varieties, and a generalization of it due to Levine. These are then used to prove the so-called “Fundamental Theorem” (9.8), which computes K i (A[t, t −1]), and to relate the study of 0-cycles on normal surfaces to modules of finite length and finite projective dimension over the local rings at singular points. We begin with Quillen’s localization theorem, proved in Higher Algebraic K-Theory II.KeywordsExact SequenceVector BundleLocal RingFull SubcategoryExceptional DivisorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.