Modules Of Finite Projective Dimension Research Articles

The grothendieck group of the category of modules of finite projective dimension over certain weakly triangular algebras

In this paper we study the category of finitely generated modules of finite projective dimension over a class of weakly triangular algebras, which includes the algebras whose idempotent ideals have finite projective dimension. In particular, we prove that the relations given by the (relative) almost split sequences generate the group of all relations for the Grothendieck group of P <∞(Λ) if and only if P <∞(Λ) is of finite type. A similar statement is known to hold for the category of all finitely generated modules over an artin algebra, and was proven by C.M.Butler and M. Auslander ( [B] and [A]).

A Homological Bridge Between Finite and Infinite-Dimensional Representations of Algebras ?

Given a finite-dimensional algebra Λ, we show that a frequently satisfied finiteness condition for the category $$P^\infty$$ Λ-mod) of all finitely generated (left) Λ-modules of finite projective dimension,namely contravariant finiteness of $$P^\infty$$ (Λ-mod) in Λ-mod, forces arbitrary modules of finite projective dimension to be direct limits of objects in $$P^\infty$$ (Λ-mod). Among numerous applications, this yields an encompassing sufficient condition for the validity of the first finitistic dimensionconjecture, that is, for the little finitistic dimension of Λ to coincide with the big (this is well known to fail overfinite-dimensional algebras in general).

Covers and envelopes over Gorenstein rings

A module over a Gorenstein ring is said to be Gorenstein injective if it splits under all modules of finite projective dimension. We show that over a Gorenstein ring every module has a Gorenstein injective envelope. We apply this result to the group algebra $\hat{Z}_{\rho}G$ (with $G$ a finite group and $\hat{Z}_{p}$ the ring of $p$-adic integers for some prime $p$) and show that ever finitely generated $\hat{Z}_{p}G$-module has a cover by a lattice. This gives a way of lifting finite dimensional representations of $G$ over $Z/(p)$ to modular representations of $G$ over $\hat{Z}_{p}$.

Modules of finite projective dimension and cocovers

Modules of finite projective dimension and cocovers

A remarkable property of the (co) syzygy modules of the residue field of a nonregular local ring

Let R be a commutative noetherian local ring with residue field k. We introduce two new invariants of R-modules and compute them for the module k. As a consequence, we show, when R is nonregular, that no direct sum of the syzygy modules of k surjects onto a nonzero module of finite projective dimension. A dual statement is also true: no direct sum of the cosyzygy modules of k (in an injective resolution) contains a nonzero module of finite injective dimension with nonzero scale.

Approximating Modules by Modules of Finite Projective Dimension

Approximating Modules by Modules of Finite Projective Dimension

Zero Cycles and Projective Modules

Let A be a reduced affine k-algebra of dimension n over an algebraically closed field k. Let FnKo(A) denote the sub-group of Ko(A) generated by the images of all the residue fields of all smooth maximal ideals of height n. For a module of finite projective dimension M, let (M) denote the image of M in Ko(A). For a projective A-module P of rank n, we define the nth Chern class of P to be: Cn(P) = E(-l)i(AiP*), where P* is the dual of P. Suppose FnKo(A) has no (n 1)! torsion; then our main result (Th. 3.8) is that P has a free direct summand of rank one if and only if Cn(P) = 0. When the characteristic of k is zero or A is normal and n > 3, it is known that FnKo(A) is torsion-free ([Le], [Sr]). Hence our theorem is applicable in these cases. Also when A is regular, FnKo(A) coincides with CHn(X) the Chow group of zero cycles of Spec A and Cn(P) coincides with the usual nth Chern class as defined by Grothendieck (see [Fu]). When n < 3 and A is regular, this result was proved in [MKM]. When n = 3 and the characteristic of k is not equal to two, this result in [MKM] was extended to the singular case by M. Levine. In this paper, we extend the results of [MKM] to all dimensions. With the assumption stated above on torsion in FnKo(A), we first show that (Cor. 3.4) if I C A is a local complete intersection of height n, then (A/I) is zero in Ko(A) if and only if I is a complete intersection. Corollary 3.4 together with a result of Mohan Kumar ([MK], Cor. 1.9 here) at once gives Theorem 3.8. In Section 1, we give some preliminaries and generalize results in [MK2]. These results are crucially used in the rest of the paper. The basic ideas in Section 1 are all taken from [MK2]. In Section 2, we prove Theorem 2.2, which strengthens a result of Boratynski [Bo] for local complete intersections with trivial co-normal bundle. Theorem 2.2 is one of the crucial ingredients in the proofs of the results in Section 3. As an amusing application of Theorem 2.2, we give a new proof of a theorem of Srinivas ([Sr]) about torsion in zero cycles

Finistic dimensions of serial-like rings

Bounds are determined for the finitistic dimensions of one-sided almost serial rings in terms of the projective dimensions of the simple modules of finite projective dimension and in terms of the number of the simple modules, and we discuss when the little finitistic dimension and the big finitistic dimension (to be defined below) are equal for some serial-like rings.

On constructing stably equivalent functors

In this case, the left module BT becomes again a generalized tilting module with End(,T) = A [8]. It should be noted here that the notion of generalized tilting modules is fairly general and the class of such modules contains all tilting modules of finite projective dimension in the sense of Miyashita [6] and also all cotilting modules of finite injective dimension. Especially, the injective cogenerator DA is always a generalized tilting module even if the regular module A has infinite injective dimension on each side. In the joint work with Tachikawa [7], we proved the existence of a stably equivalent functor &-T(A) %&-T(B) for any genuine tilting module defined by Happel and Ringle [S] by introducing torsion and torsion-free resolutions of modules. We have tried to generalize the existence

The MacRae invariant and the first local Chern character

Let R be a commutative Noetherian ring, and let M be a finitely generated module of finite projective dimension. In his investigation of invariants of these modules, MacRae [6] constructed an invertible ideal G(M) associated to M which describes th; part of the support of M of codimension 1. The fact that G(M) is invertible implies many properties of the support of M, and it has recently been used by Foxby [3] to prove some conjectures on intersection multiplicities of modules of finite projective dimension with modules of Krull dimension one. In this paper we generalize the construction of MacRae to a bounded complex E* of locally free sheaves on a Noetherian scheme Y and show that this can be used to describe the first local Chern character of E*. A bounded complex of locally free sheavesoof finite rank will be called a perfect complex. For technical reasons, we assume that Y is connected and quasi-projective over an affine scheme. Let X be the support of E*, denoted Supp(E*); this can be defined as the set of points of Y where E* is not exact, or, equivalently, as the union of the supports of the homology modules Hi(E*). We assume that X is contained in some Cartier divisor. Locally, this means that the ideal defining X contains a non-zero-divisor, and if E* is a resolution of a module, our assumption follows whenever X is a proper subset of Y. An equivalent formulation of this condition is that X contains no points y of Y such that the local ring dy has depth zero; such a point will be called a point of depth zero. In this situation we construct a Cartier divisor G(E*) on Y generalizing the MacRae invariant. If Y is quasi-projective over a regular local ring (this includes, among others, the case where Y= SpecR and R is a complete local ring), there is a theory of local Chern characters defined for perfect complexes on Y. For a complex E* with support X as above, for any scheme Y' together with a map of finite type f: Y' Y,

Tilting modules of finite projective dimension

Tilting modules are well known, and are useful in the representation theory of artinian algebras ([2-4, 6, 11]). The purpose of this paper is to define tilting modules of finite projective dimension and prove some fundamental results. A unital left A-module A T over a ring A is said to be a tilting module of projective dimension 0 is an integer) if it satisfies the following three conditions: (1) A T has a projective resolution

Modules of finite projective dimension with negative intersection multiplicities

Let x~ be the image of X~ in R. The main object of this paper is to study modules of finite length and finite projective dimension over R. One consequence of our study will be counterexamples to several conjectures concerning the behavior of intersection multiplicities for modules of finite projective dimension. Let / denote length. If S is a Noetherian ring, M, N are finitely generated S-modules, l(M| ) is finite, and d=pdsM<~, we may define, following Serre IS] d Zs(M, N)= z(M, N) = ~ ( 1)'/(TorS(M, N)).

Patching modules of finite projective dimension

(1985). Patching modules of finite projective dimension. Communications in Algebra: Vol. 13, No. 7, pp. 1461-1473.

On Finite Weak and Injective Dimension

An extension is given of the theorem relating projective dimension 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.

On finite weak and injective dimension

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.

The construction of a module of finite projective dimension from a finitely generated module of finite injective dimension

The construction of a module of finite projective dimension from a finitely generated module of finite injective dimension

Adic approximation of complexes, and multiplicities

In [2, Section 1.6] Peskine and Szpiro prove a theorem on adic approximations of finite free resolutions over local rings which, together with M. Artin’s Approximation Theorem [1], allows them to “descend” modules of finite projective dimension over the completions of certain local rings to modules of finite projective dimension over finite étale extensions of those rings. In this note we will prove a more general result, which deals with the change in homology under an adic approximation of any complex of finitely generated modules over a noetherian ring, and which allows one to descend not only modules of finite projective dimension, but also the Euler characteristic or intersection multiplicity of two such modules.

𝑛-Gorenstein rings

The object of this note is to study commutative noetherian n-Gorenstein rings. The first result is: if each module satisfying Samuel’s conditions ( a i ) ({{\text {a}}_i}) for some i ≦ n i \leqq n is an ith syzygy, then the ring is n-Gorenstein. This is the converse to a theorem of Ischebeck. The next result characterizes n-Gorenstein rings in terms of commutativity of certain rings of endomorphisms. This answers a question of Vasconcelos. Finally the last result deals with embedding of finitely generated modules into finitely generated modules of finite projective dimension.

Homological dimension of simple pro-$p$-Iwahori–Hecke modules

Let $G$ be a split connected reductive group defined over a nonarchimedean local field of residual characteristic $p$, and let $\mathcal{H}$ be the pro-$p$-Iwahori--Hecke algebra associated to a fixed choice of pro-$p$-Iwahori subgroup. We explore projective resolutions of simple right $\mathcal{H}$-modules. In particular, subject to a mild condition on $p$, we give a classification of simple right $\mathcal{H}$-modules of finite projective dimension, and consequently show that most simple modules have infinite projective dimension.