Projective Indecomposable Modules Research Articles

K-Gorenstein algebras and syzygy modules

We investigate for an artin algebra Λ when categories of dth syzygy modules are closed under extensions. In this case they are functorially finite resolving, and have an associated cotilting module. We show that this holds for all d for algebras Λ which are k-Gorenstein for all k, that is, in a minimal injective resolution 0 → Λ → I 0 → I 1 → ⋯ → I j → ⋯ we have pd Λ I j ⩽ j for all j. from this we get a correspondence between indecomposable projective and indecomposable injective modules over these algebras, which can be applied to prove that if Λ is k-Gorenstein for all k and id Λ Λ < ∞, then id Λ Λ < ∞.

The structure of the projective indecomposable modules of 3̂𝑀₂₂ in characteristic 2

This paper presents the socle series of the projective indecomposable modules for the triple cover 3 ^ M 22 \hat 3{M_{22}} in characteristic 2. The results are obtained by computational means; the methods as well as the constructive approach are explained.

The structure of the projective indecomposable modules of the Mathieu groupM 12 in characteristic 2 and 3

The structure of the projective indecomposable modules of the Mathieu groupM 12 in characteristic 2 and 3

Supplement to The Structure of the Projective Indecomposable Modules of the Suzuki Group Sz(8) in Characteristic 2

Supplement to The Structure of the Projective Indecomposable Modules of the Suzuki Group Sz(8) in Characteristic 2

The projective indecomposable modules for the higman-sims group in characteristic 3

The projective indecomposable modules for the higman-sims group in characteristic 3

The loewy structure of the projective indecomposable modules for the mathieu groups in characteristic 3

The loewy structure of the projective indecomposable modules for the mathieu groups in characteristic 3

The structure of the projective indecomposable modules of the Suzuki group 𝑆𝑧(8) in characteristic 2

This paper describes the socle series of the projective indecomposable modules and of tensor products of simple modules for the simple group Sz ( 8 ) {\text {Sz}}(8) in characteristic 2. The results have been obtained by computational means and the various steps are described. The main algorithm was modified to allow for parallel execution on a network of workstations. This made possible the effective handling of modules of degree 4030.

An Effective criterion for uniqueness of the tilting module

Throughout this note, we assume that k is an algebraically closed field and A is a basic connected finite-dimensional algebra over k (associative, with identity). All modules over an algebra A are finitely generated left A-modules. We usually denote, up to isomorphism, by {P_A(a)|a∈I} the set of all indecomposable projective modules, by {E_A(a)|a∈I} the set of all indecomposable injective modules, and by {S_A(a)|a∈I} the set of all simple modules, where

Universally reflexive algebras

Finite dimensional algebras whose representations are all reflexive are characterized as those over which the homomorphisms between any pair of indecomposable projective modules form a vector space of dimension at most one. This includes incidence algebras over finite preordered sets.

Schurian sp-representation-finite right peak PI-rings and their indecomposable socle projective modules

such that soc(R,) is an essential right ideal in R and it is isomorphic to a direct sum of finitely many copies of P, (called a right peak of R). Here F, , . . . . F, are local rings, F= F, is a division ring, iMj is an ( Fi Fj)-bimodule, and the multiplication in R is given by (Fi F,)-bimodule maps cii,: iMj@iM, + jM, satisfying the natural associativity conditions. We denote by P,, . . . . P,, P, the right indecomposable row ideals of R.

On Loewy series of group algebras of some solvable groups

Loewy series of projective indecomposable modules of group algebras of some solvable groups have been studied by several authors (see [ 1,2,4-63). The purpose of this paper is to calculate Loewy series of projective indecomposable modules of the following group G (see [3]). Throughout this paper, we shall use the next notations. Let p be a fixed prime, let F be a finite field of order qp where q = pr, and let S be the subgroup of the multiplicative group F* of F with order h = (q+’ l)/(q 1). We consider permutation groups on F:

A note on three artinian rings

A module over an artinian ring is uniserial if it has a unique composition series, and an artinian ring is serial if each of its indecomposable projective modules is uniserial. Fuller [4, Theorem 5.4] showed that an artinian ring R is serial if and only if each of left indecomposable projective and injective R-modules is uniserial. The following question was raised in 4, p.134: Is an artinian ring R necessarily serial if each of its indecomposable injective modules is uniserial? Example 1 in this note answers this question in the negative

Generic Cartan invariants for Frobenius kernels and Chevalley groups

Chastkofsky [2] described the generic behavior of the composition factor multiplicities (Cartan invariants) of indecomposable projective modules for finite groups of Lie type in the defining characteristic. More recently Ye [lo] provided a different approach, along with interesting refinements. In each case the reason for the existence of generic Cartan invariants is somewhat hidden behind the complicated character manipulations involved. Here we offer a more conceptual treatment, emphasizing the much simpler picture for the group schemes G,, T associated with the Frobenius kernels G, in the ambient algebraic group. This is closer in spirit to [lo] than to [2], but avoids the messy explicit calculations found in both papers.

Projectives of Large Uniform-Rank, in Krull Dimension 1

It is shown that for every positive integer r there is a (left and right) noetherian domain, of Krull dimension 1, that has an indecomposable projective module of uniform-rank r. Direct-sum decompositions of free modules over this domain need not satisfy uniqueness of the number of indecomposable summands. If desired, the domain can be taken to be an order over a discrete valuation ring, in a finite-dimensional division algebra over a global field.

Left serial rings over which every right module with homogeneous top is a direct sum of hollow modules

Let R be a left and right artinian ring with identity, and / the Jacobson radical of R. In [4], M. Harada has considered a left serial ring R satisfying a condition (*, 2) that every maximal submodule of a direct sum of any two hollow modules is also a direct sum of hollow modules, and characterized such a ring by the structure of eR for each primitive idempotent e. Further it has been shown that the condition (*, 2) is equivalent to saying that every factor module of ejQ)eR is a direct sum of hollow modules for every primitive idempotent e. Modifying this, we here consider the following condition on a projective indecomposable right module eR over a ring R.

On Schur algebras and related algebras, II

This paper is a continuation of [7]. The purpose of [7] was to set up, in the context of reductive algebraic groups, a theory of “generalized Schur algebras.” A special case of our construction gives the Schur algebras S(n, r) studied extensively in [lo]. We concentrate here on the Schur algebras themselves and show that our construction yields new results in their representation theory. By means of the Schur functor (see [ 10, Chap. 6]), we obtain new results in the representation theory of the symmetric groups. The main results are as follows. The Schur algebras S,(n, r), defined over a principal ideal domain R, have finite global dimension (see also [ 1 I). Over a field or complete valuation ring R, every finitely generated projective S,(n, r)-module has a Weyl module filtration. We prove the irreducibility of the socle of certain Weyl modules (see also [15]). We describe a connection between weights of projective indecomposable modules for the Schur algebras in terms of the involution on irreducible representations of symmetric groups given by tensoring with the sign representation. We show that, over a complete valuation ring or a field, every Young module for a symmetric group has a Specht series, in particular the projective indecomposables have Specht series. Finally, we show that, for r <n and a complete valuation ring or field R, the blocks of the Schur algebra S,(n, r) are given by the rule embodied in Nakayama’s conjecture. The notations and conventions are as in the first part of this work [7].

The loewy structure of the projective indecomposable modules for sl(3,3) And its automorphism group in characteristic 3

The loewy structure of the projective indecomposable modules for sl(3,3) And its automorphism group in characteristic 3

On artinian $QF-3$,$1$-Gorenstein rings

A noetherian ring is called 1-Gorenstein if it has the self-injective dimension at most one on both sides. A well known example of artinian QF-3, 1-Gorenstein rings is the triangular matrix ring over a QF ring, which is QF-2, that is, every indecomposable projective module has the simple socle. Conversely Sumioka [11] characterized such a ring as an artinian QF-3, 1-Gorenstein ring with QF maximal quotient ring. But an artinian QF-3, 1Gorenstein ring has not necessarily the QF maximal quotient ring (see §4).On the other hand, Sumioka's result is a generalization of Harada's characterization of artinian QF-3 hereditary rings, which states that a connected artinian ring is QF-3 hereditary if and only if it is Morita equivalent to a triangular matrix ring over a division ring (cf. [3]). Our results in the present paper are closely related to their results mentioned above. First we shall deal with artinian QF-3 hereditary rings, which were investigated by Harada [3] and Iwanaga [41. Our result is as follows.

On the cartan invariants of SL(2,pm)

A new description of the characters of the protective indecomposable modules for SL(2,pm) in characteristic p is given.From this, one can get the Car tan invariants of these groups by computing the products of certain matrices.This method explains why previous complete descriptions of these invariants are so complicated.We also answer a problem of Upadhyaya concerning the total number of composition factors in one particular projective indecomposable module.

On the Cartan matrix of an artin algebra of global dimension two

The purpose of this paper is to show that if n is an artin algebra of g!obal dimension (81 dim) 2, then the determinant of its Cartan matrix equals + I. This generalizes previous results of Donovan and Freislich [2], Igusa and Todorov ]4 ] and Wilson [5]. We recall that if n is an artin algebra (for instance, a finite-dimensional algebra over a field), there are finitely many nonisomorphic indecomposable projective rl modules P,, P ,,..., P,,. We denote by /End,(P.) Hom,,(P,, P,,) the length of Hom,(P,. PI) as an End,(P,) module. Then the h x n matrix C(n j, having knd.,cPi, Hom.,(P,, P,i) as the (i,j)th entry is called the Cartan matrix of rl. Let ,d be now an artin algebra of finite representation typethat is, having finitely many nonisomorphic indecomposable ‘4 modules. Let H= u End,(,tr) Hom.,(M, N)) be the matrix with integral entries, where M, N range through the set of the nonisomorphic indecomposable II modules. Donovan and Freislich [2], have shown that if .4 = kG is a group algebra of finite representation type, then det H = + 1. Then, using associated graded algebras, Wilson showed that det H = + 1 if A is a finite-dimensional algebra of finite-representation type over an algebraically closed field [ 5 IO Independently the same result was proved by Igusa and Todorov [4] without making any assumption on the ground field. Let P be the Auslander algebra of n (see [ 1 ] for further details). that is. r = End,,(M, 0 MT . . . @ M,JoD, where M, ,..., M, are the nonisomorphic indecomposable n modules. Then, it has been shown that H = C(T), where. C(T) denotes the Cartan matrix of the Auslander algebra r. It is known that gl dim r 2. [We recall that