Loewy Length Research Articles

An elementary construction of tilting complexes

Let A be an artin algebra and e∈ A an idempotent with add( eA A )=add( D( A Ae)). Then a projective resolution of Ae eAe gives rise to tilting complexes {P(l) •} l⩾1 for A, where P( l) • is of term length l+1. In particular, if A is self-injective, then End K( Mod-A) (P(l) •) is self-injective and has the same Nakayama permutation as A. In case A is a finite dimensional algebra over a field and eAe is a Nakayama algebra, a projective resolution of eAe over the enveloping algebra of eAe gives rise to two-sided tilting complexes { T(2 l) •} l⩾1 for A, where T(2 l) • is of term length 2 l+1. In particular, if eAe is of Loewy length two, then we get tilting complexes { T( l) •} l⩾1 for A, where T( l) • is of term length l+1.

On the Loewy Length and the Noetherian Dimension of Artinian Modules

(2002). On the Loewy Length and the Noetherian Dimension of Artinian Modules. Communications in Algebra: Vol. 30, No. 3, pp. 1077-1084.

DONOVAN CONJECTURE AND LOEWY LENGTH FOR PRINCIPAL 3-BLOCKS OF FINITE GROUPS WITH ELEMENTARY ABELIAN SYLOW 3-SUBGROUP OF ORDER 9

In modular representation theory of finite groups there has been a well-known conjecture due to P. Donovan. Donovan conjecture is on blocks of group algebras of finite groups over an algebraically closed field k of prime characteristic p, which says that, for any given finite p-group P, up to Morita equivalence, there are only finitely many block algebras with defect group P. We prove that Donovan conjecture holds for principal block algebras in the case where P is elementary abelian 3-group of order 9. Moreover, under the same assumption, namely, if G is a finite group with elementary abelian Sylow 3-subgroup P of order 9, then the Loewy length of the principal block algebra B 0 (kG) of the group algebra kG is 5 or 7. The results here depend on the classification of finite simple groups.

Defect 3 Blocks of Symmetric Group Algebras, I

In this paper, we study if the following properties are present in defect 3 blocks of symmetric group algebras kGn, where k has characteristic p (≥5):•the Ext-quiver is bipartite;•the p.i.m.'s have a common Loewy length 7.We obtain a criteria for a defect 3 block B of kGn to inherit these properties from a defect 3 block B̃ of kGn−1, with which it forms a [3:1]-pair. We also verify that the principal blocks of kGn (3p≤n≤4p−1) have these properties.

Defect 3 Blocks of Symmetric Group Algebras

Abstract In this paper, we study if the following properties are present in defect 3 blocks of symmetric group algebras k G n , where k has characteristic p ( ≥ 5): • the Ext-quiver is bipartite; • the p.i.m.'s have a common Loewy length 7. We obtain a criteria for a defect 3 block B of k G n to inherit these properties from a defect 3 block B of k G n − 1 , with which it forms a [3 : 1]-pair. We also verify that the principal blocks of k G n (3 p ≤ n ≤ 4 p − 1) have these properties.

Loewy series, V-modules and trace ideals

If R Vis a V-module and (R V W S) is a Morita context in which (S/WV) s is flat, then the trace ideal WVis left V-module over S. If, in additionS:(S/WV) is flat and S/WVis a fully left idempotent ring, then Sis also fully left idempotent. The lower (upper) Loewy length of R Vprovides an upper bound for the corresponding Loewy length of s(WV).

Some Remarks on Index and Generalized Loewy Length of a Gorenstein Local Ring

Some Remarks on Index and Generalized Loewy Length of a Gorenstein Local Ring

Cartan invariants of group algebras of finite groups

We give a result on Cartan invariants of the group algebra k G kG of a finite group G G over an algebraically closed field k k , which implies that if the Loewy length (socle length) of the projective indecomposable k G kG -module corresponding to the trivial k G kG -module is four, then k k has characteristic 2. The proof is independent of the classification of finite simple groups.

Semiartinian V-Rings and Semiartinian Von Neumann Regular Rings

Semiartinian right V-rings, which we call right SV-rings, form a special class of Von Neumann regular rings. We characterize these rings by the fact that every factor ring imbeds as a subring in a direct product of right full linear rings containing the socle. If R is a semiartinian ring with all primitive factor rings artinian, then the condition of being an SV-ring is right/left symmetrical and is equivalent to being regular. On the other hand, if R is a right and left SV-ring, then all primitive factor rings of R are artinian. For right SV-rings whose proper ideals are prime we show that the condition of being unit-regular is equivalent to being directly finite. On the other hand we show that there exists a directly finite right SV-ring which is not unit-regular. Furthermore we provide two constructions. For any given ordinal ξ, the first one gives a prime, unit-regular right SV-ring of Loewy length ξ + 1, which is not a left V-ring, and is hereditary if ξ is a natural number; the second one gives a directly infinite right SV-ring, not a left V-ring, whose Loewy length is ξ + 2 if ξ is a natural number and is ξ + 1 otherwise. These constructions are general enough to produce a wide supply of SV-rings, starting from given ones.

The Associated Graded Ring and the Index of a Gorenstein Local Ring

Let $(R,\mathfrak {m},k)$ be a Gorenstein local ring. It is shown that if the associated graded ring $G(R)$ of $R$ is Cohen-Macaulay, then the index of $R$ is equal to the generalized Loewy length of $R$.

Projective modules of finite groups with elementary abelian Sylow 3-subgroups of order 9 in characteristic 3

Let G be any finite group with elementary abelian Sylow 3-subgroups of order 9, and let F be any field of characteristic 3. Then, the Loewy length of the projective cover of the trivial FG-module is at least 5. This lower bound is the best possible.

The associated graded ring and the index of a Gorenstein local ring

Let ( R , m , k ) (R,\mathfrak {m},k) be a Gorenstein local ring. It is shown that if the associated graded ring G ( R ) G(R) of R R is Cohen-Macaulay, then the index of R R is equal to the generalized Loewy length of R R .

Algebras with Large Homological Dimensions

An example is given of a semiprimary ring with infinite finitistic dimension. The construction shows that the global dimensions of finite dimensional algebras of finite global dimension cannot be bounded by a function of only Loewy length and the number of nonisomorphic simple modules.

Algebras with large homological dimensions

An example is given of a semiprimary ring with infinite finitistic dimension. The construction shows that the global dimensions of finite dimensional algebras of finite global dimension cannot be bounded by a function of only Loewy length and the number of nonisomorphic simple modules.

The endomorphism ring of a finite-length module

Let M be an R-module of finite length. For a simple R-module A, let ℓA denote the nuber of times the isomorphism type of A appears in a composition chain of M, and let σ denote the maxinium of the ℓA, A ranging over all simple submodules of M. Let S be the endomorphism ring of M. We show that the Loewy length of S is bounded by σ.

A remark on the Loewy-layers of the modular group algebra of a finite p-group

We consider the dimensions c i of the Loewy factors of the modular group algebra KG of a finite p-group G. It is well-known that c i = c s−i where s is the Loewy length of KG. For p = 3 and 5 we describe p-groups with the property that c i ⩽ c i−1 for some i⩽ 1 2 S thus settling an old open problem. Also, the similar question for restricted Lie algebras with nilpotent p-map is studied.

An upper bound for Loewy lengths of projective modules in p-solvable groups

An upper bound for Loewy lengths of projective modules in p-solvable groups

Artin rings of global dimension two

In this paper, we show that a left artin ring A of global dimension at most two has Loewy length at most 2” - 1, where n is the number of simple components of A/rad(A). We assume that the reader is familiar with the trends (but not necessarily the details) of contemporary representation theory of artin rings. Let us record a few conventions. The rings discussed here will be left artinian, unless otherwise specified, and will sometimes be finite-dimensional algebras over an algebraically closed field, henceforth denoted by k. Modules will be finitely generated left modules unless otherwise specified. The radical of A will usually be denoted by r; the Loewy

Projective modules in the category 𝒪_{𝒮}: Loewy series

Let g \mathfrak {g} be a complex, semisimple Lie algebra with a parabolic subalgebra p S {\mathfrak {p}_S} . The Loewy lengths and Loewy series of generalized Verma modules and of their projective covers in O S {\mathcal {O}_S} are studied with primary emphasis on the case in which p S {\mathfrak {p}_S} is a Borel subalgebra and O S {\mathcal {O}_S} is the category O \mathcal {O} . An examination of the change in Loewy length of modules under translation leads to the calculation of Loewy length for Verma modules and for self-dual projectives in O \mathcal {O} , assuming the Kazhdan-Lusztig conjecture (in an equivalent formulation due to Vogan). In turn, it is shown that the Loewy length results imply Vogan’s statement, and lead to the determination of Loewy length for the self-dual projectives and certain generalized Verma modules in O S {\mathcal {O}_S} . Under the stronger assumption of Jantzen’s conjecture, the radical and socle series are computed for self-dual projectives in O \mathcal {O} . An analogous result is formulated for self-dual projectives in O S {\mathcal {O}_S} and proved in certain cases.

On loewy lengths of projective modules for p-solvable groups

On loewy lengths of projective modules for p-solvable groups