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Our purpose in these notes is to present a result for a specific Nakayama Algebra. In essence, it affirms that for any order of simple modules, the cyclic Nakayama Algebras with relations (i.e. ) are not standardly stratified or costandardly stratified.

The aim of this paper is to present a series of results obtained in relation to a particular class of Nakayama algebras. We will begin by recalling the fundamental notions and results of standarly stratified and almost hereditary algebras theory, which will be our main tool.

The concept of standardly stratified algebras emerged as a natural generalization of quasi-hereditary algebras. The class of quasi-hereditary algebras was introduced by Cline, Parshall and Scott in connection with their study of highest weight categories arising in the representation theory of semisimple complex Lie algebras and alge- braic groups.

We present our first result, which allows to obtain the main theorem of the article as an immediate con- sequence.

Theorem 1. Let L be an algebra, such that all non trivial quotient of indecomposable projective has infinite projective dimension, then L is not a standardly stratified algebra for any order of simple modules, unless

Later, we introduce some notions of uniserial algebras and uniserial modules. In section 4, we introduce Nakayama algebras, also known as uniserial generalised Algebras that are studied by Tadasi Nakayama in [

We conclude by presenting a special class of Nakayama algebras, for which is the main result of this paper that we quote below:

Theorem 2. There is no simple order of simple modules for which the cyclical Nakayama Algebras with rela- tions

The following concepts will allow us to define the notions of projective dimension, injective dimension and global dimension; which we will be very useful in demonstrating the fundamental result of this paper.

Definition 1. Let M be an L-module. A projective resolution of M is a complex P whit

where

It is possible show that being L a K-algebra, it follows that every

Definition 2. Let M be an L-module. A minimal projective resolution of M is a projective resolution of M such that

Dually we define concepts injective resolution and minimal injective resolution. Is possible also prove that if L is a finite dimensional algebra then all module in

Definition 3. Projective dimension of L-module M is the number

M of length n and M does not have projective resolution of length

Dually it has the injective dimension of a L-module M.

Definition 4. Let L be an finite dimensional K-algebra. The global dimension

Let R be a commutative Artin ring and L a basic Artin algebra over R. As further we assume full subcategories of

The principal results of this section can be find in [

Definition 5. Let L be a Artin algebra and

1.

2.

In the following we consider that L denote an

to

Definition 6. Let M be a L-module. A normal series in M is a sequence of submodules

The number t is called the length of the series. The quotients

of composition is a normal series whose factors are simple modules, i.e., a normal serie which can not be refined to another longest.

If X is a L-module, we denote by

For

Definition 7. Standard module

Note that the above definition implies that

Definition 8. An algebra L is said standardly stratified if

Note that if L is standardly stratified the projective modules are in

The following example will allow us to understand the theory discussed above.

Example 3.1. Let L be the algebra given by the following quiver whit relations

We have to:

It is not difficult to check that this algebra is standardly stratified and costandardly stratified only at orders for respective simple modules given below:

1. To order

2. To order

We denote by

Proposition 3. Let L be an standardly stratified algebra, then

The following theorem is the first result that present us in this paper. It will allow us to obtain, as an immediate consequence, our main result.

Theorem 4. Let L be an algebra, such that all non trivial quotient of indecomposable projective has infinite projective dimension, then L is not standardly stratified algebra for any order of simple modules, unless

Proof. It’s clear that

Throughout, L is assumed to be a finite dimensional K-algebra, defined over an algebraically closed field K.

The principal results of this section can be find in [

Definition 9. Let M be a L-module. Radical series M is defined as follows:

We agree to denote by

We can define inductively soclo series for module M as:

where

We denote

Note that if

series M is finite. How

We can observe that

is finit.

Proposition 5. Let

Definition 10. Let

Is necessary introduce new notion for define the Nakayama Algebras.

Definition 11. Let M be an L-module. We say that M is uniserial if M possesses exactly one composition series.

Lemma 4.1 Let M be an L-module. Next conditions are equivalents.

1. M is uniserial;

2. Radical series of M is a composition serie;

3. Soclo series of M is a composition serie;

4.

Definition 12. Let L be an K = - algebra. L is right serial if all right indecomposable projective is a uniserial L-module. Dually define us the left indecomposable projective notion.

If

Theorem 6. A basic K-algebra L is left serial izquierda if and only if for each vertex

Corollary 1. A basic K-algebra L is right serial if and only if for each vertex

Definition 13. The algebra L is a Nakayama Algebra if every projective indecomposable and every injective indecomposable L-module is uniserial.

It is possible to characterize Nakayama Algebras through its underlying quiver.

Theorem 7. A basic and connected algebra L is a Nakayama Algebra if and only if

or (cyclical)

Proof. Immediately of Theorem 6 and Corollary 1.

Let L be a Nakayama algebra. We denote

In [

Theorem 8. There is no simple order of simple modules for which the cyclical Nakayama Algebras with rela- tions

Proof. It is easy to see that every projective module

Note that the length of L-module

in which, again, we note us that the length of L-module

Inductively, given a module

Commentary 5.1. Generally Nakayama algebras that are not

We thank the Editor and the referee for their comments.

José Fidel Hernández Advíncula,Rafael Francisco Ochoa de la Cruz, (2015) Stratification Analysis of Certain Nakayama Algebras. Advances in Pure Mathematics,05,850-855. doi: 10.4236/apm.2015.514079