Journal of Applied Mathematics and Physics
Vol.05 No.01(2017), Article ID:73865,15 pages
10.4236/jamp.2017.51016
Homological Properties in Relation to Nakayama Algebras
Kwasi Baah Gyamfi, Abraham Aidoo, Dickson Y. B. Annor
Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana

Copyright © 2017 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/



Received: October 30, 2016; Accepted: January 22, 2017; Published: January 25, 2017
ABSTRACT
The goal of this paper is to investigate whether the Ext-groups of all pairs
of modules over Nakayama algebras of type
satisfy the condition
for
for
. We achieve that by discussing the Ext-groups of Nakayama algebra with projectives of lengths
and
using combinations of modules of different lengths.
Keywords:
Quivers, Path algebras, Ext-Groups, Projective Resolutions

1. Introduction
In this paper, we describe homological properties of Nakayama algebras. The algebra Ʌ is a Nakayama algebra if every projective indecomposable and every injective indecomposable Ʌ-module is uniserial. In other words, these modules have a unique composition series, (see Schröer [1] ). Nakayama algebras are finite dimensional and representation-finite algebras that have a nice representation theory in the sense that the finite-dimensional indecomposable modules are easy to describe.
The main contribution of this paper is to investigate whether the Ext-groups of all pairs
of modules over Nakayama algebras of type
satisfy the condition
for
for
.
2. Preliminary Notes
This section will briefly discuss the properties of Nakayama algebra and some related propositions. We consider also proofs of some of the propositions. All the information presented here can be found in deeper details in the book from Rotman [2] .
Definition: Let
be an artin algebra. A
-module
is called a uniserial module if the set of submodules is totally ordered by inclusion.
Proposition 1: The following are equivalent for
-module
1. A is uniserial.
2. There is only one composition series for
.
3. The radical filtration of
is a composition series for
.
4. The socle filtration of
is a composition series for
.
5.
, where
is the length of
and
is the radical length of
.
Proof: See Rotman [2] for the details.
Proposition 2: The following are equivalent for an artin algebra
.
1.
is a sum of uniserial modules .
2.
is a sum of uniserial modules for all ideals
of
.
3.
is a sum uniserial modules.
Proof:
and
are trivial. If
is the sum of uniserial modules, then
and
which are factors of
are also a sum of uniserial modules.
Let
be an indecomposable projective
-module. We show that
is uniserial by induction on
when
. When
, there is nothing to prove. Suppose
. Let the radical filtration of
be;
such that
is simple for
.
When
, we have
. Hence by induction hypothesis,
is uniserial. Considering the exact sequence
, which also implies that
is uniserial, hence
is also uniserial.
If
, then
is clearly uniserial, so we have to assume that
. From proposition 1, it follows that
is simple for
. To show that
is uniserial, then it is sufficient by proposition 1 to prove that
is also simple.
Let
be a projective cover. Since
is simple,
must be indecomposable and so
is uniserial. But we have an epimorphism
which shows that
is simple.
Proposition3: Proposition 2 (1).
Let
be a
Tr-orbit of
. Suppose there is a projective module
in
. Then we have the following
1.
of non-zero objects in
.
2.
is finite if and only if
is injective for some
in
. Moreover, if
is injective, then
.
Proof: By proposition 1,
if and only if
is projective. Since
is projective module in
for all
. Hence the claim in 2 (1).
We claim that if
with
we have
which implies that
which is not possible since
.
can therefore be finite if
for some
. Since
then
is injective in
. We know therefore that if
is injective, then
.
3. Results
The main goal of this work is to investigate whether the Ext-groups of all pairs (M, N) of modules over Nakayama algebras of type (n, n, n) satisfy the condition:
for
for
, where
is a positive integer. We discuss the Ext-groups of Nakayama algebras with projectives of lengths
and
using combinations of modules of different lengths. Reader may refer to Auslander, M et al. [3] for ideas illustrated in this section.
We begin with the Ext-groups of Nakayama algebra with projectives of length
using the combinations
, and
where
and
are modules of lengths
and
respectively,
is also module of length
.
Let
be a path with the relations
and
where the length of each relation is
. Let
The projectives of the above path algebra are as follows:
The above projectives
and
each has length
. The minimal projective resolution of the module
of length
is given as;
where
and
for
.
The combinations have been reduced to four class because in the above minimal projective resolution of the module of length
, we see that it is in the same group as the module of length
. The modules of lengths
and
therefore have the same properties with respect to the conditions;
for
for
.
From the above resolution, we have;
Let
, the pd
since the resolution is periodic. The period is 6. The truncation of the resolution is given as;
The map
is the multiplication by
is the multiplication by
is multiplication by
is the multiplication by
is the multiplication by
and
is the multiplication by
.
Applying
where
is the module
of length
we have;
where
and
.
We have the following Ext-groups;
where
and
are modules of length
and
respectively. We first compute the kernel for
We have
where
is in the coordinate number
, therefore
is in the coordinate number
. Hence
and consequently
,
Next we compute the image of
We have that
where
is in the coordinate number
. This shows that
Hence
, for
. We compute
We compute the kernel for
we have
where
is the coordinate number
. This shows that
where
is in the coordinate number
. Next we compute the image of
We see that
where
is in the coordinate number
and therefore
is in the coordinate number
. This implies
and hence
,
Hence
for
.
We compute
First we compute the kernel for
We have
where
is in the coordinate number
and therefore
is in the coordinate number
. Hence
and consequently
. This shows that
We compute the image of
to have
where
is in the coordinate number
. This shows that
Hence
for
.
Next we have
Similarly, we have
where
is in the coordinate number
, then
where
is in the coordinate number
. Hence
Again we have
We compute the kernel for
to have
where
is in the coordinate number
and therefore
is in the coordinate number
and consequently
This show that
.
We compute the image for
to get
where
is in the coordinate number
. This shows that
Hence
. This shows that
for
and
for
which implies
and
It follows from the computation of the fifth Ext-group that the condition
for
for
holds.
We calculate the Ext-groups for the combination
.
where
is the module of length
We compute the kernel for
we have
where
is in the coordinate number
and therefore
is in the coordinate number
. Hence
and consequently
. This shows that
Computing the image for
we have
where
is in the coordinate number
. This shows that
We therefore have
for
.
Next we have
We compute the kernel for
we have
where
is in the coordinate number
. This shows that
where
is in the coordinate number
. We compute the image for
We have
where
is in the coordinate number
and therefore
is in the coordinate number
. Hence
. This shows that
We therefore have
for
. Next we have
Similarly, we have;
where
is in the coordinate number
. We also have
where
is in the coordinate number
. We therefore have
for
. Next we have
We compute the kernel for
We have
where
is in the coordinate number
. This shows that
where
is in the coordinate number
. We compute the image for
We have
where
is in the coordinate number
and therefore
is in the coordinate number
and therefore
,
We therefore have
for
. Finally, we have
We compute the kernel for
we have
where
is in the coordinate number
and therefore
is in the coordinate number
and consequently
. This shows that
We compute the image for
we have
where
is the
coordinate number
and hence
We therefore have
for
for
,
for
and
for
. We therefore have
and
.
It follows from the computations of the forth and fifth Ext-groups that the condition
for
for
holds.
We now discuss the Ext-groups of the Nakayama algebras with the projectives of length
using the combination
, where
and
are also modules of length
and
respectively.
Let
be a path with the relations
and
where the length of each relation is
is a positive integer. Let
The projectives of the above path algebra are as follows:
The above projectives
each has length
. The minimal projective resolution of module
of length
is given as;
where
is the module of length
and
for
and
is a multiplication by
is a multiplication by
is a multiplication by
is a multiplication by
is a multiplication by
and
is multiplication by
.
The combinations have been reduced to four class because in the above resolution, we see that the module of length
is in the same group with the module of length
. The two modules therefore have the same properties with respect to the condition
for
for
.
From the above resolution we have,
The
since the resolution is periodic. The period is 6. The truncation of the above resolution is given as:
Applying
where
is the module of length
, we have
where
and
.
We discuss the Ext-groups of the Nakayama algebras with projectives of length
by considering the case:
.
is the module
of length
and
is the module
of length
.
The minimal projective resolution of the module
of length
is given as;
where
and
for
is a multiplication by
is a multiplication by
is a multiplication by
is a multiplication by
is a multiplication by
and
is a multiplication by
and
is the module of length
.
Considering the above resolution, we have
The
since the resolution is periodic. The period ic 6. The truncation of the above resolution is given as;
Applying
where
is in the module
of length
, we have
where
and
.
We compute the Ext-groups for
. The first Ext-group is;
where
is the module of length
. We compute the kernel for
We have
where
is in the coordinate number
and therefore
is in the coordinate number
and consequently,
. This shows that
We compute the image
We have
where
is in the coordinate number
. Hence
We have
for
. Next, we compute the second Ext-group.
We compute the kernel for
We have
where
is in the coordinate number
. This shows that
where
is in the coordinate number
. We compute the image for
We have
where
is in the coordinate number
and therefore
is the coordinate number
and
. This implies that
where
is in the coordinate number
. We have
for
. We compute the third Ext-group.
We compute the kernel for
We have
where
is in the coordinate number
and therefore
is in the coordinate number
and consequently,
. Hence
We compute the image for
We have
where
is in the coordinate number
. This implies that
We therefore have
for
. We discuss the fourth Ext-group.
We compute the kernel for
We have
where
is in the coordinate number
. Hence
where
is in the coordinate number
. We compute the image for
to have
where
is in the coordinate number
and therefore
is in the coordinate number
. Consequently,
. This shows that
where
is in the coordinate number
. We have
for
. We discuss the fifth Ext-group.
We compute the kernel for
We have
where
is in the coordinate number
and therefore
is in the coordinate number
and consequently,
. This shows that
We compute the image for
We have
where
is in the coordinate number
. Hence
We therefore have
for
.
for
and 4, and
for
. We therefore have
and
.
It follows from the fifth Ext-group that the condition
for
for
holds.
4. Conclusion
The study found that the Ext-groups of pairs
of modules over Naka- yama algebras of type
satisfy the condition
for
for
and justified the claim with projectives of lengths
and
by using combinations of modules of different lengths.
Acknowledgements
We are grateful to the referee for many useful suggestions.
Cite this paper
Gyamfi, K.B., Aidoo, A. and Annor, D.Y.B. (2017) Homological Properties in Relation to Nakayama Algebras. Journal of Applied Mathematics and Physics, 5, 153-167. http://dx.doi.org/10.4236/jamp.2017.51016
References
- 1. Schröer, J. (2016) Nakayama Algebras, 2016. http://www.math.un-bonn.de/people/Schroer/fd-atlas/FD-NakayamaAlgebras.pdf
- 2. Rotman, J.J. (2009) An Introduction to Homological Algebra. 2nd Edition, Universitex, Springer.
- 3. Auslander, M., Reiten, I. and Smalø, S. (1995) Representation Theory of Artin Algebras. Series: Cambridge Studies in Advanced Mathematics (No. 36). Cambridge University Press, Cambridge.