The Second Hochschild Cohomology Group for One-Parametric Self-Injective Algebras ()
1. Introduction
This paper determines the second Hochschild cohomology group for all standard one-parametric but not weakly symmetric self-injective algebras of tame representation type. Bocian, Holm and Skowroński give, in [1], a classification of these algebras by quiver and relations up to derived equivalence. The algebras in [1] are divided into two types, namely the algebra 
where 
are integers such that p, 
and 
and the algebra 
where
. Thus the second Hochschild cohomology group will be known for all the classes of the algebras given in [1]. We remark that an algebra of the type 
is never isomorphic to an algebra of the type 
as their stable Auslander-Reiten quivers are not isomorphic. We refer the reader to [1] which gives precise conditions for two algebras of the same type 
or 
to be isomorphic.
We start, in Section 2, by introducing the algebras
, for both types, by quiver and relations. Section 3 of this paper describes the projective resolution of [2] which we use to find
. In the third section, we determine 
for the algebra
, considering separately the cases 
and
. The main result in this section is Theorem 4.9, which shows that 
has dimension 1 for
. This group measures the infinitesimal deformations of the algebra
; that is, if 
then 
has no non-trivial deformations, which is not the case here. We include, in Section 4, Theorem 4.10 where we find a non-trivial deformation 
of 
associated to our nonzero element 
in
. This illustrates the connection between the second Hochschild cohomology group and deformation theory. In the final section, we determine 
for
. The main result in Section 5 is Theorem 5.4 which shows that
. The results we found in this paper are in contrast to the majority of self-injective algebras of finite representation type (see [3]). Since Hochschild cohomology is invariant under derived equivalence, the second Hochschild cohomology group is now known for the standard one-parametric but not weakly symmetric self-injective algebras of tame representation type which are derived equivalent to the algebra of the type 
or
.
2. The One-Parametric Self-Injective Algebras
In this chapter we describe the algebras of [1]. We start with the algebra
. Let K be an algebraically closed field and let 
be integers such that p, 
and
. From [1, Section 5], 
has quiver
:
where, for any
, 
denotes the path
and 
denotes the path
Then 
where 
is the ideal generated by the relations
• 
, for
• 
, for
• 
for
, 
,
for
, 
,
for
, and
where
.
Next we describe the algebra 
For
, 
is given in [1, Section 6] by the quiver
:
Then 
where 
is the ideal generated by the relations:
1) 
2) 
3) for all 
Note that we write our paths from left to right.
In order to compute
, the next section gives the necessary background required to find the first terms of the projective resolution of 
as a 
-bimodule. Section 4 and Section 5 uses this part of a minimal projective bimodule resolution for our algebras to determine the second Hochschild cohomology group and provides the main results of this paper.
3. Projective Resolutions
To find the second Hochschild cohomology group
, we could use the bar resolution given in [4]. This bar resolution is not a minimal projective resolution of 
as 
-bimodule. In practice, it is easier to compute the Hochschild cohomology group if we use a minimal projective resolution. So here we use the projective resolution of [2]. More generally, let 
be a finite dimensional algebra, where K is an algebraically closed field, 
is a quiver, and I is an admissible ideal of
. Fix a minimal set 
of generators for the ideal I. Let
. Then
, that is, x is a linear combination of paths 
for 
and 
and there are unique vertices v and w such that each path 
starts at v and ends at w for all j. We write 
and 
Similarly 
is the origin of the arrow a and 
is the end of a.
In [2, Theorem 2.9], it is shown that there is a minimal projective resolution of 
as a 
-bimodule which begins:
where the projective 
-bimodules 
are given by
and the maps
, 
and 
are 
-bimodule homomorphisms, defined as follows. The map 
is the multiplication map so is given by
. The map 
is given by
for each arrow
. With the notation for 
given above, the map 
is given by
where
.
In order to describe the projective bimodule 
and the map 
in the 
-bimodule resolution of 
in [2], we need to introduce some notation from [5]. Recall that an element 
is uniform if there are vertices 
such that 
We write 
and
. In [5], Green, Solberg and Zacharia show that there are sets 
in
, for
, consisting of uniform elements 
such that
for unique elements 
such that
. These sets have special properties related to a minimal projective 
-resolution of
, where 
is the Jacobson radical of
. Specifically the n-th projective in the minimal projective 
-resolution of 
is
In particular, to determine the set
, we follow explicitly the construction given in [5, §1]. Let 
denote the set of arrows of
. Consider the intersection
. Set this intersection equal to some
. We then discard all elements of the form 
that are in
; the remaining ones form precisely the set
.
Thus, for 
we have that
. So we may write
with
, such that 
are in the ideal generated by the arrows of
, and 
unique. Then [2] gives that
and, for 
in the notation above, the component of 
in the summand 
of 
is
Applying 
to this part of a minimal projective bimodule resolution of 
gives us the complex
where 
is the map induced from 
for
. Then 
Throughout, all tensor products are tensor products over
, and we write 
for
. When considering an element of the projective 
-bimodule
it is important to keep track of the individual summands of
. So to avoid confusion we usually denote an element in the summand 
by 
using the subscript “a” to remind us in which summand this element lies. Similarly, an element 
lies in the summand
of 
and an element 
lies in the summand 
of
. We keep this notation for the rest of the paper.
4. 
for 
We have given 
by quiver and relations in Section 2. However, these relations are not minimal. So next we will find a minimal set of relations 
for this algebra.
Let
The remaining relations given in Section 2 are all linear combinations of the above relations. For example, the relation 
can be written as
So this relation is in I and is not in
.
Proposition 4.1 For 
and with the above notation, the minimal set of relations is
In contrast to the majority of self-injective algebras of finite representation type, we will show that the algebra 
has non-zero second Hochschild cohomology group (see [3, Theorem 6.5]). Recall that
, where
is induced by
.
First we will find
. Since
let 
so that
. We consider the cases 
and 
separately.
Let 
and
where all coefficients 
for 
for 
Now we find
.
First we have,
Similarly for
,
For the remaining terms, 
where 
for all
,
and
.
Let
for 
and
for 
Thus for 
and
, fA2 is given by
where 
with
. So
For
, we let
where for all 
the coefficients 
for 
for 
are in 
Then we can find 
for 
in the same way as the previous case to see that it is given by
where 
with
. Note that there is no dependency between the 
So 
Proposition 4.2 If
, we have 
If
, we have 
Next we find 
and again consider the two cases separately. Let 
and 
. Then 
is defined by
where
.
Therefore 
Hence, 
For 
and
, 
is given by
where 
are in K for 
Thus 
Proposition 4.3 If
, we have
If
,
Corollary 4.4 If
, we have
. If
, 
In order to find Kerd3 and hence determine 
we start by giving a non-zero element in 
for all s.
Proposition 4.5 Define 
by
Then 
is in
.
Proof. We note that 
so 
is a non-zero map. To show that 
we show that
. First, observe that 
and 
Hence
. Similarly we have 
Recall that 
where
and 
are in the ideal generated by the arrows. For 
the component of
in 
is
Then
Thus
As 
is in the arrow ideal of
, 
So we have 
Similarly
as 
Therefore
for all 
so
. Thus 
as required.
Theorem 4.6 For 
where 
are positive integers, 
, 
with
and
, we have
.
Proof. Consider the element 
of 
where 
is given as in Proposition 4.5 by
Suppose for contradiction that 
Then
. So 
and so
. Also 
where 
Then 
where 
But this contradicts having
. Therefore
, that is,
. So 
is a nonzero element in
□
Note that we can also define maps 
by
for
. However, 
all represent the same element 
of
.
As we have found a non-zero element in 
we know that
. In the case
we have the following result, the proof of which is immediate from Proposition 4.2, Corollary 4.4 and Theorem 4.6.
Proposition 4.7 For 
where
, we have 
and
For the case
, we need more details to find
. Following [5] we may choose the set 
to consist of the following elements:
where
Thus the projective bimodule 
is 
Now we determine 
in the case
. Let
, so 
and
. Recall that for
, 
is given by
where 
are in
.
Then for
, we have 
In a similar way we can show that
.
For
, we have 
As 
we have 
for
.
Similarly it can be shown that
so that
.
We also have 
for
and 
Finally, putting
does not give any new information for
,
.
Thus h is given by
where 
for 
are in K. It is clear that there is no dependency between
, and therefore
.
Proposition 4.8 For 
and
, we have 
Using Propositions 4.2, 4.7, 4.8 and Theorem 4.6 we get the main result of this section.
Theorem 4.9 For 
where p, q, s, k are integers such that p, 
and
, we have 
We conclude this section by giving a deformation of 
which arises from the non-zero element 
in
.
Let
. Recall that
. We introduce a new parameter 
and define the algebra 
to be the algebra 
where 
is the ideal generated by the following elements:
1) 
where 
2) for all
, 
where 
3) 
for all arrows a with
4) 
for all arrows a with 
We now need to show that 
to verify that 
is indeed a deformation of
. First of all, it is clear that 
for all t and for all vertices ei with
. Now we consider 
and 
with
, and 
with
. These projective modules are described as follows:
In each case we see that
for all t. Hence
. Moreover, when 
the algebras 
and 
are not isomorphic since, in this case, 
is not self-injective. Thus we have found a non-trivial deformation of
.
Theorem 4.10 With 
and 
as defined above, then 
is a non-trivial deformation of
. Moreover, the algebras 
and 
are socle equivalent.
5. 
for 
We have given the algebra 
by quiver and relations in Section 2. Note that these relations are not minimal. So we will find a minimal set of relations 
for this algebra.
Let
The remaining relation 
can be written as
. So this relation is in I and is not in
.
Proposition 5.1 For 
and with the above notation, the minimal set of relations is
Recall that the projective
. Thus we have
(We note that the projective 
is also described in [4] although Happel gives no description of the maps in the 
-projective resolution of
.) Following [2], and with the notation introduced in Section 3, we may choose the set 
to consist of the following elements:
with 
where
We know that
. First we will find
. Let 
and so write
where 
Now we find
. We have
Also
We can show by direct calculation that
for all
.
Thus 
is given by
So
.
Proposition 5.2 For
, we have 
Now we determine
. Let
, so 
and
. Then 
is given by
for some 
for 
Then
As 
we have 
and 
As 
we have 
and
. So 
and 
Next,
So we have 
and hence 
Therefore 
as 
Thus again we have 
As 
above, we have 
as we already know.
Also
So we have 
and 
Finally, for
, we have
Therefore we have 
and
. Hence 
and 
for 
as we have above 
and 
Thus 
is given by
for some 
Proposition 5.3 For
, we have 
Therefore
and a basis is given by the maps 
and 
where 
is given by
is given by
From Proposition 5.2 and Proposition 5.3 we get the main result of this section.
Theorem 5.4 For 
with 
we have 
To connect this with deformations we use a similar discussion as Section 4. We introduce the parameter 
and define the algebra 
to be the algebra 
where 
is the ideal generated by the following elements:
1) 
2) 
3) 
4) 
We can show that
. Hence this algebra has no non-trivial deformation.
From Theorem 4.9 and Theorem 5.4 we have now found 
for all standard one-parametric but not weakly symmetric self-injective algebras of tame representation type.
6. Acknowledgements
I thank Prof. Nicole Snashall for her encouragement and helpful comments.