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Revista de la Unión Matemática Argentina
Print version ISSN 0041-6932On-line version ISSN 1669-9637
Rev. Unión Mat. Argent. vol.48 no.2 Bahía Blanca July/Dec. 2007
Lectures on algebras
Sverre O. Smalø
Mar del Plata, Argentina, March 2006
Abstract: The purpose of this note is to give a fast introduction to some problems of homological and geometrical nature related to finitely dimensional representations of finitely generated, and especially, finitely dimensional algebras over a field. Some of these results can also be extended to the situation where the field is not algebraically closed, and some of the results can even be extended to the situation where one is considering algebras over a commutative artin ring. For the results which hold true in the most general situation the proofs become most elegant since they depend on using length arguments only and thereby forgetting about the nature of a field altogether.
2000 Mathematics Subject Classification 16D10, 16E30, 16G10, 16G20
1. Introduction and notation
Let be an algebraically closed field and let
a finitely generated associative
-algebra with 1.
Examples:
1. , the field itself which is the simplest example of a
-algebra.
2. , the polynomial algebra in one variable over the field
.
3. , the free algebra in two non-commuting variables over the field
.
4. , the polynomial algebra in
commuting variables over the field
.
5. , the algebra of
-matrices over the field
.
6. For a -vector space
,
is the algebra of
-linear endomorphisms of
where addition is the usual addition of
-linear homomorphisms and the multiplication is given by composition of
-linear homomorphisms.
7. For a -vector space
and
elements of
, one lets
be the subalgebra of
generated by the elements
.
8. For a -vector space V and a subspace
, one can look at the algebra
. By choosing a complement
to
in
and decomposing
as
as a
-vector space, one can identify this algebra
with the matrix algebra
![K](/img/revistas/ruma/v48n2/2a0340x.png)
![K](/img/revistas/ruma/v48n2/2a0341x.png)
9. Now, this last example can be extended to more than one subspace. So let be a
-vector space and
subspaces for
; then
is defined as the subalgebra
for
of
. However, it is not so easy in general to describe this algebra as an algebra of matrices.
10. If one in example 9 has a -vector space
and subspaces
and the dimension of the
-vector spaces are given by
, then this algebra can be identified with the algebra of lower triangular matrices over
. This identification can be obtained by choosing a basis for
by starting with a basis for
and then successively extending a basis from
to
until one has a basis for the whole space
.
Finitely generated -algebras can also be given by generators and relations over the ground field
. This way one obtains the algebra as a quotient of a finitely generated free algebra
over the ground field
by an ideal
.
11. which is isomorphic to
, the polynomial algebra in two variables over the field
.
12. which is isomorphic to
, the exterior algebra in two variables over the field
.
13. which is called the first Weil algebra over the field
. This
-algebra is isomorphic to the
-subalgebra of the endomorphism algebra of the
-vector space
generated by the two
-linear endomorphisms where one is given as multiplication by
, and the other one is derivation with respect to
.
Another way of giving an algebra is to start with a vector space over with a basis
, and then give a multiplication table for the base elements, and then extend the multiplication by bilinearity to all elements of the
-vector space
. Let
be a
-vector space with a basis. A multiplication table is then given as
with the elements
where for each pair of indices
the set
is finite. For this to become an associative
-algebra with 1, severe restrictions on the
, the structure constants, have to be imposed.
14. Let be a
-vector space with
where
runs through the nonnegative integers, as a basis. Define a multiplication on the base elements by
and extend this multiplication by bilinearity to all elements of
. This makes
into an associative
-algebra with
as the identity element. By identifying
with
one gets an isomorphism between the
-algebra
and
, the polynomial algebra in one variable over the field
.
15. Another source of this form of -algebras is the group algebras. Let
be a group written multiplicatively. Let
be the
-vector space of all maps
from
to
such that
is finite. Then for each
in
one has the map
given by
for
and zero for
. The collection of all these maps as
runs through
becomes a basis for
. One can then form the multiplication table
which makes
into a
-algebra with
as the unit element, where
is the identity element of the group
. Usually one writes an element of
as
where
are elements of
and each
is substituted with g, and only a finite number of the elements
as
runs through
are nonzero. The notation used for the group algebra is often
or
.
16. Another important source of examples of algebras where the multiplication is given by a multiplication table for a basis, is the path algebra of a quiver.
A quiver is an oriented graph. Here one lets
denote the set of vertices, and one lets
denote the set of oriented edges. The oriented edges are also often called arrows as the name indicates and one usually represents a quiver by a collection of vertices and arrows drawn in the plain.
Example:
Now one can look at the set of all oriented paths in this oriented graph including the paths of length zero at each vertex, and use them as a basis for an algebra. By concatenating paths one makes a multiplication table for these base elements and in this way one obtains the path algebra.
For the example of the quiver above, this will be a six dimensional algebra, with basis ,
and
. Here
,
and
represent the paths of length zero at the vertices 1, 2 and 3 respectively. One has to make a convention about how to represent a path and here one is using the convention that an oriented path is ordered from right to left. The multiplication table for this algebra is rather long, but for the convenience of the reader the complete table is included.
![e1 + e2 + e3](/img/revistas/ruma/v48n2/2a03157x.png)
For this simple example, the path algebra is isomorphic to the -algebra of lower three by three matrices over
. To see this let
be the matrix with 1 in place
and zero otherwise. Then an isomorphism can be given by sending
in the path algebra to the matrix
,
in the path algebra to the matrix
,
in the path algebra to the matrix
,
in the path algebra to the matrix
,
in the path algebra to the matrix
and
in the path algebra to the matrix
. An easy calculation now shows that this is a
-algebra isomorphism from the path algebra of this quiver to the algebra of lower three by three matrices over
.
An important problem which often appears in representation theory is to determine whether two algebras are isomorphic or not isomorphic. This can sometimes be decided by giving an explicit algebra isomorphism between them, but sometimes one can still prove that two algebras have to be isomorphic without ever in principle being able to give such an isomorphism explicitly.
Example. The algebra is isomorphic to the algebra
. However, finding an explicit isomorphism requires that one can find the zeros of the polynomial
. However, this is one of the degree five complex polynomials where the zeros cannot be expressed with the help of radicals of the coefficients. By calculating the difference between this polynomial and its derivative one obtains the following:
. Therefore the polynomial
and its derivative have no common factors. This implies that the polynomial
has no multiple roots, and therefore by the Chinese remainder theorem there is an isomorphism between the algebra
and the algebra
, as claimed.
If an algebra is given, one way or another, one wants to get as much information about the algebra as possible. Some of this information can be obtained by considering the algebra homomorphisms from
to other better understood algebras. For example knowing all algebra homomorphisms from
to
, the ring of
-matrices over
, gives a lot of information.
Examples: 1. . There is only one
-algebra homomorphism
for each
, and that is given by
where
is the
-identity matrix.
2. . A
-algebra homomorphism from
to
is completely determined by the image of
, and any matrix in
can be the image of
. And hence,
a K-algebra homomorphism
can be identified with
.
3. . A
-algebra homomorphism
is completely determined by its value at
and
, and any two matrices
and
in
can be such an image. Hence the set of
-algebra homomorphisms can be identified with
.
4. . Now a
-algebra homomorphism is completely determined by the images
and
of
and
respectively, and in addition these two matrices
and
have to commute. Hence the set of
-algebra homomorphism from
to
can be identified with the set
.
For this gives all ordered pairs of matrices since any pair of
-matrices commutes. And for
this gives the set of ordered pairs of
-matrices
These four polynomial equations in the 8 unknowns and
,
are expressing exactly that the two matrices
and
are commuting.
Here the first and the fourth equation becomes the same, so the system consisting of these four equations is redundant.
For a general for this example one gets
quadratic equations in
unknowns, and one obtains a bijection between the set of
-algebra homomorphisms from
to
and the set of zeros of these polynomial equations, i.e. the set of
-algebra homomorphisms is in a bijection with a subvariety of the set of pairs of matrices defined as the set of common zeros of all these quadratic polynomials.
5. . Let
be an invertible
-matrix and define
by
. Then
is a
-algebra homomorphism, and by Skolem-Noether's theorem all
-algebra homomorphisms from
to
are given in this way. Since
they are all isomorphisms.
Letting denote the group of invertible
-matrices, one gets a group homomorphism from
, the group of invertible
-matrices, onto the group of
-algebra automorphisms of
. The kernel of this group homomorphism is the set of invertible matrices commuting with all invertible matrices. This is the set of nonzero scalar matrices. Hence the group of
-algebra automorphisms of
is naturally isomorphic to
.
13. . A
-algebra homomorphism
from
to
is completely determined by the values
and
of
and
respectively, and in addition these two matrices
and
have to satisfy the equation
![Id](/img/revistas/ruma/v48n2/2a03283x.png)
![d × d](/img/revistas/ruma/v48n2/2a03284x.png)
![φ(X )φ(Y ) - φ (Y )φ(X )](/img/revistas/ruma/v48n2/2a03285x.png)
![d × d](/img/revistas/ruma/v48n2/2a03286x.png)
![K](/img/revistas/ruma/v48n2/2a03287x.png)
![d](/img/revistas/ruma/v48n2/2a03288x.png)
![K](/img/revistas/ruma/v48n2/2a03289x.png)
![d](/img/revistas/ruma/v48n2/2a03290x.png)
![K](/img/revistas/ruma/v48n2/2a03291x.png)
![Λ](/img/revistas/ruma/v48n2/2a03292x.png)
![Md (K )](/img/revistas/ruma/v48n2/2a03293x.png)
![K](/img/revistas/ruma/v48n2/2a03294x.png)
![d](/img/revistas/ruma/v48n2/2a03295x.png)
![K](/img/revistas/ruma/v48n2/2a03296x.png)
![Λ](/img/revistas/ruma/v48n2/2a03297x.png)
![Md (K )](/img/revistas/ruma/v48n2/2a03298x.png)
In this section the action of the group on the varieties of representations will be introduced, and some facts about this action will be given.
For a finitely generated
-algebra and a fixed natural number
, let
a
-algebra homomorphism
.
Examples:
1. . One has that
consists of one element, the
-algebra homomorphism sending
to
where
is the
-identity matrix.
2. . One has that
where the map from the right to the left is given by evaluation, i.e. a matrix
is sent to the
-algebra homomorphism which takes the value
on a given polynomial
.
3. . One has that
where the map from the right to the left takes a pair of matrices
to the
-algebra homomorphism sending a given polynomial
in two non commuting variables to the matrix
.
4. . One has that
, where again the map from right to left is given by sending a polynomial
to the matrix obtained by evaluating the polynomial
in two commuting variables in the pair of matrices
obtaining
.
13. . If
then
is empty. In general one has that
is nonempty if and only the characteristic of the field
divides the number
.
5. . One has that
where the map from right to left is given by sending a coset
of an invertible matrix
to the
-algebra homomorphism given by sending a matrix
to the matric
.
Since composition of two -algebra homomorphisms is again a
-algebra homomorphism, one can let
act on
just by composition with the corresponding automorphism on
.
is given by
where
where
is any representative from the coset
.
Often one suppresses and one is speaking about an action of
on
.
Now let be a finitely generated
-algebra. Then one has that
is isomorphic to
for an ideal
in the free algebra
. Using such an isomorphism as an identification, an element
of
is completely determined by the values on the elements
where one denotes the residues of the variables
also with the symbol
respectively. Therefore the
-tuple
can be identified with
. In the Examples 2, 3, 4 and 13,
has been 1, 2, 2 and 2 respectively. In Examples 2 and 3, one obtained the whole space of matrices and the whole space of ordered pairs of matrices as possible values respectively, while in example 4 one obtained that the set of possible ordered pairs of matrices were given as the common zeros of
quadratic polynomials in
variables. While in Example 13 with
one obtained the empty set, which is also the common zeros of a set of polynomials.
In general, the possible -tuples in
which represent
-algebra homomorphisms are given by polynomial equations. These can be described in the following way: For each
,
gives a set of polynomial equations on the
entries in the matrices
. Next taking all these equations as
runs through the ideal
, gives (usually) an infinite collection of polynomial equations on the entries in the matrices
, and these equations determine an affine variety which is in bijection with
. Hence
![repd Λ](/img/revistas/ruma/v48n2/2a03387x.png)
![n Md (K )](/img/revistas/ruma/v48n2/2a03388x.png)
![repdΛ](/img/revistas/ruma/v48n2/2a03389x.png)
![n](/img/revistas/ruma/v48n2/2a03390x.png)
![f ∈ repdΛ](/img/revistas/ruma/v48n2/2a03391x.png)
![n](/img/revistas/ruma/v48n2/2a03392x.png)
![(f(X ),f(X ),...,f(X )) 1 2 n](/img/revistas/ruma/v48n2/2a03393x.png)
![A ∈ Gl (K ) d](/img/revistas/ruma/v48n2/2a03394x.png)
![n](/img/revistas/ruma/v48n2/2a03395x.png)
![K](/img/revistas/ruma/v48n2/2a03396x.png)
![φA ∘ f](/img/revistas/ruma/v48n2/2a03397x.png)
![n](/img/revistas/ruma/v48n2/2a03398x.png)
![-1 - 1 -1 (Af (X1)A ,Af (X2 )A ,...,Af (Xn )A )](/img/revistas/ruma/v48n2/2a03399x.png)
![n](/img/revistas/ruma/v48n2/2a03400x.png)
![K](/img/revistas/ruma/v48n2/2a03401x.png)
![Λ](/img/revistas/ruma/v48n2/2a03402x.png)
![Md (K )](/img/revistas/ruma/v48n2/2a03403x.png)
![Gld(K )](/img/revistas/ruma/v48n2/2a03404x.png)
![rep Λ d](/img/revistas/ruma/v48n2/2a03405x.png)
![Gl (K ) d](/img/revistas/ruma/v48n2/2a03406x.png)
![K](/img/revistas/ruma/v48n2/2a03407x.png)
![Gld (K )](/img/revistas/ruma/v48n2/2a03408x.png)
![Gld (G )](/img/revistas/ruma/v48n2/2a03409x.png)
For each there is associated a
-dimensional
-module
. This module is
as a
-vector space, and for each
and
the multiplication of
with
is given as
where the multiplication on the right side is regular matrix multiplication when each element of
is considered as a
-matrix.
Two elements and
in
represent isomorphic
-modules
and
if and only if
and
belong to the same orbit under the action of
on
. Hence, one obtains a bijection between the set of isomorphism classes of
-modules of dimension
as
-vector spaces and the set of
-orbits in
.
Here are some facts about the set of orbits in
for a
-algebra
generated by
elements.
1. For each in
the orbit
is open in its Zariski closure
whenever
is a polynomial in
variables such that
.
2. For each in
the closure of the orbit of
,
, is a union of orbits.
3. The dimension of the complement of an orbit in its closure is less than the dimension of the orbit, i.e. . The following formula for the dimension hold:
. Here dimension is referring to the Krull-dimension of the varieties.
Now one looks at isomorphism classes of -modules having dimension
as
-vector spaces, and denotes each of them also by a simple letter
. For two isomorphism classes
and
, one says that M degenerates to
if the orbit in
corresponding to
is contained in the closure of the orbit of the isomorphism class corresponding to
. One writes this by
. Because of statement 3 above, this becomes a partial order on the set of isomorphism classes of
-modules having dimension
as
-vector spaces.
Example:
2. and
. Then
consists of all 2 by 2 matrices over
. The orbits under the
-action are basically determined by the eigenvalues of the matrices. Most of the orbits can be parameterized by
where the correspondence is given in a one-to-two fashion by taking the orbit of the matrix
which has dimension
according to the above formula. These orbits are closed.
Then to each in
there are two orbits, the orbit of the matrix
, which consists of a single element and is hence of dimension
, and the orbit of the matrix
. This orbit is of dimension
, and its closure contains in addition to the orbit itself, the orbit (one single element) of
.
If one wants to look at higher dimensions than two for this example, one can list the eigenvalues of the matrix, and since conjugate matrices have the same eigenvalues, all matrices in an orbit have the same eigenvalues counted with multiplicities. The eigenvalues together with their multiplicity will therefore parameterize the orbits in a certain combinatorial way.
To do this, one first has to look at the situation where there is only one eigenvalue. Letting , one then has the set
which is the subset of
consisting of matrices having
as the sole eigenvalue. Then the orbits in
will correspond to partitions of
,
where
. Each
corresponds to a Jordan normal form of a matrix of size
with eigenvalue
, and the partition
then corresponds to the representation of dimension
with Jordan blocks of size
with eigenvalue
along the diagonal. Or equivalently, this corresponds to the direct sum of representations of dimension
given by a Jordan normal form of size
with eigenvalue
. Then a
-module corresponding to the partition
degenerates to the
-module corresponding to the partition
if and only if
for all
.
Now this information can be put together by first sorting by eigenvalues and their multiplicities, and then using the order given by partitions for each eigenvalue individually, to get a complete picture of the degeneration order in for any
as the example with
illustrates.
4. Degeneration and exact sequences
As one shall see in this section, there is a close relationship between exact sequences and degenerations. The starting point for this relationship is the following old result which is included in order to illustrate some of the techniques used to obtain degenerations.
Proposition 4.1. Given a finitely generated -algebra
and an exact sequence
of
-modules which are finitely dimensional as
-vector spaces, then
.
Proof. Let be a presentation of the finitely generated
-algebra. Assume the dimension of
as a
-vector space is
which is then the same as the dimension of
as a
-vector space. Produce a basis for
by first choosing a basis for
, and then extend this basis to a basis for the whole of
. This produces a complement
of
in
such that
as a
-vector space. Then identify the part of the basis for
coming from
with a basis for
through the map from
to
given in the exact sequence. Since
is closed with respect to multiplication with each elements from
one gets that
for each
in
. Further,
induces a
-module structure on
which by the identification of the basis in
with a basis of
makes
equal to
as a
-module. Hence
so the matrix representing multiplication with each
in
is with respect to this basis given as
for some
-linear homomorphism
. Therefore the
-tuple of matrices representing the module
is
![t ⁄= 0](/img/revistas/ruma/v48n2/2a03561x.png)
![K](/img/revistas/ruma/v48n2/2a03562x.png)
![φM](/img/revistas/ruma/v48n2/2a03563x.png)
![( ) t-1Id′ 0 0 Id′′](/img/revistas/ruma/v48n2/2a03564x.png)
![′ d](/img/revistas/ruma/v48n2/2a03565x.png)
![N ′](/img/revistas/ruma/v48n2/2a03566x.png)
![K](/img/revistas/ruma/v48n2/2a03567x.png)
![d ′′](/img/revistas/ruma/v48n2/2a03568x.png)
![V](/img/revistas/ruma/v48n2/2a03569x.png)
![K](/img/revistas/ruma/v48n2/2a03570x.png)
![αXi](/img/revistas/ruma/v48n2/2a03571x.png)
![t ⁄= 0](/img/revistas/ruma/v48n2/2a03573x.png)
![M](/img/revistas/ruma/v48n2/2a03574x.png)
![n](/img/revistas/ruma/v48n2/2a03575x.png)
![t = 0](/img/revistas/ruma/v48n2/2a03576x.png)
![t = 0](/img/revistas/ruma/v48n2/2a03577x.png)
![repdΛ](/img/revistas/ruma/v48n2/2a03578x.png)
![M](/img/revistas/ruma/v48n2/2a03579x.png)
![t = 0](/img/revistas/ruma/v48n2/2a03580x.png)
![K](/img/revistas/ruma/v48n2/2a03582x.png)
![Λ](/img/revistas/ruma/v48n2/2a03583x.png)
![N ′](/img/revistas/ruma/v48n2/2a03584x.png)
![N](/img/revistas/ruma/v48n2/2a03585x.png)
![□](/img/revistas/ruma/v48n2/2a03586x.png)
This result was the starting point for the degeneration theory of modules and goes back to M. Artin in the 60'ties.
As a corollary of this result one obtains the following.
Corollary 4.2. Let be a finitely generated
-algebra and
a
-module in
. Then the module
is semisimple if the orbit of
in
is closed.
Proof. Assume that the orbit of in
is closed. Then for each
-submodule
of
, the module
is in the closure of the orbit of
by Artin's result. By assumption the orbit of
is its own closure. Therefore
has to be isomorphic to
. However, this forces
to be a direct summand of
and hence
is semisimple since each submodule of
is then a direct summand.
Later on, one will see that the result of this corollary can be extended to give a complete characterization of the closed orbits in for a finitely generated
-algebra
since the closed orbits in
are exactly the orbits of semisimple
-modules which has dimension
as
-vector spaces.
However, not all degenerations can be obtained in this way since there are examples where an indecomposable -module degenerates properly into another indecomposable
-module, while each degeneration
coming from a short exact sequence with all terms being non-zero as above, have the property that
have to be a decomposable
-module.
Proposition 4.3. Let be a finitely generated
-algebra. If there exists an exact sequence
of
-modules with
and
finitely dimensional as
-modules, then
and
.
This result was obtained by Chr. Riedtmann and could be used directly to give examples where an indecomposable -module was degenerating properly into some other indecomposable
-module.
An outline of the proof of this proposition will be postponed. Instead a couple of examples are given in order to show how this result can be applied to obtain degenerations.
The first application is an example originally coming from representations of algebra given by a quiver which is illustrating that an indecomposable -module can degenerate properly into another indecomposable
-module.
Example: Consider the -subalgebra
of the
-algebra of lower
-matrices over the field
given as the set of matrices
![I](/img/revistas/ruma/v48n2/2a03641x.png)
![( ) | 0 0 0| | 1 0 0| ( 0 1 0)](/img/revistas/ruma/v48n2/2a03642x.png)
![Λ](/img/revistas/ruma/v48n2/2a03643x.png)
![Λ](/img/revistas/ruma/v48n2/2a03644x.png)
![( 0 0 0) ( ) P2 = Λ 0 1 0 0 0 1](/img/revistas/ruma/v48n2/2a03645x.png)
![Λ = P1 ⊕ P2](/img/revistas/ruma/v48n2/2a03648x.png)
![I](/img/revistas/ruma/v48n2/2a03649x.png)
![P2](/img/revistas/ruma/v48n2/2a03650x.png)
![Λ](/img/revistas/ruma/v48n2/2a03651x.png)
![P1 ≤deg Λ∕I](/img/revistas/ruma/v48n2/2a03653x.png)
![P1](/img/revistas/ruma/v48n2/2a03654x.png)
![Λ∕I](/img/revistas/ruma/v48n2/2a03655x.png)
![Λ](/img/revistas/ruma/v48n2/2a03656x.png)
![I](/img/revistas/ruma/v48n2/2a03657x.png)
![Λ](/img/revistas/ruma/v48n2/2a03658x.png)
![Λ∕I](/img/revistas/ruma/v48n2/2a03659x.png)
![Λ](/img/revistas/ruma/v48n2/2a03660x.png)
![P 1](/img/revistas/ruma/v48n2/2a03661x.png)
![Λ∕I](/img/revistas/ruma/v48n2/2a03662x.png)
![Λ](/img/revistas/ruma/v48n2/2a03663x.png)
Here is another example where one can use the result of Riedtmann to prove that any finitely dimensional -module degenerates to a semisimple
-module.
Let be a finitely generated
-algebra, and let
be a
-module which is finitely dimensional as a
-vector space. Let
where
denotes the
-th power of the radical of the
-module
and
. Consider now the exact sequence
![Λ](/img/revistas/ruma/v48n2/2a03679x.png)
![f](/img/revistas/ruma/v48n2/2a03680x.png)
![radiM ⊂ radi- 1M](/img/revistas/ruma/v48n2/2a03681x.png)
![i = 1, 2, ...,n - 1](/img/revistas/ruma/v48n2/2a03682x.png)
![M](/img/revistas/ruma/v48n2/2a03683x.png)
![Λ](/img/revistas/ruma/v48n2/2a03684x.png)
![M](/img/revistas/ruma/v48n2/2a03685x.png)
One could also have seen this last result by an inductive argument using just exact sequences as in the result of M. Artin.
Proof. Here is an indication of a proof of Riedtmann's result in Proposition 4.3
Consider an exact sequence
![f : A → A](/img/revistas/ruma/v48n2/2a03687x.png)
![g : A → M](/img/revistas/ruma/v48n2/2a03688x.png)
![Λ](/img/revistas/ruma/v48n2/2a03689x.png)
![λ](/img/revistas/ruma/v48n2/2a03690x.png)
![K](/img/revistas/ruma/v48n2/2a03691x.png)
![Λ](/img/revistas/ruma/v48n2/2a03693x.png)
![IA](/img/revistas/ruma/v48n2/2a03694x.png)
![A](/img/revistas/ruma/v48n2/2a03695x.png)
![Nλ](/img/revistas/ruma/v48n2/2a03696x.png)
![Λ](/img/revistas/ruma/v48n2/2a03697x.png)
![(f-λIA ) g](/img/revistas/ruma/v48n2/2a03698x.png)
![(f-λIA ) g](/img/revistas/ruma/v48n2/2a03699x.png)
![λ](/img/revistas/ruma/v48n2/2a03700x.png)
![K](/img/revistas/ruma/v48n2/2a03701x.png)
![f - λIA](/img/revistas/ruma/v48n2/2a03702x.png)
![λ](/img/revistas/ruma/v48n2/2a03703x.png)
![f](/img/revistas/ruma/v48n2/2a03704x.png)
![A ⊕ M](/img/revistas/ruma/v48n2/2a03705x.png)
![M](/img/revistas/ruma/v48n2/2a03706x.png)
![( ) f-λgIA A](/img/revistas/ruma/v48n2/2a03707x.png)
![λ = 0.](/img/revistas/ruma/v48n2/2a03708x.png)
![λ](/img/revistas/ruma/v48n2/2a03709x.png)
![K](/img/revistas/ruma/v48n2/2a03710x.png)
![(f- λIA )A g](/img/revistas/ruma/v48n2/2a03711x.png)
![V](/img/revistas/ruma/v48n2/2a03712x.png)
![Λ](/img/revistas/ruma/v48n2/2a03713x.png)
![V](/img/revistas/ruma/v48n2/2a03714x.png)
![A ⊕ M](/img/revistas/ruma/v48n2/2a03715x.png)
![V](/img/revistas/ruma/v48n2/2a03716x.png)
![(f-λIA) g A](/img/revistas/ruma/v48n2/2a03717x.png)
![λ](/img/revistas/ruma/v48n2/2a03718x.png)
![f](/img/revistas/ruma/v48n2/2a03719x.png)
![Λ](/img/revistas/ruma/v48n2/2a03720x.png)
![V](/img/revistas/ruma/v48n2/2a03721x.png)
![M](/img/revistas/ruma/v48n2/2a03722x.png)
![λ = 0](/img/revistas/ruma/v48n2/2a03723x.png)
![Λ](/img/revistas/ruma/v48n2/2a03724x.png)
![N](/img/revistas/ruma/v48n2/2a03725x.png)
![M ≤ N deg](/img/revistas/ruma/v48n2/2a03726x.png)
![□](/img/revistas/ruma/v48n2/2a03727x.png)
The converse of Riedtmann's result also holds as has been proved by G. Zwara. For a proof of this statement given in the next proposition, the reader is referred to the paper [Z2].
Proposition 4.4. Let be a finitely generated
-algebra and
and
in
with
. Then there exists an exact sequence
of
-modules where
is finitely dimensional as a
-module.
These two results give a complete algebraical description of degeneration, and hence one can take these as the starting point generalizing the notion of degeneration also to the situations where one is not working over an algebraically closed field, or even to the situations considering just a finitely generated -algebra over a commutative artin ring
, and where one is using lengths instead of dimensions. So from now on, unless otherwise stated,
will be a commutative artin ring,
will be a finitely generated algebra over
, and the dimension as a
-vector space will be substituted by the length as a
-module and denoted by
or just
if there can be no confusion. One can also loosely introduce the set
as the set of
-modules of length
considered as
-modules.
In this note the version of the exact sequence
In this section the notion of degeneration is generalized. Since the Krull-Remark-Schmidt-theorem holds for finite length modules, one has uniqueness in decompositions of such a module into indecomposable summands up to isomorphism. However, as one will soon see, this decomposition does not behave nicely with respect to degenerations. This is because one cannot cancel common direct summands from modules and
when
, and obtain a degeneration of the remaining complements. Here is an interesting example that demonstrates this.
Consider where
is an algebraically closed field and
a non-zero element of
.
is then a self injective local
-algebra of dimension four. Then look at
. The regular representation
itself belongs to
and the dimension of its orbit under the action of
is given as
. Next consider, for each
, the 2-dimensional
-module
where also the residue of
and
in
are denoted by
and
respectively. Now
is isomorphic to
if and only if the determinant of the matrix
is zero. Further, if the determinant of the matrix
is non-zero, the dimension of the endomorphism ring of
is 6. Hence, the orbit of each of these
-modules
when the determinant of the matrix
is nonzero, has dimension
. This together with the two parameters
and
shows that all these orbits make up a geometric object of dimension 12. Therefore not all orbits of these
-modules
can be in the closure of the orbit corresponding to
since this orbit is also, as calculated above, of dimension 12. However, for each
there is an exact sequence
![S](/img/revistas/ruma/v48n2/2a03791x.png)
![Λq](/img/revistas/ruma/v48n2/2a03792x.png)
![soc Λq](/img/revistas/ruma/v48n2/2a03793x.png)
![Λq](/img/revistas/ruma/v48n2/2a03794x.png)
![S](/img/revistas/ruma/v48n2/2a03795x.png)
![Λq ⊕ S ≤deg Ma,b ⊕ (Λq ∕soc Λq)](/img/revistas/ruma/v48n2/2a03796x.png)
![Ma ′,b′](/img/revistas/ruma/v48n2/2a03797x.png)
![socΛ q](/img/revistas/ruma/v48n2/2a03798x.png)
![Λq ⊕ S ≤deg Ma,b ⊕ Ma ′,b′ ⊕ S](/img/revistas/ruma/v48n2/2a03800x.png)
![0 ⁄= (a,b)](/img/revistas/ruma/v48n2/2a03801x.png)
![0 ⁄= (a′,b′)](/img/revistas/ruma/v48n2/2a03802x.png)
![Λ ≤ M ⊕ M ′′ q deg a,b a,b](/img/revistas/ruma/v48n2/2a03803x.png)
This example was first considered for this purpose by J. Carlson where he used the case .
Let again be a finitely generated
-algebra with
an algebraically closed field. One says that a
-module
in
virtually degenerates to a
-module
in
if there exists a finitely dimensional
-module
such that
degenerates to
.
By Zwara's result, this is the same as saying that there exact an sequence of
-modules where
and
are finitely dimensional as
-vector spaces.
Again, using the characterization of virtual degenerations given by short exact sequences, one can extend the notion of virtual degeneration to situations where one is working over a finitely generated algebra over a commutative artin ring
instead of working with finitely generated algebras over an algebraically closed field.
Proposition 5.1. Let be a finitely generated
-algebra. If the
-module
virtually degenerates to the
-module
, i.e. there is an exact sequence of
-modules
![K](/img/revistas/ruma/v48n2/2a03833x.png)
![ℓ(Hom Λ(M, X )) ≤ ℓ(Hom Λ(N, X ))](/img/revistas/ruma/v48n2/2a03834x.png)
![Λ](/img/revistas/ruma/v48n2/2a03835x.png)
![X](/img/revistas/ruma/v48n2/2a03836x.png)
![K](/img/revistas/ruma/v48n2/2a03837x.png)
Proof. Consider an exact sequence
![X](/img/revistas/ruma/v48n2/2a03839x.png)
![Λ](/img/revistas/ruma/v48n2/2a03840x.png)
![K](/img/revistas/ruma/v48n2/2a03841x.png)
![Hom Λ( ,X )](/img/revistas/ruma/v48n2/2a03842x.png)
![K](/img/revistas/ruma/v48n2/2a03844x.png)
![K](/img/revistas/ruma/v48n2/2a03845x.png)
![ℓ(Hom Λ (A ⊕ B, X ))](/img/revistas/ruma/v48n2/2a03847x.png)
![ℓ(Hom (M, X )) ≤ ℓ(Hom (N, X )) Λ Λ](/img/revistas/ruma/v48n2/2a03848x.png)
![Λ](/img/revistas/ruma/v48n2/2a03849x.png)
![X](/img/revistas/ruma/v48n2/2a03850x.png)
![K](/img/revistas/ruma/v48n2/2a03851x.png)
![□](/img/revistas/ruma/v48n2/2a03852x.png)
One can also obtain this result from geometric considerations in the case one is working over an algebraically closed field.
In light of this result one introduces another relation on , called the
order, which also turns out to be a partial order on the set
by a result of M. Auslander.
For two -modules
and
in
one says that
if
for all
-modules
of finite length as
-modules.
As already mentioned, this order is called the hom order. This order is also symmetric in the sense that for all
-modules
of finite length as
-modules if and only if
for all
-module
of finite length as
-modules.
¿From the propositions above one then has the following implications: . By the example of J. Carlson one knows that the first implication is not an equivalence in general. So far no one has come up with an example showing that the last implication is not an equivalence, and hence it is an open problem if the second implication is in fact an equivalence.
Recall that a finitely generated -algebra
with
a commutative artin ring is called an artin algebra if
as a
-module has finite length. Such an artin algebra is said to be of finite representation type if there are only a finite number of isomorphism classes of indecomposable
-modules. The algebra in the example of J. Carlson is artinian, but it is not of finite representation type. One can see that directly. However, that the
s above are not of finite representation type can also be deduced from the next result stating that for artin algebras of finite representation type the degeneration order, the virtual degeneration order and the hom order all agree. This was first proved by G. Zwara. For general background on representation theory of artin algebras see [ARS].
Theorem 5.2. If is an artin
-algebra of finite representation type, and
and
are two
-modules of the same length as
-modules, then the three following statements are equivalent:
1.
2.
3.
Proof. A complete proof of this is based on the study of the finitely presented functors on the category of finitely generated -modules and hence belongs to the Auslander-Reiten theory for artin algebras. Here only a sketch will be given. That statement 1 implies statement 2 is obvious, and that statement 2 implies statement 3 has already been proven, so the only part left which has to be proved is that statement 3 implies statement 1 when
is an artin
-algebra of finite representation type.
So, assume there are two -modules
and
of the same length as
-modules such that
. If
is isomorphic to
there is nothing to prove. Hence one can assume that
and
are not isomorphic modules. Then one can consider the contravariant functors
and
on the category of finitely generated
-modules. Since
is of finite representation type, these functors are of finite length and hence artinian. Next consider the category
of finitely presented contravariant functors on the category of finitely generated
-modules. For each
in
one lets
denote the element associated with the functor
in the Grothendieck group of
. Since
is of finite representation type, there is a finitely presented functor
such that
. Now a fact which will be proven in the next lemma is that for a pair of nonzero
-modules
and
of the same length as
-modules, with no common nonzero direct summand and with
, there is always a direct summand
of
such that
. Using this fact and that the functor
is artinian, one can now prove that the functor
has a subfunctor
with the same composition factors as
counted with multiplicity, i.e.
. To see this, consider all subfunctors
of
such that
for all
-modules
of finite length as
-module. This set is nonempty since
is a member, and hence this set has a minimal element. Let
be one of these minimal elements. Choose a minimal projective presentation of the functor
which has projective dimension one since it is a submodule of a projective functor,
. If
, one obtains that
with
. But then, by first cancelling common direct summands one gets from the fact which is proved in the next lemma, that there exists an indecomposable summand
of
where one has that
. Hence, since
is contained in the radical of the functor
, the image
of the subfunctor
of
in
when
, also have the property that
for all
-modules
of finite length as an
-module. This contradicts the minimality of
and completes the proof of the claim that
has a subfunctor
such that
.
Now since the global dimension of the category of finitely presented functors is at most two, one takes a minimal projective resolution of and obtains an exact sequence
![′′ N](/img/revistas/ruma/v48n2/2a03962x.png)
![N](/img/revistas/ruma/v48n2/2a03963x.png)
![[Hom Λ( ,A)]](/img/revistas/ruma/v48n2/2a03969x.png)
![A](/img/revistas/ruma/v48n2/2a03970x.png)
![Λ](/img/revistas/ruma/v48n2/2a03971x.png)
![N](/img/revistas/ruma/v48n2/2a03973x.png)
![M](/img/revistas/ruma/v48n2/2a03974x.png)
![M](/img/revistas/ruma/v48n2/2a03975x.png)
![Y](/img/revistas/ruma/v48n2/2a03976x.png)
![′ ′′ X = X ⊕ X](/img/revistas/ruma/v48n2/2a03977x.png)
![′′ ′′ N ≃ X ⊕ N](/img/revistas/ruma/v48n2/2a03978x.png)
![′ Y ≃ M ⊕ X](/img/revistas/ruma/v48n2/2a03979x.png)
![X ′′](/img/revistas/ruma/v48n2/2a03980x.png)
![Y](/img/revistas/ruma/v48n2/2a03981x.png)
![N ′′](/img/revistas/ruma/v48n2/2a03982x.png)
![X ′′](/img/revistas/ruma/v48n2/2a03983x.png)
![0 → X → X ⊕ M → N → 0](/img/revistas/ruma/v48n2/2a03984x.png)
![M ≤ N deg](/img/revistas/ruma/v48n2/2a03985x.png)
![□](/img/revistas/ruma/v48n2/2a03986x.png)
Here is the statement, including a proof, of the proper inequality in the length of homomorphism spaces which was used in the proof of the theorem above. This result is due to K. Bongatz.
Lemma 5.3. Let be an finitely generated
-algebra and
and
two nonzero
-modules in
such that
and such that
and
have no nonzero common direct summand. Then there exist an indecomposable summand
of
such that
.
Proof. Let be a generating set for the finite length
-module
and consider the exact sequence
![C](/img/revistas/ruma/v48n2/2a031003x.png)
![fi](/img/revistas/ruma/v48n2/2a031004x.png)
![i = 1,2,...,n](/img/revistas/ruma/v48n2/2a031005x.png)
![Hom Λ( ,M )](/img/revistas/ruma/v48n2/2a031006x.png)
![Hom Λ( ,N )](/img/revistas/ruma/v48n2/2a031007x.png)
![K](/img/revistas/ruma/v48n2/2a031010x.png)
![Hom Λ ((f1,f2,...,fn)tr,N )](/img/revistas/ruma/v48n2/2a031011x.png)
![ℓ(Hom Λ(N, N )) = ℓ(Hom Λ (M, N ))](/img/revistas/ruma/v48n2/2a031012x.png)
![tr Im (Hom Λ ((f1,f2,...,fn) ,M ))](/img/revistas/ruma/v48n2/2a031013x.png)
![Hom Λ((f1,f2,...,fn)tr,M )](/img/revistas/ruma/v48n2/2a031014x.png)
![ℓ Im (Hom Λ((f1,f2,...,fn )tr,M )) = ℓ(Hom Λ(N n,M )) - ℓ(Hom Λ(C, M )) ≥ ℓ(Hom Λ(N n,N )) - ℓ(Hom Λ (C,N )) = ℓ(Hom Λ(M, N](/img/revistas/ruma/v48n2/2a031015x.png)
![Hom ((f ,f ,...,f )tr,M ) Λ 1 2 n](/img/revistas/ruma/v48n2/2a031016x.png)
![M](/img/revistas/ruma/v48n2/2a031017x.png)
![M](/img/revistas/ruma/v48n2/2a031018x.png)
![tr (f1,f2,...,fn)](/img/revistas/ruma/v48n2/2a031019x.png)
![tr (f1,f2,...,fn)](/img/revistas/ruma/v48n2/2a031020x.png)
![M](/img/revistas/ruma/v48n2/2a031021x.png)
![N](/img/revistas/ruma/v48n2/2a031022x.png)
![M](/img/revistas/ruma/v48n2/2a031023x.png)
![N](/img/revistas/ruma/v48n2/2a031024x.png)
![M ≤hom N](/img/revistas/ruma/v48n2/2a031025x.png)
![ℓ(Hom (N, M )) < ℓ(Hom (N, N )) Λ Λ](/img/revistas/ruma/v48n2/2a031026x.png)
![′ N](/img/revistas/ruma/v48n2/2a031027x.png)
![N](/img/revistas/ruma/v48n2/2a031028x.png)
![′ ′ ℓ(Hom Λ (N ,M )) < ℓ(Hom Λ(N ,N ))](/img/revistas/ruma/v48n2/2a031029x.png)
![□](/img/revistas/ruma/v48n2/2a031030x.png)
If one analysis the proof of the theorem above, one also get that if and
is satisfied for all but a finite number of indecomposable
-modules
which are of finite length as
-modules, then
.
Also for the algebra , the degeneration order and the hom order coincides on
. This can be seen from the description on the degeneration order given earlier in this note and a simple calculation of the hom order.
One also has that the hom order descends to all extension groups for an artin -algebra
as the next result shows.
Proposition 5.4. Let be an artin
-algebra and
and
two
-modules of the same length as
-modules. Then
implies that
and
for all
-modules
and
of finite length and for all
.
Proof. Since is assumed to be an artin
-algebra, one can consider an exact sequence
![P](/img/revistas/ruma/v48n2/2a031057x.png)
![P](/img/revistas/ruma/v48n2/2a031058x.png)
![ΩX](/img/revistas/ruma/v48n2/2a031059x.png)
![Hom Λ( ,M )](/img/revistas/ruma/v48n2/2a031060x.png)
![Hom Λ( ,N )](/img/revistas/ruma/v48n2/2a031061x.png)
![M](/img/revistas/ruma/v48n2/2a031066x.png)
![N](/img/revistas/ruma/v48n2/2a031067x.png)
![rep Λ d](/img/revistas/ruma/v48n2/2a031068x.png)
![M ≤ N hom](/img/revistas/ruma/v48n2/2a031069x.png)
![ℓ(Hom Λ(P,M )) = ℓ(Hom Λ (P, N ))](/img/revistas/ruma/v48n2/2a031070x.png)
![Λ](/img/revistas/ruma/v48n2/2a031071x.png)
![ℓ(Ext1Λ(X, M )) ≤ ℓ(Ext1Λ(X, N ))](/img/revistas/ruma/v48n2/2a031072x.png)
![Λ](/img/revistas/ruma/v48n2/2a031073x.png)
![X](/img/revistas/ruma/v48n2/2a031074x.png)
![ℓ(ExtiΛ (X, M )) ≤ ℓ(ExtiΛ(X, N ))](/img/revistas/ruma/v48n2/2a031075x.png)
![i ≥ 0](/img/revistas/ruma/v48n2/2a031076x.png)
![□](/img/revistas/ruma/v48n2/2a031077x.png)
This result is also due to K. Bongartz.
The following consequences on the projective and injective dimensions of -modules are immediate:
Corollary 5.5. Let be an artin
-algebra and
and
two finitely generated
-modules of the same length as
-modules. If
, then the projective dimension of
is less than or equal to the projective dimension of
, and the injective dimension of
is less than or equal to the injective dimension of
.
For those who want to work with tensor products and torsion groups instead of groups of homomorphisms and extensions, the following translation may be of interest.
Proposition 5.6. Let be a finitely generated
-algebra and
and
two nonzero
-modules of the same length as
-modules. Then
if and only if
for all right
-modules
of finite length as
-modules.
Proof. Let be a right
-module which has finite length as a
-module. Then one has the following equality for all left
-modules
which have finite length as
-modules.
where
is a duality on the category of finitely generated
-modules over the commutative artin ring
. Using the isomorphisms
and
one then gets
and
. The statement in the proposition is then an immediate consequence of this.
Restricting to artin -algebras
this inequality also descends to all torsion groups which is now stated without a proof.
Proposition 5.7. Let be an artin
-algebra and
and
two
-modules of the same length as
-modules such that
, then
for all finitely generated right
-modules
and all
.
6. Degenerations, hom order and discrete invariants
The degeneration order and the hom order has also some impact on discrete invariants associated with modules. One has already observed that projective dimension and injective dimension behaves nicely with respect to these orders as a corollary of Bongatz' result. These numerical invariants are rather coarse, and some interesting cases of degenerations occur between modules which cannot be separated by these invariants. Therefore it is interesting to look at the induced degeneration order, the induced virtual degeneration order and the induced hom orders on sets of modules where these discrete invariants stay the same.
Some consequences of the hom order, and which are therefore also consequences of the degeneration and the virtual degeneration orders, are the following.
Proposition 6.1. Let be a finitely generated
-algebra and
and
two
-modules of finite length as
-modules such that
Then the following holds true:
1. The annihilator of is contained in the annihilator of
.
2. for all
.
3. for all
. Here
denotes the
'th socle of the
-module.
Proof. From both general considerations and the assumption of the proposition one has the following equalities and inequalities:
![Hom Λ (Λ ∕ann M, N ) ≃ {n ∈ N | (ann M )n = 0}](/img/revistas/ruma/v48n2/2a031146x.png)
![N](/img/revistas/ruma/v48n2/2a031147x.png)
![(ann M )N = 0](/img/revistas/ruma/v48n2/2a031148x.png)
For the -module
, one can by evaluating a
-homomorphism
in
in the residue of the identity, identify
with
for all
-modules
annihilated by
. Applying this to
and
using 1, one gets that part 3 is satisfied.
Part 2 is proven in a similar way. Let and consider the
-module
. Then from Proposition 5.6 one obtains that
![□](/img/revistas/ruma/v48n2/2a031166x.png)
This shows that when one wants to consider the hom order, one can always cut down to an artin algebra since will always be an artin algebra when
is a
-module of finite length as a
-module.
Let
![Λ](/img/revistas/ruma/v48n2/2a031172x.png)
![K](/img/revistas/ruma/v48n2/2a031173x.png)
![Γ = Λ ∕ann M](/img/revistas/ruma/v48n2/2a031174x.png)
![Γ ⊗ Λ M](/img/revistas/ruma/v48n2/2a031175x.png)
![M](/img/revistas/ruma/v48n2/2a031176x.png)
![Γ ⊗ Λ N](/img/revistas/ruma/v48n2/2a031177x.png)
![N](/img/revistas/ruma/v48n2/2a031178x.png)
![Γ ⊗ Λ -](/img/revistas/ruma/v48n2/2a031179x.png)
![Γ ⊗ Λ X ≃ X ∕ann M X](/img/revistas/ruma/v48n2/2a031181x.png)
![K](/img/revistas/ruma/v48n2/2a031183x.png)
![Γ](/img/revistas/ruma/v48n2/2a031185x.png)
![K](/img/revistas/ruma/v48n2/2a031186x.png)
![M ≤deg N](/img/revistas/ruma/v48n2/2a031187x.png)
![M ≤vgen N](/img/revistas/ruma/v48n2/2a031188x.png)
![Λ ∕ann M](/img/revistas/ruma/v48n2/2a031189x.png)
One can now give the characterization of closed orbits in for a finitely generated algebra over an algebraically closed field
.
Corollary 6.2. Let be a finitely generated
-algebra where
is an algebraically closed field and
a natural number. Then the orbit of a
-module
in
is closed if and only if
is semisimple.
Proof. That the orbit of is closed in
forces
to be semisimple is already established as a corollary of Artin's result. One has that if
is a semisimple module, then
is a semisimple ring. Therefore if
is semisimple and
degenerates into
, then
is annihilated by
and is hence itself semisimple. From the comments above one then has that
degenerates into
as a
-module. Hence, there is an exact sequence
as
-modules, which have to split since
is semisimple. This forces
to be isomorphic to
, and therefore the orbit of
is closed in
.
For modules and
in
one has the following list of partial orders on subsets of
:
1. if
and
is isomorphic to
, i.e the
-modules
and
have isomorphic tops and
degenerates to
.
2. if
and
is isomorphic to
for all
, i.e. the
-modules
and
have isomorphic radical layers and
degenerates to
.
The notation and
is defined in a similar way using the virtual degeneration order and hom order instead of degeneration order respectively.
This list is not exhaustive since one could also add socle layers or even use the subsets of having the same lengths for the extension groups with a fixed set of
-modules on one or the other side, or both sides. Here the semisimple
-module
is the most obvious one to be used. This will be giving subsets of
-modules of a fixed length as
-modules where the terms in the projective and the injective resolutions are the same respectively, but where the
-homomorphisms in these resolutions may be different. These are generalizations of the requirement that the tops and the socles being isomorphic for the two
-modules being compared, respectively.
There is the following result obtained by B. Huisgen-Zimmermann regarding degenerations between two -modules
and
which have simple tops and where the radical layers of the two modules are isomorphic. [H-Z]
Proposition 6.3. If is a
-module of finite length as a
-module with
simple and
, then
is isomorphic to
.
This result can be deduced as a corollary of the following more general result using the hom order instead of the degeneration order.
Proposition 6.4. If is a
-module with
simple and
, then
is isomorphic to
.
Proof. The proof of this proposition goes by induction on the Loewy length of .
If the Loewy length of is one, then
is simple and hence also
has to be simple. Further since
, there is at least one nonzero
-homomorphism from
to
. This is then showing that
is isomorphic to
.
So, let the Loewy length of be
and assume by induction that the statement hold for each
with Loewy length less than or equal to
. Now assume
with
having simple top. As has already been observed,
so one can reduce to the artin
-algebra
.
Now consider the two -modules
and
of Loewy length
. Then from the assumption in the proposition one has that
is isomorphic to
for all
. One also has that for each
-module
of finite length as a
-module that the following equalities and inequalities hold:
![M ′ ≤lhom N ′](/img/revistas/ruma/v48n2/2a031301x.png)
![M ′](/img/revistas/ruma/v48n2/2a031302x.png)
![N ′](/img/revistas/ruma/v48n2/2a031303x.png)
Next, since the top of is simple the following equalities and inequalities hold:
![N](/img/revistas/ruma/v48n2/2a031309x.png)
![M](/img/revistas/ruma/v48n2/2a031310x.png)
![M](/img/revistas/ruma/v48n2/2a031311x.png)
![N ′](/img/revistas/ruma/v48n2/2a031312x.png)
![M ′](/img/revistas/ruma/v48n2/2a031313x.png)
![M](/img/revistas/ruma/v48n2/2a031316x.png)
![n + 1](/img/revistas/ruma/v48n2/2a031317x.png)
![ℓ(Hom (M ′,rad M )) Λ](/img/revistas/ruma/v48n2/2a031318x.png)
![ℓ(Hom (M, M )) Λ](/img/revistas/ruma/v48n2/2a031319x.png)
![ℓ(Hom Λ(N, M )](/img/revistas/ruma/v48n2/2a031320x.png)
![ℓ(Hom Λ(M, M )](/img/revistas/ruma/v48n2/2a031321x.png)
![M ≤hom N](/img/revistas/ruma/v48n2/2a031322x.png)
![M](/img/revistas/ruma/v48n2/2a031323x.png)
![N](/img/revistas/ruma/v48n2/2a031324x.png)
![□](/img/revistas/ruma/v48n2/2a031325x.png)
This result about the hom order was obtained in a joint paper with Anita Valenta [SV].
7. Modules determined by their tops and first syzygies
One has seen in the previous section that sometimes some invariants are completely determining a module up to isomorphism. In this section one will look at a result of this type for artin algebras obtained by R. Bautista and E. Perez and also present a generalization of their result discovered by C. M. Ringel at this conference.
Let be an artin
-algebra, and let
denote the radical of
. For a
-module
, let
denote its first syzygy, i.e. there is an exact sequence
![P → M](/img/revistas/ruma/v48n2/2a031334x.png)
Theorem 7.1. Let be an artin
-algebra,
and
two finitely generated
-modules such that
and
. Then
is isomorphic to
if and only if
is isomorphic to
and
is isomorphic to
.
The proof of this fact presented here is based on elementary homological algebra and length considerations over the commutative artin ring .
The next lemma is a consequence of Nakayama's lemma and is well known.
Lemma 7.2. Let be a non-zero finitely generated
-module and
a
-epimorphism. Then there is an indecomposable
-module
and
-homomorphisms
and
such that
, the identity on
.
This lemma can be applied to give the result after the following proposition has been proven.
Proposition 7.3. Let and
be finitely generated
-modules such that
,
,
is isomorphic to
and
is isomorphic to
. Then there is a
-epimorphism
for some natural number
.
Proof. Let and
satisfy the assumption in the proposition. Without loss of generality one can assume that
(and hence
) is non-zero. One has exact sequences
and
of
-modules with
a projective
-module. Then for each finitely generated
-module
one obtains exact sequences
![Λ](/img/revistas/ruma/v48n2/2a031390x.png)
![X](/img/revistas/ruma/v48n2/2a031391x.png)
Next, let be the trace of
in
which is the sum of all images of all
-homomorphisms from
to
. Then there is an
such that there is an exact sequence
![Hom (M, - ) Λ](/img/revistas/ruma/v48n2/2a031400x.png)
![K](/img/revistas/ruma/v48n2/2a031403x.png)
![Ext1Λ(M, K ) = 0](/img/revistas/ruma/v48n2/2a031404x.png)
Next, apply the functor to the sequence (2) and one obtains the exact sequence
![ℓ(Hom Λ(N, τM (N ))) ≥ ℓ(Hom Λ (N, M n)) - ℓ(Ext1Λ(N, M n )) - (ℓ(Hom Λ(N, K )) - ℓ(Ext1Λ(N, K )))](/img/revistas/ruma/v48n2/2a031409x.png)
![X = M n](/img/revistas/ruma/v48n2/2a031410x.png)
![X = K](/img/revistas/ruma/v48n2/2a031411x.png)
![n 1 n 1 ℓ(Hom Λ(M, M )) - ℓ(Ext Λ(M, M )) - (ℓ(Hom Λ (M, K )) - ℓ(Ext Λ(M, K )))](/img/revistas/ruma/v48n2/2a031412x.png)
![n ℓ(Hom Λ(M, M )) - ℓ(Hom Λ(M, K ))](/img/revistas/ruma/v48n2/2a031413x.png)
![ℓ(Ext1Λ (M, M n)) = 0](/img/revistas/ruma/v48n2/2a031414x.png)
![ℓ(Ext1Λ (M, K )) = 0](/img/revistas/ruma/v48n2/2a031415x.png)
![ℓ(Hom Λ(M, M n)) - ℓ(Hom Λ(M, K ))](/img/revistas/ruma/v48n2/2a031416x.png)
![ℓ(Hom Λ(M, τM (N )))](/img/revistas/ruma/v48n2/2a031417x.png)
![Hom (M, τ (N )) Λ M](/img/revistas/ruma/v48n2/2a031418x.png)
![Hom (M, N ) Λ](/img/revistas/ruma/v48n2/2a031419x.png)
![τM (N ) ⊆ N](/img/revistas/ruma/v48n2/2a031420x.png)
![X = N](/img/revistas/ruma/v48n2/2a031421x.png)
![ℓ(Hom Λ(M, N )) ≥ ℓ(Hom Λ(N, N ))](/img/revistas/ruma/v48n2/2a031422x.png)
![ℓ(Hom Λ(N, τM (N ))) ≥ ℓ(Hom Λ (M, τM(N ))) = ℓ(Hom Λ(M, N )) ≥ ℓ(Hom Λ(N, N ))](/img/revistas/ruma/v48n2/2a031423x.png)
![τM N](/img/revistas/ruma/v48n2/2a031424x.png)
![Λ](/img/revistas/ruma/v48n2/2a031425x.png)
![N](/img/revistas/ruma/v48n2/2a031426x.png)
![ℓ(Hom (N, τ (N ))) ≤ ℓ(Hom (N, N )) Λ M Λ](/img/revistas/ruma/v48n2/2a031427x.png)
![ℓ(Hom Λ(N, τM (N ))) = ℓ(Hom Λ (N, N ))](/img/revistas/ruma/v48n2/2a031428x.png)
![Λ](/img/revistas/ruma/v48n2/2a031429x.png)
![N](/img/revistas/ruma/v48n2/2a031430x.png)
![N](/img/revistas/ruma/v48n2/2a031431x.png)
![τM N](/img/revistas/ruma/v48n2/2a031432x.png)
![N](/img/revistas/ruma/v48n2/2a031433x.png)
![N](/img/revistas/ruma/v48n2/2a031434x.png)
![τM N](/img/revistas/ruma/v48n2/2a031435x.png)
![τM (N ) = N](/img/revistas/ruma/v48n2/2a031436x.png)
![□](/img/revistas/ruma/v48n2/2a031437x.png)
One can now give the promised proof of the theorem of R. Bautista and E. Perez.
Proof. So let and
be two
-modules such that
and
. If
is isomorphic to
then clearly
is isomorphic to
and
is isomorphic to
.
For the converse, assume that is isomorphic to
and that
is isomorphic to
. One can without loss of generality assume that
(and hence also
) is non-zero. From Proposition 7.3 one then has an
-epimorphism
for some natural number
and also an
-epimorphism
for some natural number
. Hence, there is a composition of
-epimorphisms
. Then Lemma 7.2 ensures that there is an indecomposable
-module
and
-homomorphisms
and
such that the composition
![X](/img/revistas/ruma/v48n2/2a031469x.png)
![X](/img/revistas/ruma/v48n2/2a031470x.png)
![M](/img/revistas/ruma/v48n2/2a031471x.png)
![N](/img/revistas/ruma/v48n2/2a031472x.png)
![M](/img/revistas/ruma/v48n2/2a031473x.png)
![N](/img/revistas/ruma/v48n2/2a031474x.png)
![M](/img/revistas/ruma/v48n2/2a031475x.png)
![□](/img/revistas/ruma/v48n2/2a031476x.png)
During the meeting in Mar del Plata, C. M. Ringel observed that the following more general statement is valid.
Theorem 7.4. Let be an artin
-algebra and
a finitely generated
-module such that
. If
is a finitely generated
-module such that there exists finitely generated
-modules
and
, and exact sequences
and
, then there exists a finitely generated
-module
and an exact sequence
, i.e.
.
[ARS] Auslander, M., Reiten, I. and Smalø, S. O., Representation Theory of Artin Algebras. Cambridge studies in advanced mathematics, 36, 1995. [ Links ]
[BP] Bautista, Raymundo; Pérez, Efrén On modules and complexes without self-extensions. Comm. Algebra 34 (2006), no. 9, 3139-3152. [ Links ]
[H-Z] Huisgen-Zimmermann, B., Top-stable and layer-stable degenerations by way of Grassmannians. Preprint [ Links ]
[R] Riedtmann, Chr., Degenerations for representations of quivers with relations. Ann. Sci. Ecole Nomale Sup. 4, 275-301, 1986. [ Links ]
[S] Smalø, S. O. Modules determined by their tops and first syzygies Comm. Algebra 35 (2007) no.4, 1223-1225. [ Links ]
[SV] Smalø, S. O.; Valenta, A. Top-stable and layer-stable degenerations and hom-order. Colloq. Math. 108 (2007), no. 1, 63-71. [ Links ]
[V] Voigt, D., Induzierte Darstellungen in der Theorie der endlichen algebraischen Gruppen, LNM 592, Springer Berlin 1977, MR 58:5949. [ Links ]
[Z1] Zwara, G., A degeneration-like order for modules. Arch. Math 71 437-444 1998. [ Links ]
[Z2] Zwara, G., Degenerations of finite-dimensional modules are given by extensions Composito Math. 121, 205-218, 2000. [ Links ]
Sverre O. Smalø
Norwegian University of Science and Technology
Department of mathematical sciences
N-7491 Trondheim, Norway.
sverresm@math.ntnu.no
Recibido: 7 de noviembre de 2006
Aceptado: 3 de junio de 2007