Interest (was Project):
Given a short exact sequence 0->U->M->V->0 of modules it is easy to see that M degenerates to the direct sum U+V. The partial order generated by this is called the extension order; we write M <=ext N.
On the other hand the dimension of homorphism spaces behave nice under degeneration. Let T be a test module and M degenerate to N. Then dim Hom(T,M) <= dim Hom(T,N). (In the simplest case such a dimension is the multiplicity of an eigenvalue. And that should increase when degenerating.) We obtain another (chain-finite) partial order on the isoclasses: M <=hom N iff for all modules T we have dim Hom(T,M) <= dim Hom(T,N). By theorem of AUSLANDER and REITEN this is a partial order. Moreover, the dual condition is equivalent to this one.
In total we have
M <=ext N ===> M <=deg N ===> M <=hom N.
All this is known from work done by ABEASIS & DEL FRA, RIEDTMANN, BONGARTZ, and others. However, the question raises whether these partial orders coincide for a given algebra A.
ABEASIS & DEL FRA (1985) classified degenerations for modules over quiver algebras of type An and over equivoriented quiver algebras of type Dn. In these cases all partial order actually coincide. C. RIEDTMANN (1986) extended these results to certain algebras with relations of the same type. She showed this with the help of almost split sequences (aka AUSLANDER-REITEN-sequences). And K. BONGARTZ (1995) succeeded in proving that for all tame quiver algebras the degeneration and the homomorphism order coincide. His major tools apart from the above conditions was a cancellation theorem giving conditions under which M+X <=deg N+X implies M <=deg N. (BONGARTZ (1994), Theorem 1)
When I started to work in this field BONGARTZ's result was not known (to me) yet. So I reproved that for the KRONECKER algebra (the quiver algebra to the two point quiver with two parallel arrows) the three relations coincide. For the (self-injective) algebra C := k[X,Y]/(X2,Y2) there is an example due to J. CARLSON that shows that the orders need not always coincide. It turns out that here the three partial orders are all different. A complete classification of the degenerations of this algebra can be found in NÜSKEN (1996). K. BONGARTZ observed that all minimal degenerations used there stem from short exact sequences in a more general sense: Given a short exact sequence 0->U->M->V->0 and an endomorphism of U we obtain a pushout sequence 0->U->N->V->0. Then M degenerates to N. (See BONGARTZ (1996), Lemma 1.1.) This generates another partial order situated between <=ext and <=deg; we write M <=PO N. Then for the algebra C this pushout order and the degeneration order coincide. Dually the pullback order and the degeneration coincide too. Now it may be asked whether this is true for arbitrary algebras:
Question Is <=PO equivalent to <=deg for any algebra A? Or, for which algebras A does this hold?
ZWARA (1998) answered this question: No, this is already wrong for some representation finite algebra. Moreover, he characterized degenerations by means of representation theory. To this end he introduced a new partial order based on a lemma of RIEDTMANN (1986): Write M <=Z N if there is a module Z and a short exact sequence 0->N->M+Z->Z->0. The cited lemma now says: M <=Z N implies M <=deg N. Actually, the other implication holds, too, see ZWARA (1998):
M <=Z N <===> M <=deg N.
Of course all this can be dualized, but ZWARA observed that this leads to the same order.
It seems that the problem is settled now, but there is only very few control on the module Z or even on its dimension. And we can still ask:
Question How can degenerations be characterized in terms of homorphism space dimensions?
This is possible at least in principle, since - due to results of AUSLANDER & REITEN - the isomorphism type of a module can be characterized by these dimensions.