Jacob Skarby - Auslander Reiten theorem

Abstract

Talk

We recall the Auslander-Reiten quiver.

Definition

The Auslander-Reiten-quiver ${\Gamma }_A$ for a given Algebra $A$ has as vertices the representatives of indecomposable modules (called $indA$) in $modA$. We insert an arrow if there is an irreducible morphism between the two

$indA$ is a Spectroid which allowed us to construct an associate quiver.

We imagine projective objects as being objects being easy to map out of. In the catergory of groups, those would be the free groups. In the category of modules, an example would be given by paths inside the Aulander-Reiten-quiver which “go from the bottom to the top”

We quickly define what a Projective cover is and when it is considered minimal. But we’re allowed to black box the concepts. Next we define the Nakayama functor on a projective cover and the Nakayama translate. The Nakayama swaps injective and surjective objects. On the Auslander Reiten quiver, it acts by mirroring.

The Nakayama translate ${\tau }_A$ can be seen as a translation in the Auslander-Reiten-quiver. There is an “almost” inverse ${\tau }_A^{-}$.

Properties

1. Independent of minimal projective presentation (even though we had some choice in definitions)
2. ${\tau }_A$ iff $M$ is projective.
3. It induces a bijection between the non-projective and the non-injective.

The Nakayama functor is more complicated than a flip, but it does map projective to injectives.

Lemma

The Nakayama functor preserves exact sequences.

Proposition

For any $A$-Module (Algebra) $L,M$ there is a natural map. ${\alpha }_{(L,M)}:DHo{m}_A(M,L)\to Ho{m}_A(L,{\nu }_{A}M),\phi \mapsto \phi (mu{l}_l\circ \cdot )$. $DHo{m}_A(M,L)$ is the dual space of the morphisms between $M$ and $L$. (“Things that take morphisms and spit out smth. probably in $A$”)

The kernel of this map are all $\phi$, which map all $pro{j}_A(M,L)$ (“morphisms factorin over projectives”) to $0$. Some calculations give us as the kernel: $DHo{m}_A(M,L)$ So just all of the pre-image??

Definition

The Auslander-Reiten sequence is defined by some decoration of a short exact sequence by Radical morphisms.

THEOREM (AR)

1. Take $L,M$ be $A$-modules. There are bijections Ext_{A}^{1}(L, \tau_{A}M) \to D\underline{Hom}_{A}(M, L)$$$$Ext_{A}^{1}(\tau^{-}_{A}L, M) \to D\overline{Hom}_{A}(M, L)
2. There is also a second and third statement, which is also hard to follow
   What is Ext??

I just checked a typical resource where I’d find this kind of information. It turns out that $Ext$ is a concept taken from homological algebra rather than category theory. I checked a standard textbook on homological algebra. It turns out that Ext is mentioned at around page 100.

The next 30 minutes where spent on understanding the proof.

I am told that $Ex{t}_A^1(L,M)$ corresponds to short exact sequences $0\to M\to K\to L\to 0$. This means the theorem gives us an identification of short exact sequences.