Kadush ? - Modules and Homological Algebra

Abstract

Talk

Definition (Module)

$M$ is a left $R$-module if $(M,+)$-ab group. If we have a multiplication $T\times M\to M$ with

1. $r(x+y)=rx+ry$
2. $(r+s)x=rx=sx$
3. $(rs)x=r(sx)$
4. $1x=x$

So its a generalisation of a Vector space, but $R$ doesn’t need to be a field.

Definition

$modA$ is the category of all $A$-modules with $A$ a finitely dimensional $K$-algebra. (Algebras are rings)

Definition

We definine when an object is projective. Similarily, we define the concept of injective

Definition

If $p=modA$ then $p$ is called a projective $A$-module.

Projective means “free”

All of this is used to show that epimorphisms and monomorphisms in the usual sense just mean projections and injections in the categorical sense.

Example

Consider $A$ as left $A$-module, then $A$ is projective mod $A$.

Lemma

If $P=P_1\oplus P_2$ then $P$ is projective iff. $P_1$ and $P_2$ are projective.

Proposition

Each indecomposable projective $A$-module is isomorphic to a direct summand $A_i$ of $A=⨁A_i$.

• What is a decomposable module?

Corollary

Let $A$ be a basic, fin-dim $K$-alg. If $\{e_1,...,e_t\}$ is a complete pairwise orthogonal, primitive, idempotents of $A$, then $A{e}_1,...,A{e}_t$ is a complete list of pairwise non-isomorphic projective $A$-modules up to isomorphism.

• But aren’t the $e_i$ elements in $A$.

Proposition

A fin-dim left algebra $A$-module is indecomposable iff. $En{d}_A(M)$ is local.

Corollary

The algebra $eAe$ is local iff. $e$ is primitive

• What does local mean?

A very long proof followed where I completely lost track.

Definition

A spectroid is a $K$-category s.t. it satisfies

1. the morphisms $C(x,y)$ are finite dimensional
2. the endomorphism algebra $C(x,x)$ is local
3. object are pairwise non-isomorphic

Lemma (Yoneda's Lemma)

Let $A$ be a finite dimensional $K$-algebra and $\{e_1,...,e_t\}$-complete set. (?) Then a lot of words followed…

If you have a quiver with no cycles, then the module

What to review:

• Quiver and Path algebras
• Projective Resolutions
• Baer sums
• Radical
• Nakayama
• Short exact sequences