Hello,

We have the following assertion:

In an abelian category (edit: that has enough projectives), indecomposable projectives of finite length correspond bijectively to irreducibles (both up to isomorphism).

I see how to construct a well defined map from irreducibles (up to iso.) to indecomposable projectives, and to show that it is injective. What I do not know is how to show the following:

Given an indecomposable projective of finite length, how to show that it has a unique irreducible quoteint?

Thanks, Sasha

Hello,

We have the following assertion:

In an abelian category that has enough projectives and in which every object has finite length, indecomposable projectives correspond bijectively to irreducibles (both up to isomorphism).

I see how to construct a well defined map from irreducibles (up to iso.) to indecomposable projectives, and to show that it is injective. What I do not know is how to show the following:

Given an indecomposable projective of finite length, how to show that it has a unique irreducible quoteint?

(I have edited the question; Sorry about the initial imprecision).

Thanks, Sasha

Hello,

We have the following assertion:

In an abelian category, indecomposable projectives of finite length correspond bijectively to irreducibles (both up to isomorphism).

I see how to construct a well defined map from irreducibles (up to iso.) to indecomposable projectives, and to show that it is injective. What I do not know is how to show the following:

Given an indecomposable projective of finite length, how to show that it has a unique irreducible quoteint?

Thanks, Sasha

Hello,

We have the following assertion:

In an abelian category (edit: that has enough projectives), indecomposable projectives of finite length correspond bijectively to irreducibles (both up to isomorphism).

I see how to construct a well defined map from irreducibles (up to iso.) to indecomposable projectives, and to show that it is injective. What I do not know is how to show the following:

Given an indecomposable projective of finite length, how to show that it has a unique irreducible quoteint?

Thanks, Sasha