= Ext-connected components determine blocks
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For a finite-dimensional <associative algebra>, form a graph on its <simple modules>, joining $S,T$ when $\operatorname{Ext}^1(S,T)$ or $\operatorname{Ext}^1(T,S)$ is nonzero. Its connected components are exactly the <simple modules> belonging to each <block of a finite-dimensional algebra>. They are also the components generated by sharing a <Jordan–Hölder factor> occurrence in an <indecomposable representation> that is a <projective module>.
Indeed, <Ext separation of finite-length modules> would split the regular module of a block into canonical summands if that block had two graph components. Right multiplication preserves these summands, so its projection supplies a nontrivial central <idempotent>, contradicting the definition of a block. An indecomposable projective belongs to one block. Finally, a nonsplit extension $0\to U\to V\to W\to0$ of <simple modules> is a quotient of the <projective cover> of $W$: a lift onto $W$ must contain $U$, since otherwise the extension splits. Thus its two endpoints share factors in an indecomposable projective.
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