{c}

We prove $(i)\Rightarrow(iii)$, the main step in <Ext-connected components determine blocks>. Suppose a <block of a finite-dimensional algebra> $B=Re$ had more than one component of the graph of nonsplit extensions between its <simple modules>. Partition its simple types into one component and the union of the others. There are no cross-extensions, so part (a) decomposes the left regular $B$-module as $B=U_1\oplus U_2$. Both summands are nonzero: every simple $B$-module is a quotient of the regular module and therefore occurs among its composition factors.

The maximality in (a) makes these summands canonical. Every right multiplication is a left $B$-module endomorphism, so it preserves them. Thus they are two-sided ideals, and their projection $p:B\to U_1$ commutes with both left and right multiplication. Putting $f=p(e)$ gives
$$
p(x)=xf=fx\quad(x\in B),\qquad f^2=f.
$$
Since both summands are nonzero, $f\ne0,e$. This contradicts the primitivity of the central block idempotent $e$. There is therefore just one Ext component within each block, proving that simples in the same block satisfy $\boxed{S\sim T}$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 138 / 5 / b / i / Solution

{c}

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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