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I have the following question : in a Krull-Schmidt category (say the category of finite length left modules over a ring, this is the case which interests me), is it possible to relate the number of indecomposable summands of an extension $E$ of $M$ by $N$ with the number of indecomposables of $M$ and $N$ ?

More precisely, if we write $M=M_1\oplus...\oplus M_r $ and $N=N_1\oplus ...\oplus N_s$ where all the $M_i$ and $N_i$ are indecomposable, and given an exact sequence $0\longrightarrow N \longrightarrow E\longrightarrow M\longrightarrow 0$ and $E=E_1\oplus...\oplus E_q$ with $E_i$ indecomposable again, do we have $q\leq r+s$ ?

I am interested in the equality case too. In case we have $q=r+s$, is $E$ isomorphic to the direct sum $M\oplus N$ ?

I would like to tell you that in case of $\mathbf{Z}$-modules of finite length, all the answers are positive :just consider good bases for the inclusion $N\longrightarrow E$ to see that it is sufficient to treat the case where $M,E,N$ are cyclic.

Thank you !

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    $\begingroup$ It may be useful to comment on the special case of finite dimensional modular representations of a simple algebraic group (or its Lie algebra). The tensor product of two irreducible modules (where $r=s=1$) very often has a large number of composition factors, and usually this kind of module does not decompose neatly into a direct sum of irreducibles. (Look for example at the case where one of the two modules is a Steinberg module.) In this set-up, or the somewhat similar one of a finite group of Lie type, one sees many concrete examples where both of your questions have answer no. $\endgroup$
    – Jim Humphreys
    Commented Apr 25, 2018 at 14:43
  • $\begingroup$ P.S. Here I'm of course working over a field, of prime characteristic. $\endgroup$
    – Jim Humphreys
    Commented Apr 25, 2018 at 15:28

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The answers to both questions are no in general, with a counter-example being given by the path algebra of a quiver of type $\mathsf{D}_4$—the category of left modules over this algebra is a Hom-finite Krull–Schmidt category.

First, there is an (Auslander–Reiten) short exact sequence

$$0\to\begin{smallmatrix}0\\0&1\\0\end{smallmatrix}\to\begin{smallmatrix}1\\0&1\\0\end{smallmatrix}\oplus\begin{smallmatrix}0\\1&1\\0\end{smallmatrix}\oplus\begin{smallmatrix}0\\0&1\\1\end{smallmatrix}\to\begin{smallmatrix}1\\1&2\\1\end{smallmatrix}\to 0$$

with indecomposable outer terms, but whose middle term is a direct sum of three indecomposables. (Here the notation indicates that vector spaces of the dimensions on the left are being included generically into a vector space of dimension on the right—such things are indecomposable when this list of dimensions is a positive root of $\mathsf{D}_4$.)

Moreover, there is a $2$-dimensional space of extensions from $\begin{smallmatrix}1\\1&1\\1\end{smallmatrix}$ to $\begin{smallmatrix}0\\0&1\\0\end{smallmatrix}$, where the middle term is generically given by the indecomposable $\begin{smallmatrix}1\\1&2\\1\end{smallmatrix}$, but there are also non-split extensions with decomposable middle term, e.g. $\begin{smallmatrix}1\\0&1\\0\end{smallmatrix}\oplus\begin{smallmatrix}0\\1&1\\1\end{smallmatrix}$.

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