In [Keller: A-infinity algebras in representation theory, Proposition 1(b)], Keller states that for an associative algebra the $Ext$-algebra of the simples is generated by $Ext^1(S,S)$ as an $A\_\infty$-algebra. 


Are there any other conditions on a module $M$ known to prove that $Ext^\*(M,M)$ is generated by $Hom(M,M)$ and $Ext^1(M,M)$ as an $A\_\infty$-algebra module $M$ over a finite dimensional associative algebra? E.g. conditions like $M$ generator of the derived category or something similar.



$Ext$-algebra generated by $Hom$ and $Ext^1$ as $A_\infty$-algebra?
Suppose $M=M\_1 \oplus \dots \oplus M\_r$ with $M\_i$ indecomposable for $1 \leq i \leq r$. Let $\mathcal F (M\_1, \dots, M\_r)$ denote the category of modules which admit a filtration with subquotients in the $M\_i$. The crucial assumption in Keller's proof is that $\mathcal F(M\_1, \dots, M\_r)$ is closed under syzygies.


With this assumption, any $n$-fold extension ($n>1$) $$\xi \in \operatorname{Ext}^n(M\_u,M\_v) \simeq \operatorname{Ext}^1(\Omega^{n-1}(M\_u),M\_v)$$ with $1 \leq u,v \leq r$ is the Yoneda splice of $n$ short-exact sequences $$0 \to X\_{i+1} \to E\_i \to X\_i \to 0 \quad (0 \leq i \leq n-1)$$ where $X\_0=M\_u$, $X\_n=M\_v$, and each $X\_i$ is an object in $\mathcal F(M\_1, \dots, M\_r)$ for $0 \leq i \leq n$.


As explained in the article you mentioned, the category $\mathcal F(M\_1, \dots, M\_r)$ is equivalent to a category of *twisted stalks*. A twisted stalk can be described as a pair $(B,\delta)$, where $B$ is a sequence of subfactors from $\{M\_1, \dots, M\_r\}$ and $\delta$ is an upper triangular matrix with entries from $\operatorname{Ext}^1(M,M)$.


If $X\_i$ corresponds to a twisted stalk $(B,\delta)$ and $X\_{i+1}$ corresponds to a twisted stalk $(B',\delta')$, then the short-exact sequence $0 \to X\_{i+1} \to E\_i \to X\_i \to 0$ corresponds to a morphism of twisted *complexes* $(B,\delta) \to (B',\delta')[1]$. Such a morphism is a matrix with entries from $\operatorname{Ext}^1(M,M)$.


So each short-exact sequence $0 \to X\_{i+1} \to E\_i \to X\_i \to 0$ can be described using only the $\operatorname{Ext}^1$ part of $\operatorname{Ext}^\ast(M,M)$. The higher multiplications enter the picture when we splice the short-exact sequences together. The rule for composing morphisms of twisted complexes involves $m\_2$ and higher multiplications of the matrices involved. When splicing all the short-exact sequences together, we end up with an expression for $\xi \in \operatorname{Ext}^n(M\_u,M\_v)$ using only elements in $\operatorname{Ext}^1(M,M)$ and $m\_2$ and higher multiplications. This shows that $\operatorname{Ext}^\ast(M,M)$ is generated in degree $0$ and $1$ as an $A\_\infty$-algebra.


A class of examples where the assumption is satisfied is when $\{M\_1, \dots, M\_r\}$ are the standard modules over a quasi-hereditary algebra.


