Let $M$ be an object in an $k$-linear abelian category with enough projectives. Then one can construct an $A\_\infty$-structure on the Ext algebra
$$Ext^\bullet(M,M)$$
as follows: One chooses projective resolution $P\rightarrow M$ and forms the Hom complex
$$Hom^\bullet (P,P)$$
Now the cohomology of this complex is the Ext algebra and in the case where $k$ is a field, one can choose a "homotopy retraction" $Ext^\bullet(M,M)\rightarrow Hom^\bullet (P,P)$ and transfer the dg-algebra structure on $Hom^\bullet (P,P)$ along it.


My questions are:


1. Why does this construction (up to $A\_\infty$-isomorphism) not depend on a choice of projective resolution?
2. One could try the same thing with an injective resolution, why is the result still the same?



$A_\infty$ structure on Ext-algebras well defined?
Let $P\to M$ be a projective resolution of $M$ and $M\to J$ be an injective resolution. Consider the composition of the morphisms of complexes $P\to M\to J$ and set $C$ to be the cone of the morphism $P\to J$. Then the complex $C$ has a subcomplex isomorphic to $J$ with the quotient complex isomorphic to $P[1]$. Moreover, the short exact sequence of complexes $J\to C\to P[1]$ splits as a short exact sequence of graded objects in your abelian category (i.e., after the differentials are forgotten).


Consider the subcomplex $D$ in the complex $Hom^\bullet(C,C)$ formed by all the homogeneous morphisms of graded objects $C\to C$ taking $J\subset C$ to $J\subset C$. This condition on morphisms $C\to C$ is preserved by the composition, so $D$ is a DG-algebra over $k$. The restriction of morphisms $C\to C$ to the subcomplex $J$ defines a DG-algebra morphism $D\to Hom^\bullet(J,J)$; and the passage to the induced morphism of the quotient complexes defines a DG-algebra morphism $D\to Hom^\bullet(P[1],P[1])\simeq Hom^\bullet(P,P)$.


It is claimed that both these DG-algebra morphisms are quasi-isomorphisms. E.g., the morphism $D\to Hom^\bullet(J,J)$ is surjective and its kernel is the complex $Hom^\bullet(P[1],C)$, which is acyclic as $P$ is a complex of projectives (bounded from above) and $C$ is acyclic. The argument for the second DG-algebra morphism is similar.


Now, being quasi-isomorphic DG-algebras, $Hom^\bullet(P,P)$ and $Hom^\bullet(J,J)$ have $A\_\infty$-isomorphic minimal $A\_\infty$-models. The proof of the independence of these from the choice of the resolution $P$ and/or $J$ is analogous.


