$A_\infty$ structure on Ext-algebras well defined?

2012-03-12T16:39:30Z

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:

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

Answer by Leonid Positselski for $A_\infty$ structure on Ext-algebras well defined?

2012-03-12T19:12:09Z

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.

$A_\infty$ structure on Ext-algebras well defined? - MathOverflow

$A_\infty$ structure on Ext-algebras well defined?

Answer by Leonid Positselski for $A_\infty$ structure on Ext-algebras well defined?