homotopy transfer for sheaves of algebras

2011-05-25T08:49:01Z

homotopy transfer for algebras

Let $A$ be a differential graded (dg) $k$-algebra, and $H(A)$ its cohomology. $H(A)$ is naturally equipped with the structure of a graded algebra. In general we don't have that $H(A)$ and $A$ are weakly equivalent (i.e. quasi-isomorphic).

Nevertheless, it is well-known that there exists an $A_\infty$-algebra structure $(m_k)_{k\geq1}$ on $H(A)$ with the following properties:

$H(A)$ equipped with this $A_\infty$-structure is weakly equivalent to $A$.
the first structure map $m_1$ (i.e. the differential) vanishes.
the second structure map $m_2$ coincide with the natural product on $H(A)$.

This structure is essentially unique: it is unique up to a unique $A_\infty$-isomorphism.

Moreover, there are explicit formulas for for the $A_\infty$-structure and the weak equivalence, in terms of planar trees. The main point is that the formula involves the choice of quasi-isomorphisms $i:H(A)\to A$ and $p:A\to H(A)$, together with an homotopy $h$ between $i\circ p$ and $id_A$.

homotopy transfer for sheaves of dg algebras?

I would be interested to know how to adapt this for sheaves. Namely, if now $A$ is a sheaf of dg algebras and $H(A)$ its cohomology sheaf. First of all one has to assume that $A$ is formal (i.e. quasi-isomorphic to $H(A)$) as a sheaf of $k$-modules. But even in this situation the existence of $i$, $p$ and $h$ is not guarantied.

the question

How does homotopy transfer works for sheaves of algebras?

possible (incomplete) answer

One can do things locally and then try to glue, the gluing condition will probably be satisfied only in a weak sens... and then my question might boil down to

what is the right definition of a (homotopy) sheaf of $A_\infty$-algeras?

A way to answer this is to use model categories. I was planning to proceed in the following way (very shortly): $A_\infty$-algebras are fibrant objects in the model category of dg coalgebras, then we have a Reedy model structure on presheaves of dg coalgebras, and sheaves of $A_\infty$-algebras can be defined as fibrant objects in this model category.

At this moment I am stuck. I am sure I am not far from the answer, but somehow I can't see the point.

Answer by Fernando Muro for homotopy transfer for sheaves of algebras

2011-05-25T10:08:45Z

If $R$ is a commutative $k$-algebra, a quasi-coherent sheaf of dg-$k$-algebras on $\operatorname{Spec}R$ would be just a dg-$R$-algebra $A$. It's known that there need not be any $R$-linear A-infinity structure on $H(A)$ quasi-isomorphic to $A$ in the $A$-infinity sense. Therefore the classical transfer theorem (which only works over fields) does not sheafify. Nevertheless, Sagave defined the notion of derived A-infinity algebra in such a way that there is a derived A-infinity structure on $H(A)$ quasi-isomorphic to $A$ in an $R$-linear and derived A-infinity sense, see :

MR2608191 (2011c:16030) Sagave, Steffen(N-OSLO) DG-algebras and derived -algebras. (English summary) J. Reine Angew. Math. 639 (2010), 73–105.

This solves the case of quasi-coherent sheaves on affine schemes, so it could be a good starting point to answer your question...

homotopy transfer for sheaves of algebras - MathOverflow

homotopy transfer for sheaves of algebras

Answer by Fernando Muro for homotopy transfer for sheaves of algebras