homotopy transfer for sheaves of algebras - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T19:54:14Z http://mathoverflow.net/feeds/question/65934 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65934/homotopy-transfer-for-sheaves-of-algebras homotopy transfer for sheaves of algebras DamienC 2011-05-25T08:49:01Z 2011-05-25T10:08:45Z <p><strong>homotopy transfer for algebras</strong></p> <p>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). </p> <p>Nevertheless, it is well-known that there exists an $A_\infty$-algebra structure $(m_k)_{k\geq1}$ on $H(A)$ with the following properties: </p> <ul> <li>$H(A)$ equipped with this $A_\infty$-structure is weakly equivalent to $A$. </li> <li>the first structure map $m_1$ (i.e. the differential) vanishes. </li> <li>the second structure map $m_2$ coincide with the natural product on $H(A)$. </li> </ul> <p>This structure is essentially unique: it is unique up to a unique $A_\infty$-isomorphism. </p> <p>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 $id_A$. </p> <p><strong>homotopy transfer for sheaves of dg algebras?</strong></p> <p>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. </p> <p><strong>the question</strong></p> <blockquote> <p>How does homotopy transfer works for sheaves of algebras? </p> </blockquote> <p><strong>possible (incomplete) answer</strong></p> <p>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 </p> <blockquote> <p>what is the right definition of a (homotopy) sheaf of $A_\infty$-algeras?</p> </blockquote> <p>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. </p> <p>At this moment I am stuck. I am sure I am not far from the answer, but somehow I can't see the point. </p> http://mathoverflow.net/questions/65934/homotopy-transfer-for-sheaves-of-algebras/65942#65942 Answer by Fernando Muro for homotopy transfer for sheaves of algebras Fernando Muro 2011-05-25T10:08:45Z 2011-05-25T10:08:45Z <p>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 <em>derived A-infinity algebra</em> 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 :</p> <p>MR2608191 (2011c:16030) Sagave, Steffen(N-OSLO) DG-algebras and derived -algebras. (English summary) J. Reine Angew. Math. 639 (2010), 73–105. </p> <p>This solves the case of quasi-coherent sheaves on affine schemes, so it could be a good starting point to answer your question...</p>