I am in the following situation. I have two (rather explicit and specific) dg commutative algebras $R,S$ over a field of characteristic $0$. In fact, $S$ is an $R$-algebra, in that I have a map $R \to S$. Because I am interested in computing some derived tensor products $S \otimes\_R$, I have worked out a "Koszul" resolution $\tilde S$ of $S$ over $R$. So all seems well.


But! Actually, $S$ and $R$ are both Gerstenhaber algebras (in dg vector spaces; the differential and the Gerstenhaber bracket point in opposite directions), and the map $R \to S$ is a homomorphism of Gerstenhaber algebras. My problem is that I have been so far unsuccessful at giving the resolution $\tilde S$ a Gerstenhaber structure such that the resolved map $R \to \tilde S$ is a homomorphism of Gerstenhaber algebras.


This leads me to two questions. The second question depends on the answer to the first.



> 
> **Question 1:** Does there necessarily exist a resolution of $S$ that computes the derived $S\otimes\_R$ and that is Gerstenhaber in a compatible way?
> 
> 
> **Question 2 if the answer to 1 is YES:** How do I construct it?
> 
> 
> **Question 2 if the answer to 1 is NO:** Certainly my homotopy equivalence $S \leftrightarrow \tilde S$ allows me to move the Gerstenhaber structure on $S$ to *something* on $\tilde S$. What structure on $\tilde S$ does it move to?
> 
> 
> 



Do I need to know what an infinity-Gerstenhaber algebra is, and if so, what is it?

> 
> Question 1: Does there necessarily
>  exist a resolution of S that computes
>  the derived $S\otimes\_R$ and that is
>  Gerstenhaber in a compatible way?
> 
> 
> 


Yes. As pointed out in the comments, the category of dg Gerstenhaber algebra admits a model structure in which the weak equivalences are the quasi-isomorphisms, fibrations are degreewise surjections, and cofibrant objects are those dg Gerstenhaber algebras that are free as graded algerbas. 


This actually works with dg algebras over any given operad $\mathcal O$ (in place of Gerstenhaber). 


This is proved in Hinich's paper (quoted by the nLab: <http://ncatlab.org/nlab/show/model+structure+on+dg-algebras+over+an+operad>). 


Then there is also a natural model structure on the category of dg Gerstenhaber $R$-algebras (there is a more general statement about existence of a model structure on the category of objects under a given one $X$ in a model category $\mathcal C$). 


So, the answer to the title of your question is that you don't "need" to know what a $G\_\infty$-algebra is. 



> 
> Question 2 if the answer to 1 is YES: How do I construct it?
> 
> 
> 


Shortly, bar-cobar. You can have a look at [Homologie et model minimal des algèbres de Gerstenhaber](https://eudml.org/doc/10501) in order to see how it works in details. 


Btw, the above paper also tells you what is the definition of a $G\_\infty$-algebra. 



> 
> Question 2 if the answer to 1 is NO:
>  Certainly my homotopy equivalence $S\leftrightarrow\widetilde{S}$
>  allows me to move the Gerstenhaber
>  structure on $S$ to something on $\widetilde{S}$.
>  What structure on $\widetilde{S}$ does it move to?
> 
> 
> 


Even though the answer to Question 1 is YES, there is still something to say here. 
There is on $\widetilde{S}$ a $G\_\infty$-structure. This is "just" homotopy transfer formula (and the use of the explicit minimal model for the Gerstenhaber operad). 


The homotopy transfer for algebras over operad $\mathcal O$, w.r.t. to a cofibrant resolution $\widetilde{\mathcal O}\to\mathcal O$ is proved in the appendix A.2 of [my paper with Van den Bergh](http://arxiv.org/abs/0710.4510) (see also Theorem 10.3.6 in [Loday-Vallette's Algebraic Operads](http://math.unice.fr/~brunov/Operads.pdf) for the Koszul case). 


