Let $k$ be a field (if necessary for my question, we can assume its characteristic to be zero). In the non derived context, we can then define deformations of an associative algebra $S$ as follows:

If $A \in \mathfrak{A}rt_k$ is an Artin ring, a deformation of $S$, is a pair $(\mathcal{S},\pi_A)$, where $\mathcal{S}$ is an $A$-algebra and $\pi_A: \mathcal{S}\otimes_A k\to S$ is an isomorphism of $k$-algebras.

Two such deformations $(\mathcal{S},\pi_A)$ and $(\mathcal{S}',\pi'_A)$ are said to be isomorphic, if there exists an isomorphism of $A$-algebras $\varphi: \mathcal{S}\to\mathcal{S}'$, such that $\pi'_A\circ (\varphi\otimes_A id_k)=\pi_A$.

Then the functor

$Def_S: \mathfrak{A}rt_k \to Set\;;\; A \mapsto \{deformations\; over\; A\}\;/\;isomorphisms$

is a classical (underived) deformation functor.

Now my question is, how exactly can we extend this deformation functor, into (a model of) the derived setting,if we allow for more general $E_n$ or $E_\infty$ deformations?

Personally, I'm most familiar with Manettis approach to the derived situation, but every explicit extension would be welcome.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.