G\'eom\'etricit\'e artinienne de l'$\infty$-champs des \'el\'ements de Maurer-Cartan

TL;DR

This paper constructs a new Artin $$-stack of Maurer-Cartan elements for a dg-category, demonstrating its formal smoothness using techniques similar to previous work on the variety of complexes.

Contribution

It explicitly constructs the Artin $$-stack $ ext{MC}_ ext{P}$ of Maurer-Cartan elements and proves the formal smoothness of the associated map.

Findings

Successfully constructs $ ext{MC}_ ext{P}$ for dg-category $ ext{P}$

Shows the map $V o  ext{MC}_ ext{P}$ is formally smooth

Extends techniques from previous work on the variety of complexes

Abstract

In this paper, based on the same techniques as in \cite{BENZ08} for the construction of a formally smooth map $V \to Perf$ from the variety of complexes to the Artin $\infty$-stack $Perf$, we explicitly construct a new Artin $ \infty $-stack of Maurer-Cartan elements of a dg-category $ \mathcal {P} $ denoted by $\mathcal{MC_P} $, with a map $ V \to \mathcal{MC_P} $ which we show to be formally smooth.