Smoothness in Relative Geometry

Abstract

In \cite{tva}, Bertrand Toen and Michel Vaquie defined a scheme theory for a closed monoidal category $(C,\otimes,1)$. In this article, we define a notion of smoothness in this relative (and not necesarilly additive) context which generalizes the notion of smoothness in the category of rings. This generalisation consists practically in changing homological finiteness conditions into homotopical ones using Dold-Kahn correspondance. To do this, we provide the category $sC$ of simplicial objects in a monoidal category and all the categories $sA-mod$, $sA-alg$ ($a\in sComm(C)$) with compatible model structures using the work of Rezk in \cite{r}. We give then a general notions of smoothness in $sComm(C)$. We prove that this notion is a generalisation of the notion of smooth morphism in the category of rings and provide some examples of smooth morphisms in $N-alg$, $Comm(Set)$ and Comm(C).