Graded loospaces in mixed characteristics and de Rham-Witt algebra ; Espaces de lacets gradués en caractéristique mixte et algèbre de de Rham-Witt

In this thesis, we study a variation of the graded loop space construction for mixed graded derived schemes endowed with a Frobenius lift. We develop a theory of derived Frobenius lifts on a derived stack which are homotopy theoretic analogues of structures for commutative rings. This graded loopspace construction is the first step towards a définition of the de Rham-Witt complex for derived schemes. In this context, a loop is given by an action of the "crystalline circle", which is a formal analogue of the topological circle, endowed with its natural endomorphism given by multiplication by p. In this language, a derived Dieudonné complex can be seen as a graded module endowed with an action of the crystalline circle. We also develop a theory of saturation of a derived Dieudonné complex which coincides with the one previously defined for p-torsion-free commutative rings. We include a short section on the study of graded functions of linear stacks where we prove that the 1-weighted functions on a linear stack recover the quasi-coherent sheaf defining the stack, under mild assumptions. We also give a full description of graded functions on a linear stack when working in characteristic zero. Finally, we give many directions for further developments regarding graded loopspaces and Frobenius lifts. More specifically, we outline a theory of symplectic forms based on our construction of the de Rham-Witt complex and similarly, a theory of Dieudonné foliations using derived Frobenius lifts. ; Cette thèse a pour but d'étudier une variation de la construction de l'espace des lacets gradués pour les schémas dérivés de caractéristique mixte, lorsqu'ils sont munis d'un relèvement du Fobenius. Nous développons une théorie de relèvement dérivé du Frobenius sur un champ dérivé : c'est un analogue homotopique de la notion de structures pour un anneau commutatif. La construction d'un espace de lacets gradués est la première étape d'une définition du complexe de De Rham-Witt pour un schéma dérivé. Dans ce contexte, un lacet est ...