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Graded loospaces in mixed characteristics and de Rham-Witt algebra

Abstract : 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 de�finition 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.
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https://theses.hal.science/tel-03848045
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Submitted on : Thursday, November 10, 2022 - 5:31:17 PM
Last modification on : Friday, November 11, 2022 - 4:06:35 AM

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2022TOU30117a.pdf
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  • HAL Id : tel-03848045, version 1

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Ludovic Monier. Graded loospaces in mixed characteristics and de Rham-Witt algebra. Algebraic Geometry [math.AG]. Université Paul Sabatier - Toulouse III, 2022. English. ⟨NNT : 2022TOU30117⟩. ⟨tel-03848045⟩

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