The main objective of this thesis is to study mixed graded complexes as a framework where to study derived deformation theory. In particular, I investigated the relationship between mixed graded complexes and derived Lie algebras. The main contributions of this thesis are the following. First, I provided the ∞-category of mixed graded complexes with a complete and non-degenerate t-structure, which exhibits such ∞-category as the left completion of the Beilinson t-structure on the filtered derived ∞-category. Secondly, I constructed a family of Chevalley-Eilenberg ∞-functors computing homology and cohomology of derived Lie algebras, endowing them of a richer structure of mixed graded complexes. Even if it is known that Chevalley-Eilenberg complexes are endowed with such structure, my construction is new and completely model-independent.
L’obiettivo principale di questa tesi è lo studio dei complessi misti graduati e del loro possibile ruolo nell’ambito della ricerca nella teoria della deformazione. In particolare, mi sono occupato della relazione tra complessi misti graduati e algebre di Lie derivate. I due contributi principali di questa tesi sono la costruzione di una t-struttura completa e non-degenere sulla ∞-categoria stabile dei complessi misti graduati, che esibisce i moduli misti graduati come il completamento sinistro della t-struttura di Beilinson sulla ∞-categoria derivata filtrata, e la costruzione di una famiglia di funtori di Chevalley-Eilenberg verso la ∞-categoria dei moduli misti graduati. Nonostante sia noto che i complessi di Chevalley-Eilenberg siano dotati di questa struttura, in letteratura tale funtore non è mai stato costruito in maniera indipendente da qualsiasi modello per le ∞-categorie.
MIXED GRADED MODULES IN HOMOTOPY LIE THEORY
PAVIA, EMANUELE
2022
Abstract
The main objective of this thesis is to study mixed graded complexes as a framework where to study derived deformation theory. In particular, I investigated the relationship between mixed graded complexes and derived Lie algebras. The main contributions of this thesis are the following. First, I provided the ∞-category of mixed graded complexes with a complete and non-degenerate t-structure, which exhibits such ∞-category as the left completion of the Beilinson t-structure on the filtered derived ∞-category. Secondly, I constructed a family of Chevalley-Eilenberg ∞-functors computing homology and cohomology of derived Lie algebras, endowing them of a richer structure of mixed graded complexes. Even if it is known that Chevalley-Eilenberg complexes are endowed with such structure, my construction is new and completely model-independent.File | Dimensione | Formato | |
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https://hdl.handle.net/20.500.14242/112581
URN:NBN:IT:UNIMI-112581