# L-infini déformations et cohomologie de Hochschild

Abstract : In the first part of this thesis, deformations of $L_\infty$-algebras are defined in such a way that the bases of deformations are also $L_\infty$-algebras. A universal and a semiuniversal deformation is constructed for $L_\infty$-algebras whose tangent complex admits a splitting. Furthermore, a minimal $L_\infty$-structure on the cohomology $H$ of a differential graded Lie algebra $L$ and an $L_\infty$-quasi-isomorphism between $H$ and $L$ is constructed explicitly. As an application, it is shown that singularities can be described by $L_\infty$-algebras, and a new proof for the existence of formal moduli spaces for isolated singularities is provided. The second part is an abstract approach to Hochschild (co)homology by means of good pairs of categories''. The classical HKR-theorem, wich gives an isomorphism between the n-th Hochschild cohomology of smooth algebras and the n-th exterior power of their module of Kähler differentials is generalized for simplicial, graded commutative algebras in good pairs of categories. This generalization is applied to complex spaces and Noetherian schemes and several theorems on the decomposition of their respective (relative) Hochschild (co)homologies are deduced.
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Document type :
Theses
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https://tel.archives-ouvertes.fr/tel-00007197
Contributor : Arlette Guttin-Lombard <>
Submitted on : Monday, October 25, 2004 - 11:29:12 AM
Last modification on : Wednesday, November 4, 2020 - 2:05:44 PM
Long-term archiving on: : Thursday, September 13, 2012 - 12:25:09 PM

### Identifiers

• HAL Id : tel-00007197, version 1

### Citation

Frank Schuhmacher. L-infini déformations et cohomologie de Hochschild. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2004. Français. ⟨tel-00007197⟩

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