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On the inverse problem for deformation rings of representations.

Deformation theory was created by B. Mazur and it was a powerful tool in the Wiles' proof of the Fermat's last theorem. In the thesis we try to understand what kind of ring can occur as universal deformation ring of a representation $\bar{\rho}\colon G\to GL_{\mathbb{F}_p}(V)$. We give three examples of representation whose universal deformation rings are: $\mathbb{F}_p,\mathbb{Z}/p^n\mathbb{Z},\mathbb{Z}_p[[t]]/(p^n,p^mt)$. In particular the last example is a negative answer to questions by M. Flach and T. Chinburg.

Mostra/Nascondi contenuto.
;; x ;; x Chapter 1 Introduction Let G be a finite group, let k be a finite field of characteristic p > 0. An n-dimensional representation of G over k is a group homomorphism : G! GL(k) n In the same way, if A is a complete Noetherian local ring with residue field k, an n-dimensional representation of G over A is a group homomorphism G! GL(A). We say that (A; ~ ) is a lift of ifA is a complete Noetherian n local ring with residue fieldk and ~ is a group homomorphism for which the diagram GL(A) n x x x x x x x x GL(k) G n commutes. Two lifts ~ ; ~ : G ! GL(A) of over A are said to be 12n equivalent if there exists a matrix K in ker(GL(A)! GL(k)) for which nn 1 K(g)K = ~ (g) for every g in G. An equivalence class of lifts is called 12 deformation of . Given a representation ofG overk and a complete Noethe- rian local ringA, we define the set Def(;A ) to be the set of all deformations of to A. For a representation : G! GL(k) the universal deformation ring R n a lift (R) for which the following universal property holds: for any lift u (A; ~ ) of there exists a unique homomorphism ’: R! A such that the following diagram: GL(R) n x x x ^ ’ x x x x x GL(A) G n u commutes, where the vertical arrow ^ ’ is the map induced by ’: R!A. //// ~ ~ ~ is ; u u

Laurea liv.II (specialistica)

Facoltà: Scienze Matematiche, Fisiche e Naturali

Autore: Raffaele Rainone Contatta »

Composta da 38 pagine.


Questa tesi ha raggiunto 45 click dal 01/02/2011.

Disponibile in PDF, la consultazione è esclusivamente in formato digitale.