Inverse Problems for deformation rings

Transactions of the American Mathematical Society (Impact Factor: 1.12). 12/2010; 365(11). DOI: 10.1090/S0002-9947-2013-05848-5
Source: arXiv


Let $\mathcal{W}$ be a complete local commutative Noetherian ring with
residue field $k$ of positive characteristic $p$. We study the inverse problem
for the versal deformation rings $R_{\mathcal{W}}(\Gamma,V)$ relative to
$\mathcal{W}$ of finite dimensional representations $V$ of a profinite group
$\Gamma$ over $k$. We show that for all $p$ and $n \ge 1$, the ring
$\mathcal{W}[[t]]/(p^n t,t^2)$ arises as a universal deformation ring. This
ring is not a complete intersection if $p^n\mathcal{W}\neq\{0\}$, so we obtain
an answer to a question of M. Flach in all characteristics. We also study the
`inverse inverse problem' for the ring $\mathcal{W}[[t]]/(p^n t,t^2)$; this is
to determine all pairs $(\Gamma, V)$ such that $R_{\mathcal{W}}(\Gamma,V)$ is
isomorphic to this ring.

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Available from: Ted Chinburg, Jun 03, 2015
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    • "This example was greatly generalised to provide a positive answer for all rings of the form W (k k k)[[t]]/(p n t, t 2 ) in [4]. Furthermore, in [5] a categorisation of all possible pairs (Γ, ρ) which have R = W (k k k)[[t]]/(p n t, t 2 ) as its universal deformation ring was given. Another class of none complete intersection rings were shown by Rainone in [13] to be universal deformation rings, namely the rings Z p [[t]]/(p n , p m t) where p > 3 and n > m are positive integers. "
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    ABSTRACT: We show the inverse deformation problem has an affirmative answer: given a complete local noetherian ring $A$ with finite residue field $\pmb{k}$, we show that there is a topologically finitely generated profinite group $\Gamma$ and an absolutely irreducible continuous representation $\bar\rho:\Gamma\to GL_n(\pmb{k})$ such that $A$ is the universal deformation ring for $\Gamma,\bar\rho$.
    Preview · Article · Jul 2013
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    • "FRAUKE M. BLEHER equivalent to a principal block, then R(G, V ) is isomorphic to a subquotient ring of W D. In particular, R(G, V )/pR(G, V ) is finite dimensional over k in these cases. Other instances where the finite dimensionality of R(G, V )/pR(G, V ) was established can be found for example in [1] [3] and their references. However, de Smit and Rainone found examples in the case when p ≥ 5 of finite groups G and kG-modules V such that R(G, V )/pR(G, V ) is isomorphic to k[[t]], and hence not finite dimensional over k (see [3, Remark 4.3] and [10]). "
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    ABSTRACT: We provide a series of examples of finite groups G and mod p representations V of G whose stable endomorphisms are all given by scalars such that V has a universal deformation ring R(G,V) which is large in the sense that R(G,V)/pR(G,V) is isomorphic to a power series algebra in one variable.
    Preview · Article · Apr 2012 · Proceedings of the American Mathematical Society
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    ABSTRACT: The versal deformation ring R(G,V) of a mod p representation V of a profinite group G encodes all isomorphism classes of lifts of V to representations of G over complete local commutative Noetherian rings. We introduce a new technique for determining R(G,V) when G is finite which involves Brauer's generalized decomposition numbers.
    Preview · Article · Mar 2012 · Transactions of the American Mathematical Society
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