Spell out existence of hull for defos of reps

examples-defos.tex

@@ -256,8 +256,6 @@ \section{Finite projective modules}
 \end{proof}
 
 
-
-
 \section{Representations of a group}
 \label{section-representations}
 
@@ -407,6 +405,69 @@ \section{Representations of a group}
 \ref{etale-cohomology-lemma-finite-dim-group-cohomology}.
 \end{proof}
 
+\noindent
+In Example \ref{example-representations} if $\Gamma$ is finitely generated
+and $(V, \rho_0)$ is a finite dimensional representation of $\Gamma$
+over $k$, then $\Deformationcategory_{V, \rho_0}$
+admits a presentation by a smooth prorepresentable groupoid in functors
+over $\mathcal{C}_\Lambda$
+and a fortiori has a (minimal) versal formal object. This follows
+from Lemmas \ref{lemma-representations-RS} and \ref{lemma-representations-TI}
+and the general discussion in Section \ref{section-general}.
+
+\begin{lemma}
+\label{lemma-representations-hull}
+In Example \ref{example-representations} assume $\Gamma$ finitely generated.
+Let $\rho_0 : \Gamma \to \text{GL}_k(V)$ be a finite dimensional representation.
+Assume $\Lambda$ is a complete local ring with residue field $k$
+(the classical case). Then the functor
+$$
+F : \mathcal{C}_\Lambda \longrightarrow \textit{Sets},\quad
+A \longmapsto \Ob(\Deformationcategory_{V, \rho_0}(A))/\cong
+$$
+of isomorphism classes of objects has a hull. If
+$H^0(\Gamma, \text{End}_k(V)) = k$, then $F$ is
+prorepresentable.
+\end{lemma}
+
+\begin{proof}
+The existence of a hull follows from Lemmas \ref{lemma-representations-RS} and
+\ref{lemma-representations-TI} and
+Formal Deformation Theory, Lemma \ref{formal-defos-lemma-RS-implies-S1-S2}
+and Remark \ref{formal-defos-remark-compose-minimal-into-iso-classes}.
+
+\medskip\noindent
+Assume $H^0(\Gamma, \text{End}_k(V)) = k$. To see that $F$
+is prorepresentable it suffices to show that $F$ is a
+deformation functor, see Formal Deformation Theory, Theorem
+\ref{formal-defos-theorem-Schlessinger-prorepresentability}.
+In other words, we have to show $F$ satisfies (RS).
+For this we can use the criterion of Formal Deformation Theory, Lemma
+\ref{formal-defos-lemma-RS-associated-functor}.
+The required surjectivity of automorphism groups will follow if we
+show that
+$$
+A \cdot \text{id}_M =
+\text{End}_{A[\Gamma]}(M)
+$$
+for any object $(A, M, \rho)$ of $\mathcal{F}$ such that
+$M \otimes_A k$ is isomorphic to $V$ as a representation of $\Gamma$.
+Since the left hand side is contained in the right hand side,
+it suffices to show
+$\text{length}_A \text{End}_{A[\Gamma]}(M) \leq \text{length}_A A$.
+Choose pairwise distinct ideals
+$(0) = I_n \subset \ldots \subset I_1 \subset A$
+with $n = \text{length}(A)$. By correspondingly filtering
+$M$, we see that it suffices to prove $\Hom_{A[\Gamma]}(M, I_tM/I_{t + 1}M)$
+has length $1$. Since $I_tM/I_{t + 1}M \cong M \otimes_A k$
+and since any $A[\Gamma]$-module map $M \to M \otimes_A k$ factors
+uniquely through the quotient map $M \to M \otimes_A k$
+to give an element of
+$$
+\text{End}_{A[\Gamma]}(M \otimes_A k) = \text{End}_{k[\Gamma]}(V) = k
+$$
+we conclude.
+\end{proof}
 
 
 
@@ -499,6 +560,39 @@ \section{Continuous representations}
 Lemma \ref{lemma-representations-TI}.
 \end{proof}
 
+\noindent
+In Example \ref{example-continuous-representations} if $\Gamma$
+is topologically finitely generated
+and $(V, \rho_0)$ is a finite dimensional continuous representation of $\Gamma$
+over $k$, then $\Deformationcategory_{V, \rho_0}$
+admits a presentation by a smooth prorepresentable groupoid in functors
+over $\mathcal{C}_\Lambda$
+and a fortiori has a (minimal) versal formal object. This follows
+from Lemmas \ref{lemma-continuous-representations-RS} and
+\ref{lemma-continuous-representations-TI}
+and the general discussion in Section \ref{section-general}.
+
+\begin{lemma}
+\label{lemma-continuous-representations-hull}
+In Example \ref{example-continuous-representations} assume $\Gamma$
+is topologically finitely generated.
+Let $\rho_0 : \Gamma \to \text{GL}_k(V)$ be a finite dimensional representation.
+Assume $\Lambda$ is a complete local ring with residue field $k$
+(the classical case). Then the functor
+$$
+F : \mathcal{C}_\Lambda \longrightarrow \textit{Sets},\quad
+A \longmapsto \Ob(\Deformationcategory_{V, \rho_0}(A))/\cong
+$$
+of isomorphism classes of objects has a hull. If
+$H^0(\Gamma, \text{End}_k(V)) = k$, then $F$ is
+prorepresentable.
+\end{lemma}
+
+\begin{proof}
+The proof is exactly the same as the proof of
+Lemma \ref{lemma-representations-hull}.
+\end{proof}
+
 
 
 \section{Graded algebras}