The Stacks project

93.5 Representations of a group

The deformation theory of representations can be very interesting.

Example 93.5.1 (Representations of a group). Let $\Gamma $ be a group. Let $\mathcal{F}$ be the category defined as follows

  1. an object is a triple $(A, M, \rho )$ consisting of an object $A$ of $\mathcal{C}_\Lambda $, a finite projective $A$-module $M$, and a homomorphism $\rho : \Gamma \to \text{GL}_ A(M)$, and

  2. a morphism $(f, g) : (B, N, \tau ) \to (A, M, \rho )$ consists of a morphism $f : B \to A$ in $\mathcal{C}_\Lambda $ together with a map $g : N \to M$ which is $f$-linear and $\Gamma $-equivariant and induces an isomorpism $N \otimes _{B, f} A \cong M$.

The functor $p : \mathcal{F} \to \mathcal{C}_\Lambda $ sends $(A, M, \rho )$ to $A$ and $(f, g)$ to $f$. It is clear that $p$ is cofibred in groupoids. Given a finite dimensional $k$-vector space $V$ and a representation $\rho _0 : \Gamma \to \text{GL}_ k(V)$, let $x_0 = (k, V, \rho _0)$ be the corresponding object of $\mathcal{F}(k)$. We set

\[ \mathcal{D}\! \mathit{ef}_{V, \rho _0} = \mathcal{F}_{x_0} \]

Since every finite projective module over a local ring is finite free (Algebra, Lemma 10.78.2) we see that

\[ \begin{matrix} \text{isomorphism classes} \\ \text{of objects of }\mathcal{F}(A) \end{matrix} = \coprod \nolimits _{n \geq 0}\quad \begin{matrix} \text{GL}_ n(A)\text{-conjugacy classes of} \\ \text{homomorphisms }\rho : \Gamma \to \text{GL}_ n(A) \end{matrix} \]

This is already more interesting than the discussion in Section 93.4.

Lemma 93.5.2. Example 93.5.1 satisfies the Rim-Schlessinger condition (RS). In particular, $\mathcal{D}\! \mathit{ef}_{V, \rho _0}$ is a deformation category for any finite dimensional representation $\rho _0 : \Gamma \to \text{GL}_ k(V)$.

Proof. Let $A_1 \to A$ and $A_2 \to A$ be morphisms of $\mathcal{C}_\Lambda $. Assume $A_2 \to A$ is surjective. According to Formal Deformation Theory, Lemma 90.16.4 it suffices to show that the functor $\mathcal{F}(A_1 \times _ A A_2) \to \mathcal{F}(A_1) \times _{\mathcal{F}(A)} \mathcal{F}(A_2)$ is an equivalence of categories.

Consider an object

\[ ((A_1, M_1, \rho _1), (A_2, M_2, \rho _2), (\text{id}_ A, \varphi )) \]

of the category $\mathcal{F}(A_1) \times _{\mathcal{F}(A)} \mathcal{F}(A_2)$. Then, as seen in the proof of Lemma 93.4.2, we can consider the finite projective $A_1 \times _ A A_2$-module $M_1 \times _\varphi M_2$. Since $\varphi $ is compatible with the given actions we obtain

\[ \rho _1 \times \rho _2 : \Gamma \longrightarrow \text{GL}_{A_1 \times _ A A_2}(M_1 \times _\varphi M_2) \]

Then $(M_1 \times _\varphi M_2, \rho _1 \times \rho _2)$ is an object of $\mathcal{F}(A_1 \times _ A A_2)$. This construction determines a quasi-inverse to our functor. $\square$

Lemma 93.5.3. In Example 93.5.1 let $\rho _0 : \Gamma \to \text{GL}_ k(V)$ be a finite dimensional representation. Then

\[ T\mathcal{D}\! \mathit{ef}_{V, \rho _0} = \mathop{\mathrm{Ext}}\nolimits ^1_{k[\Gamma ]}(V, V) = H^1(\Gamma , \text{End}_ k(V)) \quad \text{and}\quad \text{Inf}(\mathcal{D}\! \mathit{ef}_{V, \rho _0}) = H^0(\Gamma , \text{End}_ k(V)) \]

Thus $\text{Inf}(\mathcal{D}\! \mathit{ef}_{V, \rho _0})$ is always finite dimensional and $T\mathcal{D}\! \mathit{ef}_{V, \rho _0}$ is finite dimensional if $\Gamma $ is finitely generated.

Proof. We first deal with the infinitesimal automorphisms. Let $M = V \otimes _ k k[\epsilon ]$ with induced action $\rho _0' : \Gamma \to \text{GL}_ n(M)$. Then an infinitesimal automorphism, i.e., an element of $\text{Inf}(\mathcal{D}\! \mathit{ef}_{V, \rho _0})$, is given by an automorphism $\gamma = \text{id} + \epsilon \psi : M \to M$ as in the proof of Lemma 93.4.3, where moreover $\psi $ has to commute with the action of $\Gamma $ (given by $\rho _0$). Thus we see that

\[ \text{Inf}(\mathcal{D}\! \mathit{ef}_{V, \rho _0}) = H^0(\Gamma , \text{End}_ k(V)) \]

as predicted in the lemma.

Next, let $(k[\epsilon ], M, \rho )$ be an object of $\mathcal{F}$ over $k[\epsilon ]$ and let $\alpha : M \to V$ be a $\Gamma $-equivariant map inducing an isomorphism $M/\epsilon M \to V$. Since $M$ is free as a $k[\epsilon ]$-module we obtain an extension of $\Gamma $-modules

\[ 0 \to V \to M \xrightarrow {\alpha } V \to 0 \]

We omit the detailed construction of the map on the left. Conversely, if we have an extension of $\Gamma $-modules as above, then we can use this to make a $k[\epsilon ]$-module structure on $M$ and get an object of $\mathcal{F}(k[\epsilon ])$ together with a map $\alpha $ as above. It follows that

\[ T\mathcal{D}\! \mathit{ef}_{V, \rho _0} = \mathop{\mathrm{Ext}}\nolimits ^1_{k[\Gamma ]}(V, V) \]

as predicted in the lemma. This is equal to $H^1(\Gamma , \text{End}_ k(V))$ by √Čtale Cohomology, Lemma 59.57.4.

The statement on dimensions follows from √Čtale Cohomology, Lemma 59.57.5. $\square$

In Example 93.5.1 if $\Gamma $ is finitely generated and $(V, \rho _0)$ is a finite dimensional representation of $\Gamma $ over $k$, then $\mathcal{D}\! \mathit{ef}_{V, \rho _0}$ 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 93.5.2 and 93.5.3 and the general discussion in Section 93.3.

Lemma 93.5.4. In Example 93.5.1 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

\[ F : \mathcal{C}_\Lambda \longrightarrow \textit{Sets},\quad A \longmapsto \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}\! \mathit{ef}_{V, \rho _0}(A))/\cong \]

of isomorphism classes of objects has a hull. If $H^0(\Gamma , \text{End}_ k(V)) = k$, then $F$ is prorepresentable.

Proof. The existence of a hull follows from Lemmas 93.5.2 and 93.5.3 and Formal Deformation Theory, Lemma 90.16.6 and Remark 90.15.7.

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 90.18.2. In other words, we have to show $F$ satisfies (RS). For this we can use the criterion of Formal Deformation Theory, Lemma 90.16.7. 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 $\mathop{\mathrm{Hom}}\nolimits _{A[\Gamma ]}(M, I_ tM/I_{t + 1}M)$ 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. $\square$

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