Proof. By writing down a generic $d\times d$ -matrix for each $R^{\mathrm {ps}}$ -generator of E and quotienting out by the relations the generators satisfy in E, one obtains a commutative $R^{\mathrm {ps}}$ -algebra C and a homomorphism of $R^{\mathrm {ps}}$ -algebras $j: E\rightarrow M_d(C)$ . More formally, C is a quotient of $R^{\mathrm {ps}}\otimes _{\mathbb Z} \operatorname {\mathrm {Sym}}(W)$ , where W is a direct sum of n copies of $\operatorname {\mathrm {End}}(\mathrm {Std})^*$ , where $\mathrm {Std}$ is the standard representation of $\operatorname {\mathrm {GL}}_d$ over $\mathbb Z$ , n is the size of a generating set of E as an $R^{\mathrm {ps}}$ -module and $\operatorname {\mathrm {Sym}}(W)$ is the symmetric algebra over $\mathbb Z$ . If we were to only require the maps to be $R^{\mathrm {ps}}$ -algebras homomorphisms (i.e., if we did not impose the Cayley–Hamilton condition), then the map $j: E\rightarrow M_d(C)$ would satisfy the required universal property. To ensure that the Cayley–Hamilton condition is satisfied, we have to consider the quotient of C constructed as follows. Let $\Lambda _i: E\rightarrow R^{\mathrm {ps}}$ , $0\le i\le d$ be the coefficients of the characteristic polynomial of $D^u$ ; these are homogeneous polynomial laws satisfying $D^{u}(t-a)= \sum _{i=0}^n (-1)^i \Lambda _i(a) t^{d-i}$ in $R^{\mathrm {ps}}[t]$ as explained in [Reference Chenevier18, Section 1.10]. For each $a\in E$ , let $c_i(j(a))$ be the i-th coefficient of the characteristic polynomial of the matrix $j(a)\in M_d(C)$ . Let I be the ideal of C generated by $ \Lambda _i(a)-c_i(j(a))$ for all $a\in E$ and $0\le i\le d$ and let $A^{\mathrm {gen}}:= C/I$ . Since [Reference Chenevier18, Corollary 1.14] and [Reference Wang-Erickson49, 1.1.9.15] imply that the coefficients of the characteristic polynomial determine pseudo-characters uniquely, the composition $E\rightarrow M_d(C)\rightarrow M_d(A^{\mathrm {gen}})$ is a map of Cayley–Hamilton algebras, and the universal property of $j: E\rightarrow M_d(C)$ implies the universal property for $j:E\rightarrow M_d(A^{\mathrm {gen}})$ . Since E is finitely generated as $R^{\mathrm {ps}}$ -module, C and hence $A^{\mathrm {gen}}$ are of finite type over $R^{\mathrm {ps}}$ .

Proof. Since $R^{\mathrm {ps}}$ is a compact ring [Reference Arnautov and Ursul2, Corollary 1.10] implies that for every finitely generated $R^{\mathrm {ps}}$ -module M there is a unique Hausdorff topology on M making M into a topological $R^{\mathrm {ps}}$ -module.

We equip with its projective limit topology, E with the quotient topology, and its group of units $E^{\times }$ with the subspace topology via the embedding $E^{\times }\hookrightarrow E\times E$ , $x\mapsto (x, x^{-1})$ . Since the map is continuous, the map $G_F\rightarrow E^{\times }$ is also continuous. Moreover, since $\mathrm {CH}(D^u)$ is a closed ideal, the topology on E is Hausdorff.

Since E is a finitely generated $R^{\mathrm {ps}}$ -module, its topology coincides with ${\mathfrak m}_{R^{\mathrm {ps}}}$ -adic topology (this is also proved in [Reference Wang-Erickson50, Proposition 3.6]). Let $M:=f(E)\subset M_d(B)$ , let $\tau _1$ be the subspace topology on M and let $\tau _2$ be be the unique Hausdorff topology on M such that the action of $R^{\mathrm {ps}}$ is continuous. We claim that the identity map $(M, \tau _2)\rightarrow (M, \tau _1)$ is continuous. We will now prove the claim. Since B is a topological $R^{\mathrm {ps}}$ -algebra, the action of $R^{\mathrm {ps}}$ on $M_d(B)$ , and hence on M, is continuous with respect to $\tau _1$ . Since M is a finitely generated $R^{\mathrm {ps}}$ -module, we may pick a continuous surjection $\varphi : (R^{\mathrm {ps}})^n \twoheadrightarrow M$ for some $n\ge 1$ . Since $R^{\mathrm {ps}}$ is Noetherian, the kernel of $\varphi $ is finitely generated and hence a closed submodule of $(R^{\mathrm {ps}})^n$ . Thus, the quotient topology on M induced via $\varphi $ is Hausdorff and therefore must coincide with $\tau _2$ , which proves the claim.

The same argument shows that $\tau _2$ coincides with the quotient topology via $E\twoheadrightarrow M$ , and the claim implies that the map $f: E\rightarrow M_d(B)$ is continuous and hence induces a continuous group homomorphism $E^{\times }\rightarrow M_d(B)^{\times }=\operatorname {\mathrm {GL}}_d(B)$ .

Proof. Since $R^{\mathrm {ps}}[G_F]$ is dense in , its image will be dense in E for the topologies introduced in the proof of Lemma 3.2. The image is also closed, as it is an $R^{\mathrm {ps}}$ -submodule of E. Hence, the map is surjective.

Proof. Since $D^u\otimes _{R^{\mathrm {ps}}} k = \overline {D}$ , the first part follows immediately from the definition of $A^{\mathrm {gen}}$ . The second part follows from [Reference Chenevier18, Theorem 2.12]. Note that since we have assumed that all irreducible subquotients of $\overline {\rho }$ are absolutely irreducible, it is enough to prove that $\rho _x$ and $\overline {\rho }$ have the same semi-simplification after extending scalars to the algebraic closure of k.

Proof. It follows from Lemma 3.3 that the natural map $\kappa [G_F]\rightarrow E_y$ is surjective. This implies the assertion about $\operatorname {\mathrm {Hom}}$ spaces and gives an inclusion $\operatorname {\mathrm {Ext}}^1_{E_y}(\rho _i, \rho _{j})\subset \operatorname {\mathrm {Ext}}^1_{G_F}(\rho _i, \rho _{j})$ . To prove the reverse inclusion, consider an extension $0\rightarrow \rho _{j} \rightarrow W \rightarrow \rho _i \rightarrow 0$ of $G_F$ -representations and let $V= W\oplus \bigoplus _{l\neq i, {j}} \rho _l$ . As explained above, the $G_F$ -action on V will factor through the action of $E_y$ . Hence, W is a representation of $E_y$ , which implies that $\operatorname {\mathrm {Ext}}^1_{E_y}(\rho _i, \rho _{j})= \operatorname {\mathrm {Ext}}^1_{G_F}(\rho _i, \rho _{j})$ .

Proof. It follows from [Reference Seshadri47, Theorem 3] that $X^{\mathrm {gen}}_{y'}$ (and hence $X^{\mathrm {gen}}_y$ , by the remark in the paragraph above) contains a unique closed G-orbit. Thus, it is enough to show that the closure of every G-orbit will contain a semi-simple representation. If $x\in X^{\mathrm {gen}}_y(\kappa )$ , then after conjugation we may assume that x corresponds to a representation $\rho : G_F \rightarrow \operatorname {\mathrm {GL}}_d(\kappa )$ such that the image of $\rho $ is block-upper-triangular, and the blocks on the diagonal are given by $\mathrm {diag}(\rho _{\sigma (1)}(g), \ldots , \rho _{\sigma (r)}(g))$ for some permutation $\sigma \in S_r$ . By extending scalars to $\kappa [T]$ , conjugating $\rho $ by $\mathrm {diag}(T^{r-1} \operatorname {\mathrm {id}}_{d_{\sigma (1)}}, T^{r-2} \operatorname {\mathrm {id}}_{d_{\sigma (2)}}, \dots , \operatorname {\mathrm {id}}_{d_{\sigma (r)}})$ and specializing at $T=0$ , we see that the closure of the G-orbit will contain a semi-simple representation. The action of G on $X^{\mathrm {gen}}_y$ leaves the connected components invariant by Lemma 2.1. Hence, every connected component of $X^{\mathrm {gen}}_y$ will contain the closed point corresponding to the representation $g \mapsto \mathrm {diag}(\rho _{1}(g), \ldots , \rho _{r}(g))$ . Thus $X^{\mathrm {gen}}_y$ is connected.

The stabilizer of a semi-simple representation with distinct irreducible factors in $\operatorname {\mathrm {GL}}_d$ is isomorphic to $\mathbb {G}_m^r$ : a copy of $\mathbb G_m$ is embedded as scalar matrices inside of each block. Hence, the dimension of the closed G-orbit is given by $\dim \operatorname {\mathrm {GL}}_d - \dim \mathbb {G}_m^r= d^2-r$ .

Proof. The universal map $j: E\rightarrow M_d(A^{\mathrm {gen}})$ induces a map

$$\begin{align*}j_y: E_y \rightarrow M_d(A^{\mathrm{gen}} \otimes_{R^{\mathrm{ps}},y} \kappa). \end{align*}$$

Let $I_{\rho , \sigma }$ be the ideal of $A^{\mathrm {gen}} \otimes _{R^{\mathrm {ps}},y} \kappa $ generated by the matrix entries of $j_y(a)$ for all $a\in E_y$ , which lie below the diagonal blocks of P, and by all the elements on the block diagonal of the matrices $(j_y(a)- \rho _{\sigma }(a))$ for all $a\in E_y$ . Let

$$\begin{align*}X^{\mathrm{gen}}_{y,\sigma}:= \operatorname{\mathrm{Spec}} ((A^{\mathrm{gen}} \otimes_{R^{\mathrm{ps}},y} \kappa)/ I_{\rho, \sigma}). \end{align*}$$

Then $X^{\mathrm {gen}}_{y,\sigma }$ is a closed subscheme of $X^{\mathrm {gen}}_y$ , and its defining ideal $I_{\rho ,\sigma }$ was constructed precisely so that a B-point of $X^{\mathrm {gen}}_y$ factors through $X^{\mathrm {gen}}_{y,\sigma }$ if and only if it lands in ${\mathfrak p} \otimes _{\kappa } B$ and matches the $\rho _i$ on the diagonals for $1 \le i \le r$ .

Proof. This is the same proof as in Lemma 3.7 and uses the same diagonal matrix trick to kill off the unipotent part.

Proof. Using Lemma 3.9 and the decomposition ${\mathfrak p} = \mathfrak l \oplus \mathfrak n$ , we may identify $T_x(X^{\mathrm {gen}}_{y,\sigma })$ with the space of $\kappa $ -algebra homomorphisms $\varphi : E_y \rightarrow M_d(\kappa [\varepsilon ])$ , which can be written as $\varphi = \rho _{\sigma } + \varepsilon \beta $ , where $\beta $ is a $\kappa $ -linear map $\beta : E_y \rightarrow \mathfrak n$ . If $\beta : E_y\rightarrow \mathfrak n$ is any $\kappa $ -linear map, then $\varphi := \rho _{\sigma } + \varepsilon \beta $ is a homomorphism of $\kappa $ -algebras if and only if

(10) $$ \begin{align} \beta(aa') = \rho_{\sigma}(a) \beta(a')+ \beta(a) \rho_{\sigma}(a'), \quad \forall a, a'\in E_y. \end{align} $$

For $1\le i\le r$ , we let $\mathbf 1_i\in M_d(\kappa )$ be the block diagonal matrix with the identity matrix on the i-th block and zeros everywhere else. Since $\rho _{\sigma }(g)$ commutes with $\mathbf 1_i$ for all i, we have an isomorphism

$$\begin{align*}T_x(X^{\mathrm{gen}}_{y,\sigma})\cong \bigoplus_{1\le i< j\le r} V_{ij}, \end{align*}$$

where $V_{ij}$ is the space of functions $\beta : E_y \rightarrow \mathbf 1_i \mathfrak n \mathbf 1_j$ satisfying (10). We may identify $\mathbf 1_i \mathfrak n \mathbf 1_j$ with $\operatorname {\mathrm {Hom}}_{\kappa }(\rho _{\sigma (j)}, \rho _{\sigma (i)})$ . Then $V_{ij}$ is precisely the space of $1$ -cocycles for the Hochschild cohomology of $E_y$ with values in $\operatorname {\mathrm {Hom}}_{\kappa }(\rho _{\sigma (j)}, \rho _{\sigma (i)})$ . Thus,

(11) $$ \begin{align} \begin{aligned} \dim_{\kappa} V_{ij} &= \dim_{\kappa} HH^1(E_y, \operatorname{\mathrm{Hom}}_{\kappa}(\rho_{\sigma(j)}, \rho_{\sigma(i)})) + \dim_{\kappa} \operatorname{\mathrm{Hom}}_{\kappa}(\rho_{\sigma(j)}, \rho_{\sigma(i)})\\&- \dim_{\kappa} HH^0(E_y, \operatorname{\mathrm{Hom}}_{\kappa}(\rho_{\sigma(j)}, \rho_{\sigma(i)}))\\&= \dim_{\kappa}\operatorname{\mathrm{Ext}}^1_{E_y}(\rho_{\sigma(j)}, \rho_{\sigma(i)}) + d_i d_j - \dim_{\kappa} \operatorname{\mathrm{Hom}}_{E_y}(\rho_{\sigma(j)}, \rho_{\sigma(i)})\\ &= d_i d_j + [F:\mathbb{Q}_p] d_i d_j + \dim_{\kappa} \operatorname{\mathrm{Ext}}^2_{G_F}(\rho_{\sigma(j)}, \rho_{\sigma(i)}), \end{aligned} \end{align} $$

where the first equality follows from [Reference Cartan and Eilenberg15, Proposition IX.4.4.1], the second from [Reference Cartan and Eilenberg15, Corollary IX.4.4.4] and the third from Lemma 3.6 together with the local Euler–Poincaré characteristic formula in this contextFootnote ⁴ (see [Reference Böckle and Juschka9, Theorem 3.4.1 (c)]). Thus,

$$\begin{align*}\dim_{\kappa} T_x(X^{\mathrm{gen}}_{y,\sigma})= \dim \mathfrak n + (\dim \mathfrak n)[F:\mathbb{Q}_p] +\sum_{1\le i<j\le r} \dim_{\kappa} \operatorname{\mathrm{Ext}}^2_{G_F}(\rho_{\sigma(j)}, \rho_{\sigma(i)}). \end{align*}$$

It follows from the local duality (see [Reference Böckle and Juschka9, Theorem 3.4.1 (b)]) that

$$\begin{align*}\dim_{\kappa}\operatorname{\mathrm{Ext}}^2_{G_F}(\rho_{\sigma(j)}, \rho_{\sigma(i)}) = \dim_{\kappa}\operatorname{\mathrm{Hom}}_{G_F}( \rho_{\sigma(i)}, \rho_{\sigma(j)}(1)). \end{align*}$$

Thus, if this term is non-zero, then it is equal to $1$ and $\rho _{\sigma (i)}$ and $\rho _{\sigma (j)}$ belong to the same equivalence class.

Proof. This follows from Lemma 3.10 and Lemma 2.2 applied with $G = Z_L$ and $X = X^{\mathrm {gen}}_{y,\sigma }$ , noting that is a singleton.

Proof. Passing to reduced subschemes does not affect Krull dimension, so we may assume that X and Y are both reduced.

First, assume X and Y are irreducible. Pick dense open affines $U\subset Y$ , $V\subset X$ such that $f(V)\subset U$ . Since f is dominant, [48, Tag 0CC1] implies that ${A:=\mathcal {O}_Y(U)\hookrightarrow B:= \mathcal {O}_X(V)}$ is injective. Since A is an integral domain, Noether normalization [48, Tag 07NA] implies that the map factors as

$$\begin{align*}A\hookrightarrow A[x_1,\ldots, x_m]\hookrightarrow B' \hookrightarrow B, \end{align*}$$

with $B'$ finite over $A[x_1, \ldots , x_m]$ and $B^{\prime }_g \cong B_g$ for some non-zero $g \in A$ . Then [48, Tag 0DRT] and [Reference Matsumura35, 13.C, Theorem 20] imply that

$$\begin{align*}\dim X = \dim B = \dim B_g = \dim B^{\prime}_g = \dim B' = \dim A + m = \dim Y + m, \end{align*}$$

so $\dim Y \leq \dim X$ .

For the general case, we argue as in the proof of [48, Tag 01RM]. Write $X = \bigcup _j Z_j$ as the union of its irreducible components. Because f is dominant, we have $Y=\bigcup _j \overline {f(Z_j)}$ . Clearly, the $\overline {f(Z_j)}$ have to be irreducible, and so the irreducible components of Y have to be among them. The $Z_j$ and $\overline {f(Z_j)}$ are again Noetherian, Jacobson and universally catenary, and hence by the case already treated, we have

$$\begin{align*}\dim Y = \max_j \dim \overline{f(Z_j)} \leq \max_j \dim Z_{j} = \dim X. \\[-42pt] \end{align*}$$

Proof. We want to apply Lemma 3.14 to

(12) $$ \begin{align} \coprod_{\sigma\in S_r} G\times^{Z_{L_{\sigma}} N_{\sigma}} X^{\mathrm{gen}}_{y,\sigma} \rightarrow X_y^{\mathrm{gen}}, \end{align} $$

where the actions of $Z_{L_{\sigma }} N_{\sigma }$ on $X^{\mathrm {gen}}_{y,\sigma }$ and of G on $X^{\mathrm {gen}}_{y}$ are given by conjugation.

If $x\in X_y^{\mathrm {gen}}(\kappa )$ and $\varphi : E_y \rightarrow M_d(\kappa )$ is the corresponding $\kappa $ -algebra homomorphism, then there will exist $\sigma \in S_r$ such that $\kappa ^d$ will admit a filtration by subspaces $0=V_0\subset V_1\subset \ldots \subset V_r=V$ , which is invariant under the action of $E_y$ via $\varphi $ , satisfying $V_i/ V_{i-1}\cong \rho _{\sigma (i)}$ for $1\le i\le r$ . Thus, there is $g\in G(\kappa )$ such that $g \varphi g^{-1}$ will lie in $X^{\mathrm {gen}}_{y,\sigma }(\kappa )$ , and hence, (12) induces a surjection on $\kappa $ -points. But (12) is also a map of finite type $\kappa $ -schemes and therefore is a dominant map of Noetherian Jacobson universally catenary schemes, so we can apply Lemma 3.14.

The fibre bundles $G\times ^{Z_{L_{\sigma }} N_{\sigma }} X^{\mathrm {gen}}_{y,\sigma }$ have dimension equal to

$$\begin{align*}\dim G + \dim X^{\mathrm{gen}}_{y,\sigma} -\dim (Z_{L_{\sigma}} N_{\sigma}) =\dim \mathfrak{g} -r +\dim X^{\mathrm{gen}}_{y,\sigma} -\dim \mathfrak n. \end{align*}$$

The bound in Corollary 3.13 gives the required assertion.

Proof. If $r=1$ then $E_y\cong M_d(\kappa )$ and thus has a unique irreducible representation $\rho $ (up to isomorphism). Thus, all the points in $X^{\mathrm {gen}}_y(\kappa )$ lie in the same G-orbit. Fix such a point x. Since the G-stabiliser of x is equal to $Z_G$ , we obtain $\dim X^{\mathrm {gen}}_y= \dim G-\dim Z_G= \dim \mathfrak {g} -1$ .

Since $E_y$ is semi-simple, we have $\operatorname {\mathrm {Ext}}^1_{E_y}(\rho , \rho )=0$ , and thus an argument as in the proof of Proposition 3.11 gives us

$$ \begin{align*}\dim_{\kappa} T_x(X^{\mathrm{gen}}_y)= \dim_{\kappa} \operatorname{\mathrm{End}}_{\kappa}(\rho)-\dim_{\kappa} \operatorname{\mathrm{End}}_{E_y}(\rho)=\dim X_y^{\mathrm{gen}}.\end{align*} $$

Thus, x is a smooth point of $X^{\mathrm {gen}}_y$ , and since G acts transitively on $X^{\mathrm {gen}}_y(\kappa )$ , all the points in $X^{\mathrm {gen}}_y(\kappa )$ are smooth. Since $X_y^{\mathrm {gen}}$ is of finite type over $\kappa $ , we deduce that $X_y^{\mathrm {gen}}$ is smooth.

Proof. It follows from Cohen’s structure theorem that if $\mathrm {char} (R/{\mathfrak p})=0$ , then $\mathcal {O}\subset R/{\mathfrak p}$ and $R/{\mathfrak p}$ is a finitely generated $\mathcal {O}$ -module. Thus, $\kappa ({\mathfrak p})$ is a finite extension of L, and $R/{\mathfrak p}$ is contained in the integral closure of $\mathcal {O}$ in $\kappa ({\mathfrak p})$ , which is equal to $\mathcal {O}_{\kappa ({\mathfrak p})}$ . If $\mathrm {char}(R/{\mathfrak p})=p$ , then $R/{\mathfrak p}$ is finite over a subring isomorphic to , and the same argument carries over. Moreover, $\mathcal {O}_{\kappa ({\mathfrak p})}$ is a finitely generated $R/{\mathfrak p}$ -module, and this implies that the topologies coincide.

Proof. We summarize the situation in the following diagram.

We will first prove parts (1), (2) and (3). If $R=S$ and if $U=U_{\max }$ , then (1) follows from [48, Tag 02IM] and both (2) and (3) hold trivially. If $R=S$ and if U is arbitrary, then $U=V$ and (1), (2) follow from the previous case together with [48, Tag 005W] and (3) holds trivially. The case of general $\varphi $ now follows from [48, Tag 00GB] together with [48, Tag 01P4] because the map $V\to U$ induced from $\varphi $ is of finite type.

Part (4) follows from (2) applied with $S=R$ , using that ${\mathfrak m}_R$ is the unique maximal ideal of R, so that the set of closed points of $U_{\max }$ is equal to $P_1R$ .

For (5), note first that since V is open in $\operatorname {\mathrm {Spec}} S$ , the set of generic points of V is a subset of the set of generic points of $\operatorname {\mathrm {Spec}} S$ . Thus, Z is union of irreducible components of $\operatorname {\mathrm {Spec}} S$ . Let $Z'=\operatorname {\mathrm {Spec}} S'$ be an irreducible component of Z with the induced reduced subscheme structure so that $S'$ is a domain, let $V'= Z'\cap V$ and let $R'$ be the image of R in $S'$ . The rings $R'$ and $S'$ are excellent and hence universally catenary by [48, Tag 07QW]. If $\mathfrak {q}\in \operatorname {\mathrm {Spec}} S'$ and ${\mathfrak p}=\mathfrak {q} \cap R'$ , then

(13) $$ \begin{align} \begin{aligned} \dim S^{\prime}_{\mathfrak{q}}&= \dim R^{\prime}_{{\mathfrak p}} + \mathrm{trdeg}_{R'} S' - \mathrm{trdeg}_{\kappa({\mathfrak p})} \kappa(\mathfrak{q})\\ &= \dim R'+ \mathrm{trdeg}_{R'} S' -\dim R'/{\mathfrak p} -\mathrm{trdeg}_{\kappa({\mathfrak p})} \kappa(\mathfrak{q}), \end{aligned} \end{align} $$

where $\mathrm {trdeg}$ stands for transcendence degree, the first equality is [48, Tag 02IJ] and the second is [Reference Matsumura36, Theorem 31.4]. It follows from (13) that

(14) $$ \begin{align} \dim S^{\prime}_{\mathfrak{q}} \le \dim R' + \mathrm{trdeg}_{R'} S', \end{align} $$

and the equality in (14) holds if and only if $\mathfrak {q}$ maps to the maximal ideal of $R'$ and $\mathfrak {q}$ is a maximal ideal of $S'$ . Since $Z'\cap Y$ is non-empty by assumption, such $\mathfrak {q}$ exists and so

$$ \begin{align*} \dim S'= \dim R' + \mathrm{trdeg}_{R'} S'. \end{align*} $$

Let $\mathfrak {q}$ be a closed point of $V'$ and let ${\mathfrak p}=\mathfrak {q} \cap R'$ . Since $V'$ is open in $Z'$ , we have $\mathcal {O}_{V',\mathfrak {q}}= S^{\prime }_{\mathfrak {q}^{\prime }}$ . It follows from (3) that $\mathrm {trdeg}_{\kappa ({\mathfrak p})}\kappa (\mathfrak {q})=0$ and $\dim R/{\mathfrak p}=1$ . Thus, (13) gives us

$$ \begin{align*}\dim \mathcal{O}_{V',\mathfrak{q}}= \dim R' + \mathrm{trdeg}_{R'} S'-1.\end{align*} $$

Since this holds for all closed points of $V'$ , we deduce that

$$ \begin{align*}\dim V'= \dim R' + \mathrm{trdeg}_{R'} S'-1.\end{align*} $$

This implies part (5).

Let x be a closed point of V and let y be its image in U. Then y is also a closed point of U. We have

$$ \begin{align*}\dim \mathcal{O}_{V, x} \le \dim \mathcal{O}_{U, y} + \dim (\mathcal{O}_{V, x}\otimes_{\mathcal{O}_{U, y}} \kappa(y))\le \dim U + \dim \varphi^{-1}(\{y\}),\end{align*} $$

where the first inequality is given by [Reference Matsumura36, Theorem 15.1 (i)]. Since

$$ \begin{align*}\dim V = \max_x \dim \mathcal{O}_{V, x},\end{align*} $$

where the maximum is taken over all closed points x of V, we get (6).

Proof. By Lemma 2.1, each irreducible component Z of W is $\operatorname {\mathrm {GL}}_d$ -invariant. The image of Z in $X^{\mathrm {ps}}$ is closed by Corollary 2 (ii) to [Reference Seshadri47, Proposition 9], and is nonempty and so must contain ${\mathfrak m}_{R^{\mathrm {ps}}}$ because $X^{\mathrm {ps}}$ has a unique closed point. Therefore, $Y \cap Z$ is nonempty.

The claims about $\dim \mathcal {O}_{Z,x}$ follows from the proof of part (5) in Lemma 3.18.

Proof. We may assume that $W[1/p]$ is nonempty, and using Lemma 2.1, we may further assume that W is irreducible. Lemma 3.21 implies that there is a closed point $x\in W$ , which maps to the closed point in $X^{\mathrm {ps}}$ . Lemma 3.18 (5) implies that $\dim W[1/p]=\dim W -1$ .

Since W is irreducible and $W[1/p]\neq \emptyset $ , the local ring $\mathcal {O}_{W,x}$ is a domain, and multiplication by $\varpi $ is injective. Since $\mathrm {char}(\kappa (x))=p$ , $\varpi $ cannot be a unit in $\mathcal {O}_{W,x}$ . Thus, $\dim \mathcal {O}_{\overline {W}, x}= \dim \mathcal {O}_{W,x}-1$ . It follows from Lemma 3.21 that $\dim \overline {W}= \dim W-1$ .

Proof. By topological Nakayama’s lemma, it is enough to show that the fibre ring $C:=k\otimes _{R^{\mathrm {ps}}} R^{\mathrm {ps}}_{\underline {\Sigma }}$ is a finite dimensional k-vector space. Let $\mathcal F$ be the closed subfunctor of $\mathcal F_{\underline {\Sigma }}$ defined by C. If $(A, {\mathfrak m}_A)$ is a local Artinian k-algebra, then $\mathcal F(A)$ is in bijection with the set of r-tuples $(D_{1}, \ldots , D_{r})$ , each $D_{i}$ lifting $\overline {D}^{\prime }_i$ to A such that $\prod _{i=1}^r D_i= (\prod _{i=1}^r \overline {D}^{\prime }_i)\otimes _k A$ .

Since C is a complete local Noetherian ring, it is enough to show that its Krull dimension is $0$ . If this is not the case, then there is ${\mathfrak p}\in \operatorname {\mathrm {Spec}} C$ such that $\dim C/{\mathfrak p}=1$ . Let $(D_{1, y}, \ldots , D_{r,y})$ be the specialization of the universal object of $\mathcal F_{\underline {\Sigma }}$ along $y: R^{\mathrm {ps}}_{\underline {\Sigma }}\rightarrow \kappa ({\mathfrak p})$ . It follows from [Reference Chenevier18, Corollary 1.14] that the coefficients of the polynomials $D_{i,y}(t - a)$ , for all $a\in E$ and $1\le i\le r$ will generate a dense subring of $R^{\mathrm {ps}}_{\underline {\Sigma }}/{\mathfrak p}$ . Since $R^{\mathrm {ps}}_{\underline {\Sigma }}/{\mathfrak p}$ is a complete local k-algebra of dimension $1$ , there will exist $a\in E$ and index i such that the coefficients of $D_{i,y}(t - a)$ will generate a transcendental extension of k inside $\kappa ({\mathfrak p})$ . Since ${\mathfrak p}\in \operatorname {\mathrm {Spec}} C$ , we have

$$ \begin{align*}\prod_{i=1}^r D_{i,y}(t-a)= \prod_{i=1}^r \overline{D}^{\prime}_i(t-a).\end{align*} $$

Thus, all the roots of $D_{i,y}(t-a)$ in the algebraic closure of $\kappa ({\mathfrak p})$ are algebraic over k. Since $D_{i,y}(t-a)$ is a monic polynomial, we conclude that all the coefficients are also algebraic over k, giving a contradiction.

Proof. We may write $D_y=D_1+ \ldots + D_r$ with $D_i$ lifting $\overline {D}_i^{\prime }$ . We note that all the $D_i$ are absolutely irreducible since otherwise, $y \in X_{\mathcal {P}'}^{\mathrm {ps}}$ for some $\mathcal {P}'> \mathcal {P}$ . Let k and $n_i$ be the numbers defined in Section 3.2. Proposition 3.15 implies that

$$ \begin{align*} \dim X^{\mathrm{gen}}_y\le d^2-r+ n_{\mathcal{P}} [F:\mathbb{Q}_p] + \sum_{i=1}^{k} \begin{pmatrix} n_i\\ 2\end{pmatrix}. \end{align*} $$

If $D_i= D_j(m)$ for some $m \in \mathbb Z$ , then also $\overline {D}_i^{\prime }= \overline {D}_j^{\prime }(m)$ . This implies that

$$ \begin{align*}\sum_{i=1}^k \begin{pmatrix} n_i \\ 2\end{pmatrix} \le \sum_{i=1}^{k'} \begin{pmatrix} n_i^{\prime} \\ 2\end{pmatrix},\end{align*} $$

which implies the first assertion. We note that if $a_i, \ldots , a_s$ are positive integers, then

$$ \begin{align*}\sum _{i=1}^s \begin {pmatrix} a_i \\2\end {pmatrix} \le \begin {pmatrix} \sum _{i=1}^s a_i \\2\end {pmatrix}.\end{align*} $$

If $y\not \in U_{\mathcal {P}}^{ij}$ for any $i\neq j$ , then $D_i\neq D_j(1)$ for any $i\neq j$ , and the $\operatorname {\mathrm {Hom}}$ terms in (9) vanish. The assertion follows from Proposition 3.15 using this improved bound.

Proof. It follows from Lemma 3.18 (5) that the closure of $U_{\mathcal {P}}^{ij}$ has dimension $\dim U_{\mathcal {P}}^{ij} +1$ . Thus,

$$ \begin{align*}\dim U_{\mathcal{P}}^{ij} +1 \le \dim X^{\mathrm{ps}, ij}_{\mathcal{P}}\le r+ l_{\mathcal{P}}[F:\mathbb{Q}_p] - (1+[F:\mathbb{Q}_p]),\end{align*} $$

where the last inequality is (19). Parts (5) and (6) of Lemma 3.18 together with Lemma 3.25 imply that

$$ \begin{align*}\dim Z^{ij}_{\mathcal{P}}\le (r+ l_{\mathcal{P}}[F:\mathbb{Q}_p] - (1+[F:\mathbb{Q}_p]))+ (d^2-r +n_{\mathcal{P}} [F:\mathbb{Q}_p] + \sum_{i=1}^{k'} \begin{pmatrix} n_i^{\prime}\\ 2\end{pmatrix}),\end{align*} $$

which imply the assertion.

Proof. Let $U^{\prime }_{\mathcal {P}}:= U_{\mathcal {P}} \setminus \bigcup _{i\neq j} U_{\mathcal {P}}^{ij}$ , let $V^{\prime }_{\mathcal {P}}$ be the preimage of $U^{\prime }_{\mathcal {P}}$ in $\overline {X}^{\mathrm {gen}}$ and let $Z^{\prime }_P$ denote the closure of $V^{\prime }_{\mathcal {P}}$ in $\overline {X}^{\mathrm {gen}}$ . If y is a closed point of $U^{\prime }_{\mathcal {P}}$ , then $\dim \overline {X}^{\mathrm {gen}}_y \le d^2-r+ n_{\mathcal {P}} [F:\mathbb {Q}_p]$ by Lemma 3.25. Thus, Lemma 3.18 implies that

(20) $$ \begin{align} \dim Z^{\prime}_{\mathcal{P}} \le \dim \overline{X}^{\mathrm{ps}}_{\mathcal{P}} + (d^2-r+ n_{\mathcal{P}} [F:\mathbb{Q}_p])= d^2+ p_{\mathcal{P}} [F:\mathbb{Q}_p]. \end{align} $$

Since $Z_{\mathcal {P}}= Z^{\prime }_{\mathcal {P}} \cup \bigcup _{i\neq j} Z^{ij}_{\mathcal {P}}$ , we have $\dim Z_{\mathcal {P}}= \max _{i\neq j}\{ \dim Z^{\prime }_{\mathcal {P}}, \dim Z^{ij}_{\mathcal {P}}\}$ , and the assertion follows from Proposition 3.26.

Proof. In this case, $r=1$ , so $Z_{\mathcal {P}}=Z^{\prime }_{\mathcal {P}}$ , and the assertion follows from (20).

Proof. Proposition 3.27 implies that

$$ \begin{align*}d^2+ d^2[F:\mathbb{Q}_p] - \dim Z_{\mathcal{P}}\ge n_{\mathcal{P}}[F:\mathbb{Q}_p] - \delta_{\mathcal{P}}.\end{align*} $$

If $d>2$ , then $n_{\mathcal {P}}\ge 2$ , and if $d=2$ , then $n_{\mathcal {P}} =1$ , which implies the assertion if $\delta _{\mathcal {P}}=0$ . Let us assume that $\delta _{\mathcal {P}}\neq 0$ . Then using (16), we may write

$$ \begin{align*}n_{\mathcal{P}}[F:\mathbb{Q}_p] -\delta_{\mathcal{P}}= \sum_{1\le i< j\le k'} c_i c_j n_i^{\prime} n_j^{\prime} [F:\mathbb{Q}_p] + \sum_{i=1}^{k'} ( c_i^2 [F:\mathbb{Q}_p] -1) \begin{pmatrix} n_i^{\prime}\\ 2\end{pmatrix} + 1 + [F:\mathbb{Q}_p],\end{align*} $$

which implies the assertion.

Proof. Proposition 3.15 implies that

$$ \begin{align*}\dim Y \le d^2 -m + n_{\mathcal{P}_{\max}} [F:\mathbb{Q}_p]+ \sum_{i=1}^{k'} \begin{pmatrix} n_i^{\prime}\\ 2\end{pmatrix}.\end{align*} $$

As already explained in the proof of Lemma 3.29, we have $\sum _{i=1}^{k'} \begin {pmatrix} n_i^{\prime }\\ 2\end {pmatrix}\le n_{\mathcal {P}_{\max }}$ . This implies the assertion.

Proof. Since $\overline {X}^{\mathrm {ps}}= \{{\mathfrak m}_{R^{\mathrm {ps}}}\}\cup \bigcup _{\mathcal {P}} U_{\mathcal {P}}$ , we have $\overline {X}^{\mathrm {gen}}= Y \cup \bigcup _{\mathcal {P}} Z_{\mathcal {P}}$ . Since these are closed in $\overline {X}^{\mathrm {gen}}$ , we have

$$ \begin{align*}\dim \overline{X}^{\mathrm{gen}} = \max_{\mathcal{P}} \{ \dim Y, \dim Z_{\mathcal{P}}\}\le d^2+d^2[F:\mathbb{Q}_p],\end{align*} $$

by Proposition 3.28 and Lemmas 3.29 and 3.30.

Proof. If $\kappa $ is a finite field, then this is a well-known consequence of the obstruction theory due to Mazur, [Reference Mazur37, Section 1.6]. (We revisit the argument in the proof of Proposition 4.3.) If $\kappa $ is a local field, then essentially the same argument works, except that one has to work harder to justify why the $2$ -cocycle constructed out of an obstruction to lifting is continuous. Lecture 6 in [Reference Conrad21] contains a very nice exposition of the result if $\kappa $ is a finite extension of L. The same argument works if $\kappa $ is a local field of characteristic p.