hcju/stacksqa huggingface dataset:
To prove this, we may and do replace $k$ by the algebraic closure. By Lemma \ref{lemma-group-scheme-addition-tangent-vectors} the derivative of $[p]$ is multiplication by $p$ as a map $T_{A/k, e} \to T_{A/k, e}$ and hence is zero (compare with proof of Lemma \ref{lemma-abelian-variety-multiplication-by-d-etale}). Since $[p]$ commutes with translation we conclude that the derivative of $[p]$ is everywhere zero, i.e., that the induced map $[p]^*\Omega_{A/k} \to \Omega_{A/k}$ is zero. Looking at generic points, we find that the corresponding map $[p]^* : k(A) \to k(A)$ of function fields induces the zero map on $\Omega_{k(A)/k}$. Let $t_1, \ldots, t_g$ be a p-basis of $k(A)$ over $k$ (More on Algebra, Definition \ref{more-algebra-definition-p-basis} and Lemma \ref{more-algebra-lemma-p-basis}). Then $[p]^*(t_i)$ has a $p$th root by Algebra, Lemma \ref{algebra-lemma-derivative-zero-pth-power}. We conclude that $k(A)[x_1, \ldots, x_g]/(x_1^p - t_1, \ldots, x_g^p - t_g)$ is a subextension of $[p]^* : k(A) \to k(A)$. Thus we can find an affine open $U \subset A$ such that $t_i \in \mathcal{O}_A(U)$ and $x_i \in \mathcal{O}_A([p]^{-1}(U))$. We obtain a factorization $$ [p]^{-1}(U) \xrightarrow{\pi_1} \Spec(\mathcal{O}(U)[x_1, \ldots, x_g]/(x_1^p - t_1, \ldots, x_g^p - t_g)) \xrightarrow{\pi_2} U $$ of $[p]$ over $U$. After shrinking $U$ we may assume that $\pi_1$ is finite locally free (for example by generic flatness -- actually it is already finite locally free in our case). By Lemma \ref{lemma-degree-multiplication-by-d} we see that $[p]$ has degree $p^{2g}$. Since $\pi_2$ has degree $p^g$ we see that $\pi_1$ has degree $p^g$ as well. The morphism $\pi_2$ is a universal homeomorphism hence the fibres are singletons. We conclude that the (set theoretic) fibres of $[p]^{-1}(U) \to U$ are the fibres of $\pi_1$. Hence they have at most $p^g$ elements. Since $[p]$ is a homomorphism of group schemes over $k$, the fibre of $[p] : A(k) \to A(k)$ has the same cardinality for every $a \in A(k)$ and the proof is complete.
You can either prove this by repeating the arguments of Lemma \ref{lemma-complete-reducibility-Gm} for the module $M$. Alternatively, you can consider the scheme $(X', \mathcal{O}_{X'}) = (X, \mathcal{O}_X \oplus \mathcal{F})$ where $\mathcal{F}$ is viewed as an ideal of square zero. There is a natural action $a' : \mathbf{G}_m \times X' \to X'$ defined using the action on $X$ and on $\mathcal{F}$. Then apply Lemma \ref{lemma-complete-reducibility-Gm} to $X'$ and conclude. (The nice thing about this argument is that it immediately shows that the grading on $A$ and $M$ are compatible, i.e., that $M$ is a graded $A$-module.) Details omitted.
Since $U \to U'$ is quasi-compact and quasi-separated we see that $f_*$ transforms quasi-coherent sheaves into quasi-coherent sheaves (Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}). Moreover, since the squares $$ \vcenter{ \xymatrix{ R \ar[d]_t \ar[r]_f & R' \ar[d]^{t'} \\ U \ar[r]^f & U' } } \quad\text{and}\quad \vcenter{ \xymatrix{ R \ar[d]_s \ar[r]_f & R' \ar[d]^{s'} \\ U \ar[r]^f & U' } } $$ are cartesian we find that $(t')^*f_*\mathcal{F} = f_*t^*\mathcal{F}$ and $(s')^*f_*\mathcal{F} = f_*s^*\mathcal{F}$ , see Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}. Thus it makes sense to think of $f_*\alpha$ as a map $(t')^*f_*\mathcal{F} \to (s')^*f_*\mathcal{F}$. A similar argument shows that $f_*\alpha$ satisfies the cocycle condition. The functor is adjoint to the pullback functor since pullback and pushforward on modules on ringed spaces are adjoint. Some details omitted.
The pushforward of a quasi-coherent module along a quasi-compact and quasi-separated morphism is quasi-coherent, see Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}. Hence $t_*s^*\mathcal{G}$ is quasi-coherent. With notation as in Lemma \ref{lemma-diagram} we have $$ t^*t_*s^*\mathcal{G} = \text{pr}_{0, *}c^* s^*\mathcal{G} = \text{pr}_{0, *}\text{pr}_1^*s^*\mathcal{G} = s^*t_*s^*\mathcal{G} $$ The middle equality because $s \circ c = s \circ \text{pr}_1$ as morphisms $R \times_{s, U, t} R \to U$, and the first and the last equality because we know that base change and pushforward commute in these steps by Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}. \medskip\noindent To verify the cocycle condition of Definition \ref{definition-groupoid-module} for $\alpha$ and the adjointness property we describe the construction $\mathcal{G} \mapsto (\mathcal{G}, \alpha)$ in another way. Consider the groupoid scheme $(R, R \times_{s, U, s} R, \text{pr}_0, \text{pr}_1, \text{pr}_{02})$ associated to the equivalence relation $R \times_{s, U, s} R$ on $R$, see Lemma \ref{lemma-equivalence-groupoid}. There is a morphism $$ f : (R, R \times_{s, U, s} R, \text{pr}_1, \text{pr}_0, \text{pr}_{02}) \longrightarrow (U, R, s, t, c) $$ of groupoid schemes given by $t : R \to U$ and $R \times_{t, U, t} R \to R$ given by $(r_0, r_1) \mapsto r_0 \circ r_1^{-1}$ (we omit the verification of the commutativity of the required diagrams). Since $t, s : R \to U$ are quasi-compact, quasi-separated, and flat, and since we have a cartesian square $$ \xymatrix{ R \times_{s, U, s} R \ar[d]_{\text{pr}_0} \ar[rr]_-{(r_0, r_1) \mapsto r_0 \circ r_1^{-1}} & & R \ar[d]^t \\ R \ar[rr]^t & & U } $$ by Lemma \ref{lemma-diagram-pull} it follows that Lemma \ref{lemma-pushforward} applies to $f$. Note that $$ \QCoh(R, R \times_{s, U, s} R, \text{pr}_1, \text{pr}_0, \text{pr}_{02}) = \QCoh(\mathcal{O}_U) $$ by the theory of descent of quasi-coherent sheaves as $\{t : R \to U\}$ is an fpqc covering, see Descent, Proposition \ref{descent-proposition-fpqc-descent-quasi-coherent}. Observe that pullback along $f$ agrees with the forgetful functor and that pushforward agrees with the construction that assigns to $\mathcal{G}$ the pair $(\mathcal{G}, \alpha)$. We omit the precise verifications. Thus the lemma follows from Lemma \ref{lemma-pushforward}.
We will freely use the characterization of coherent modules of Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-Noetherian} as well as the fact that coherent modules form a Serre subcategory of $\QCoh(\mathcal{O}_X)$, see Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-Noetherian-quasi-coherent-sub-quotient}. If $f$ has a section $\sigma$, then we see that $\mathcal{F} \cap f_*\mathcal{G}$ is contained in the image of $\sigma^*\mathcal{G} \to \sigma^*f^*\mathcal{F} = \mathcal{F}$, hence coherent. In general, to show that $\mathcal{F} \cap f_*\mathcal{G}$ is coherent, it suffices the show that $f^*(\mathcal{F} \cap f_*\mathcal{G})$ is coherent (see Descent, Lemma \ref{descent-lemma-finite-type-descends}). Since $f$ is flat this is equal to $f^*\mathcal{F} \cap f^*f_*\mathcal{G}$. Since $f$ is flat, quasi-compact, and quasi-separated we see $f^*f_*\mathcal{G} = p_*q^*\mathcal{G}$ where $p, q : Y \times_X Y \to Y$ are the projections, see Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}. Since $p$ has a section we win.
We will use the characterization of Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-Noetherian} of coherent modules on locally Noetherian scheme without further mention. Write $\mathcal{F} = \colim \mathcal{H}_i$ with $\mathcal{H}_i$ coherent, see Properties, Lemma \ref{properties-lemma-directed-colimit-finite-presentation}. Given a quasi-coherent sheaf $\mathcal{H}$ on $U$ we denote $t_*s^*\mathcal{H}$ the quasi-coherent sheaf on $(U, R, s, t, c)$ of Lemma \ref{lemma-construct-quasi-coherent}. There is an adjunction map $\mathcal{F} \to t_*s^*\mathcal{F}$ in $\QCoh(U, R, s, t, c)$. Consider the pullback diagram $$ \xymatrix{ \mathcal{F} \ar[r] & t_*s^*\mathcal{F} \\ \mathcal{F}_i \ar[r] \ar[u] & t_*s^*\mathcal{H}_i \ar[u] } $$ in other words $\mathcal{F}_i = \mathcal{F} \cap t_*s^*\mathcal{H}_i$. Then $\mathcal{F}_i$ is coherent by Lemma \ref{lemma-push-pull}. On the other hand, the diagram above is a pullback diagram in $\QCoh(U, R, s, t, c)$ also as restriction to $U$ is an exact functor by (the proof of) Lemma \ref{lemma-abelian}. Finally, because $t$ is quasi-compact and quasi-separated we see that $t_*$ commutes with colimits (see Cohomology of Schemes, Lemma \ref{coherent-lemma-colimit-cohomology}). Hence $t_*s^*\mathcal{F} = \colim t_*\mathcal{H}_i$ and hence $\mathcal{F} = \colim \mathcal{F}_i$ as desired.
The assumption means that $\mathcal{O}_R(R)$ is a free $\mathcal{O}_U(U)$-module via $s$ with basis $e_i$. Hence for any quasi-coherent $\mathcal{O}_U$-module $\mathcal{G}$ we see that $s^*\mathcal{G}(R) = \bigoplus_i \mathcal{G}(U)e_i$. We will write $s(-)$ to indicate pullback of sections by $s$ and similarly for other morphisms. Let $(\mathcal{F}, \alpha)$ be a quasi-coherent module on $(U, R, s, t, c)$. Let $\sigma \in \mathcal{F}(U)$. By the above we can write $$ \alpha(t(\sigma)) = \sum s(\sigma_i) e_i $$ for some unique $\sigma_i \in \mathcal{F}(U)$ (all but finitely many are zero of course). We can also write $$ c(e_i) = \sum \text{pr}_1(f_{ij}) \text{pr}_0(e_j) $$ as functions on $R \times_{s, U, t}R$. Then the commutativity of the diagram in Definition \ref{definition-groupoid-module} means that $$ \sum \text{pr}_1(\alpha(t(\sigma_i))) \text{pr}_0(e_i) = \sum \text{pr}_1(s(\sigma_i)f_{ij}) \text{pr}_0(e_j) $$ (calculation omitted). Picking off the coefficients of $\text{pr}_0(e_l)$ we see that $\alpha(t(\sigma_l)) = \sum s(\sigma_i)f_{il}$. Hence the submodule $\mathcal{G} \subset \mathcal{F}$ generated by the elements $\sigma_i$ defines a finite type quasi-coherent module preserved by $\alpha$. Hence it is a subobject of $\mathcal{F}$ in $\QCoh(U, R, s, t, c)$. This submodule contains $\sigma$ (as one sees by pulling back the first relation by $e$). Hence we win.
For each quasi-coherent module $\mathcal{F}$ on $U$ there is a (possibly empty) set of maps $\alpha : t^*\mathcal{F} \to s^*\mathcal{F}$ such that $(\mathcal{F}, \alpha)$ is a quasi-coherent modules on $(U, R, s, t, c)$. By Properties, Lemma \ref{properties-lemma-set-of-iso-classes} there exists a set of isomorphism classes of $\kappa$-generated quasi-coherent $\mathcal{O}_U$-modules.
Part (1) follows from Lemmas \ref{lemma-pre-equivalence-equivalence-relation-points} and \ref{lemma-groupoid-pre-equivalence}, namely, $t(s^{-1}(W))$ is the set of points of $U$ equivalent to a point of $W$. Next, assume $s$ and $t$ open and $W \subset U$ open. Since $s$ is open the set $W' = t(s^{-1}(W))$ is an open subset of $U$. Finally, assume that $s$, $t$ are both open and quasi-compact. Then, if $W \subset U$ is a quasi-compact open, then also $W' = t(s^{-1}(W))$ is a quasi-compact open, and invariant by the discussion above. Letting $W$ range over all affine opens of $U$ we see (3).
Note that $s^{-1}(W)$ is quasi-compact open in $R$. As a continuous map $t$ maps the quasi-compact subset $s^{-1}(W)$ to a quasi-compact subset $t(s^{-1}(W))$. As $t$ is flat and $s^{-1}(W)$ is closed under generalization, so is $t(s^{-1}(W))$, see (Morphisms, Lemma \ref{morphisms-lemma-generalizations-lift-flat} and Topology, Lemma \ref{topology-lemma-lift-specializations-images}). Pick a quasi-compact open $W' \subset U$ containing $t(s^{-1}(W))$. By Properties, Lemma \ref{properties-lemma-quasi-compact-quasi-separated-spectral} we see that $W'$ is a spectral space (here we use that $U$ is quasi-separated). Then the lemma follows from Topology, Lemma \ref{topology-lemma-make-spectral-space} applied to $t(s^{-1}(W)) \subset W'$.
Set $E = t(s^{-1}(W))$. Recall that $E$ is set-theoretically $R$-invariant (Lemma \ref{lemma-constructing-invariant-opens}). By Lemma \ref{lemma-first-observation} there exists a quasi-compact open $W'$ containing $E$. Let $Z = U \setminus W'$ and consider $T = t(s^{-1}(Z))$. Observe that $Z \subset T$ and that $E \cap T = \emptyset$ because $s^{-1}(E) = t^{-1}(E)$ is disjoint from $s^{-1}(Z)$. Since $T$ is the image of the closed subset $s^{-1}(Z) \subset R$ under the quasi-compact morphism $t : R \to U$ we see that any point $\xi$ in the closure $\overline{T}$ is the specialization of a point of $T$, see Morphisms, Lemma \ref{morphisms-lemma-reach-points-scheme-theoretic-image} (and Morphisms, Lemma \ref{morphisms-lemma-quasi-compact-scheme-theoretic-image} to see that the scheme theoretic image is the closure of the image). Say $\xi' \leadsto \xi$ with $\xi' \in T$. Suppose that $r \in R$ and $s(r) = \xi$. Since $s$ is flat we can find a specialization $r' \leadsto r$ in $R$ such that $s(r') = \xi'$ (Morphisms, Lemma \ref{morphisms-lemma-generalizations-lift-flat}). Then $t(r') \leadsto t(r)$. We conclude that $t(r') \in T$ as $T$ is set-theoretically invariant by Lemma \ref{lemma-constructing-invariant-opens}. Thus $\overline{T}$ is a set-theoretically $R$-invariant closed subset and $V = U \setminus \overline{T}$ is the open we are looking for. It is contained in $W'$ which finishes the proof.
Suppose $\xi, \xi' \in (U'/R')(T)$ are sections which map to the same section of $U/R$. Then we can find a $\tau$-covering $\mathcal{T} = \{T_i \to T\}$ of $T$ such that $\xi|_{T_i}, \xi'|_{T_i}$ are given by $a_i, a_i' \in U'(T_i)$. By Lemma \ref{lemma-quotient-pre-equivalence} and the axioms of a site we may after refining $\mathcal{T}$ assume there exist morphisms $r_i : T_i \to R$ such that $g \circ a_i = s \circ r_i$, $g \circ a_i' = t \circ r_i$. Since by construction $R' = R \times_{U \times_S U} (U' \times_S U')$ we see that $(r_i, (a_i, a_i')) \in R'(T_i)$ and this shows that $a_i$ and $a_i'$ define the same section of $U'/R'$ over $T_i$. By the sheaf condition this implies $\xi = \xi'$. \medskip\noindent If $h_{U'} \to h_U$ is a surjection of sheaves, then of course $U'/R' \to U/R$ is surjective also. If $\{g : U' \to U\}$ is a $\tau$-covering, then the map of sheaves $h_{U'} \to h_U$ is surjective, see Sites, Lemma \ref{sites-lemma-covering-surjective-after-sheafification}. Hence $U'/R' \to U/R$ is surjective also in this case.
Injectivity follows on combining Lemmas \ref{lemma-groupoid-pre-equivalence} and \ref{lemma-quotient-pre-equivalence-relation-restrict}. To see surjectivity (see Sites, Section \ref{sites-section-sheaves-injective} for a characterization of surjective maps of sheaves) we argue as follows. Suppose that $T$ is a scheme and $\sigma \in U/R(T)$. There exists a covering $\{T_i \to T\}$ such that $\sigma|_{T_i}$ is the image of some element $f_i \in U(T_i)$. Hence we may assume that $\sigma$ is the image of $f \in U(T)$. By the assumption that $h$ is a surjection of sheaves, we can find a $\tau$-covering $\{\varphi_i : T_i \to T\}$ and morphisms $f_i : T_i \to U' \times_{g, U, t} R$ such that $f \circ \varphi_i = h \circ f_i$. Denote $f'_i = \text{pr}_0 \circ f_i : T_i \to U'$. Then we see that $f'_i \in U'(T_i)$ maps to $g \circ f'_i \in U(T_i)$ and that $g \circ f'_i \sim_{T_i} h \circ f_i = f \circ \varphi_i$ notation as in (\ref{equation-quotient-presheaf}). Namely, the element of $R(T_i)$ giving the relation is $\text{pr}_1 \circ f_i$. This means that the restriction of $\sigma$ to $T_i$ is in the image of $U'/R'(T_i) \to U/R(T_i)$ as desired. \medskip\noindent If $\{h\}$ is a $\tau$-covering, then it induces a surjection of sheaves, see Sites, Lemma \ref{sites-lemma-covering-surjective-after-sheafification}. If $U' \to U$ is surjective, then also $h$ is surjective as $s$ has a section (namely the neutral element $e$ of the groupoid scheme).
By Lemma \ref{lemma-quotient-pre-equivalence} the quotient sheaves have a simple description which we will use below without further mention. We first show that $$ U \longrightarrow U' \times_{U'/R'} U/R $$ is injective. Namely, assume $a, b \in U(T)$ map to the same element on the right hand side. Then $f(a) = f(b)$. After replacing $T$ by the members of a $\tau$-covering we may assume that there exists an $r \in R(T)$ such that $a = s(r)$ and $b = t(r)$. Then $r' = f(r)$ is a $T$-valued point of $R'$ with $s'(r') = t'(r')$. Hence $r' = e'(f(a))$ (where $e'$ is the identity of the groupoid scheme associated to $j'$, see Lemma \ref{lemma-equivalence-groupoid}). Because the first diagram of the lemma is cartesian this implies that $r$ has to equal $e(a)$. Thus $a = b$. \medskip\noindent Finally, we show that the displayed arrow is surjective. Let $T$ be a scheme over $S$ and let $(a', \overline{b})$ be a section of the sheaf $U' \times_{U'/R'} U/R$ over $T$. After replacing $T$ by the members of a $\tau$-covering we may assume that $\overline{b}$ is the class of an element $b \in U(T)$. After replacing $T$ by the members of a $\tau$-covering we may assume that there exists an $r' \in R'(T)$ such that $a' = t(r')$ and $s'(r') = f(b)$. Because the first diagram of the lemma is cartesian we can find $r \in R(T)$ such that $s(r) = b$ and $f(r) = r'$. Then it is clear that $a = t(r) \in U(T)$ is a section which maps to $(a', \overline{b})$.
The group scheme $G \to U$ is the base change of $R \to U \times_S U$ by the diagonal morphism $U \to U \times_S U$, see Lemma \ref{lemma-groupoid-stabilizer}. Hence if $j$ is separated (resp.\ quasi-separated), then $G \to U$ is separated (resp.\ quasi-separated). (See Schemes, Lemma \ref{schemes-lemma-separated-permanence}). Thus (a) $\Rightarrow$ (b) in both (1) and (2). \medskip\noindent If $G \to U$ is separated (resp.\ quasi-separated), then the morphism $U \to G$, as a section of the structure morphism $G \to U$ is a closed immersion (resp.\ quasi-compact), see Schemes, Lemma \ref{schemes-lemma-section-immersion}. Thus (b) $\Rightarrow$ (a) in both (1) and (2). \medskip\noindent By the result of Lemma \ref{lemma-diagram-diagonal} (and Schemes, Lemmas \ref{schemes-lemma-base-change-immersion} and \ref{schemes-lemma-quasi-compact-preserved-base-change}) we see that if $e$ is a closed immersion (resp.\ quasi-compact) $\Delta_{R/U \times_S U}$ is a closed immersion (resp.\ quasi-compact). Thus (c) $\Rightarrow$ (a) in both (1) and (2).
Consider the commutative diagram $$ \xymatrix{ & U & \\ R \ar[d]_s \ar[ru]^t & R \times_{s, U, t} R \ar[l]^-{\text{pr}_0} \ar[d]^{\text{pr}_1} \ar[r]_-c & R \ar[d]^s \ar[lu]_t \\ U & R \ar[l]_t \ar[r]^s & U } $$ of Lemma \ref{lemma-diagram}. Think of $f \in \Gamma(U, \mathcal{O}_U)$. The commutativity of the top part of the diagram shows that $\text{pr}_0^\sharp(t^\sharp(f)) = c^\sharp(t^\sharp(f))$ as elements of $\Gamma(R \times_{S, U, t} R, \mathcal{O})$. Looking at the right lower cartesian square the compatibility of the norm construction with base change shows that $s^\sharp(\text{Norm}_{s^\sharp}(t^\sharp(f))) = \text{Norm}_{\text{pr}_1^\sharp}(c^\sharp(t^\sharp(f)))$. Similarly we get $t^\sharp(\text{Norm}_{s^\sharp}(t^\sharp(f))) = \text{Norm}_{\text{pr}_1^\sharp}(\text{pr}_0^\sharp(t^\sharp(f)))$. Hence by the first equality of this proof we see that $s^\sharp(\text{Norm}_{s^\sharp}(t^\sharp(f))) = t^\sharp(\text{Norm}_{s^\sharp}(t^\sharp(f)))$ as desired.
By Morphisms, Lemma \ref{morphisms-lemma-finite-locally-free} there exists a decomposition $U = \coprod\nolimits_{r \geq 0} U_r$ such that $s : s^{-1}(U_r) \to U_r$ is finite locally free of rank $r$. As $s$ is surjective we see that $U_0 = \emptyset$. Note that $u \in U_r \Leftrightarrow$ if and only if the scheme theoretic fibre $s^{-1}(u)$ has degree $r$ over $\kappa(u)$. Now, if $z \in R$ with $s(z) = u$ and $t(z) = u'$ then using notation as in Lemma \ref{lemma-diagram} $$ \text{pr}_1^{-1}(z) \to \Spec(\kappa(z)) $$ is the base change of both $s^{-1}(u) \to \Spec(\kappa(u))$ and $s^{-1}(u') \to \Spec(\kappa(u'))$ by the lemma cited. Hence $u \in U_r \Leftrightarrow u' \in U_r$, in other words, the open subsets $U_r$ are $R$-invariant. In particular the restriction of $R$ to $U_r$ is just $s^{-1}(U_r)$ and $s : R_r \to U_r$ is finite locally free of rank $r$. As $t : R_r \to U_r$ is isomorphic to $s$ by the inverse of $R_r$ we see that it has also rank $r$.
First, by Lemma \ref{lemma-finite-locally-free-disjoint-free} we know that $(U, R, s, t, c)$ is a disjoint union of groupoid schemes $(U_r, R_r, s, t, c)$ such that each $s, t : R_r \to U_r$ has constant rank $r$. As $U$ is quasi-compact, we have $U_r = \emptyset$ for almost all $r$. It suffices to prove the lemma for each $(U_r, R_r, s, t, c)$ and hence we may assume that $s, t$ are finite locally free of rank $r$. \medskip\noindent Assume that $s, t$ are finite locally free of rank $r$. Let $f \in A$. Consider the element $x - f \in A[x]$, where we think of $x$ as the coordinate on $\mathbf{A}^1$. Since $$ (U \times \mathbf{A}^1, R \times \mathbf{A}^1, s \times \text{id}_{\mathbf{A}^1}, t \times \text{id}_{\mathbf{A}^1}, c \times \text{id}_{\mathbf{A}^1}) $$ is also a groupoid scheme with finite source and target, we may apply Lemma \ref{lemma-determinant-trick} to it and we see that $P(x) = \text{Norm}_{s^\sharp}(t^\sharp(x - f))$ is an element of $C[x]$. Because $s^\sharp : A \to B$ is finite locally free of rank $r$ we see that $P$ is monic of degree $r$. Moreover $P(f) = 0$ by Cayley-Hamilton (Algebra, Lemma \ref{algebra-lemma-charpoly}).
The proof of part (1) is omitted. Let us denote $A' = A \otimes_C C'$ and $B' = B \otimes_C C'$. Then we have $$ C^1 = \{a \in A' \mid (t')^\sharp(a) = (s')^\sharp(a) \} = \{a \in A \otimes_C C' \mid t^\sharp \otimes 1(a) = s^\sharp \otimes 1(a) \}. $$ In other words, $C^1$ is the kernel of the difference map $(t^\sharp - s^\sharp) \otimes 1$ which is just the base change of the $C$-linear map $t^\sharp - s^\sharp : A \to B$ by $C \to C'$. Hence (3) follows. \medskip\noindent Proof of part (2)(b). Since $C \to A$ is integral (Lemma \ref{lemma-integral-over-invariants}) and injective we see that $\Spec(A) \to \Spec(C)$ is surjective, see Algebra, Lemma \ref{algebra-lemma-integral-overring-surjective}. Thus also $\Spec(A') \to \Spec(C')$ is surjective as a base change of a surjective morphism (Morphisms, Lemma \ref{morphisms-lemma-base-change-surjective}). Hence $\Spec(C^1) \to \Spec(C')$ is surjective also. This implies (2)(b) holds for example by Algebra, Lemma \ref{algebra-lemma-image-dense-generic-points}. \medskip\noindent Proof of part (2)(a). By Lemma \ref{lemma-finite-locally-free-disjoint-free} our groupoid scheme $(U, R, s, t, c)$ decomposes as a finite disjoint union of groupoid schemes $(U_r, R_r, s, t, c)$ such that $s, t : R_r \to U_r$ are finite locally free of rank $r$. Pulling back by $U' = \Spec(C') \to U$ we obtain a similar decomposition of $U'$ and $U^1 = \Spec(C^1)$. We will show in the next paragraph that (2)(a) holds for the corresponding system of rings $A_r, B_r, C_r, C'_r, C^1_r$ with $n = r$. Then given $f \in C^1$ let $P_r \in C_r[x]$ be the polynomial whose image in $C^1_r[x]$ is the image of $(x - f)^r$. Choosing a sufficiently divisible integer $n$ we see that there is a polynomial $P \in C'[x]$ whose image in $C^1[x]$ is $(x - f)^n$; namely, we take $P$ to be the unique element of $C'[x]$ whose image in $C'_r[x]$ is $P_r^{n/r}$. \medskip\noindent In this paragraph we prove (2)(a) in case the ring maps $s^\sharp, t^\sharp : A \to B$ are finite locally free of a fixed rank $r$. Let $f \in C^1 \subset A' = A \otimes_C C'$. Choose a flat $C$-algebra $D$ and a surjection $D \to C'$. Choose a lift $g \in A \otimes_C D$ of $f$. Consider the polynomial $$ P = \text{Norm}_{s^\sharp \otimes 1}(t^\sharp \otimes 1(x - g)) $$ in $(A \otimes_C D)[x]$. By Lemma \ref{lemma-determinant-trick} and part (3) of the current lemma the coefficients of $P$ are in $D$ (compare with the proof of Lemma \ref{lemma-integral-over-invariants}). On the other hand, the image of $P$ in $(A \otimes_C C')[x]$ is $(x - f)^r$ because $t^\sharp \otimes 1(x - f) = s^\sharp(x - f)$ and $s^\sharp$ is finite locally free of rank $r$. This proves what we want with $P$ as in the statement (2)(a) given by the image of our $P$ under the map $D[x] \to C'[x]$.