hcju/stacksqa huggingface dataset:
The statement on base change is Lemma \ref{lemma-base-change-tensor}. Thus it suffices to show that $K$ is a pseudo-coherent object. This will follow from Lemma \ref{lemma-base-change-tensor-perfect} by approximation by perfect complexes. We encourage the reader to skip the rest of the proof. \medskip\noindent The question is \'etale local on $Y$, hence we may assume $Y$ is affine. Then $X$ is quasi-compact and quasi-separated. Moreover, there exists an integer $N$ such that total direct image $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ has cohomological dimension $N$ as explained in Lemma \ref{lemma-quasi-coherence-direct-image}. Choose an integer $b$ such that $\mathcal{G}^i = 0$ for $i > b$. It suffices to show that $K$ is $m$-pseudo-coherent for every $m$. Choose an approximation $P \to E$ by a perfect complex $P$ of $(X, E, m - N - 1 - b)$. This is possible by Theorem \ref{theorem-approximation}. Choose a distinguished triangle $$ P \to E \to C \to P[1] $$ in $D_\QCoh(\mathcal{O}_X)$. The cohomology sheaves of $C$ are zero in degrees $\geq m - N - 1 - b$. Hence the cohomology sheaves of $C \otimes^\mathbf{L} \mathcal{G}^\bullet$ are zero in degrees $\geq m - N - 1$. Thus the cohomology sheaves of $Rf_*(C \otimes^\mathbf{L} \mathcal{G})$ are zero in degrees $\geq m - 1$. Hence $$ Rf_*(P \otimes^\mathbf{L} \mathcal{G}) \to Rf_*(E \otimes^\mathbf{L} \mathcal{G}) $$ is an isomorphism on cohomology sheaves in degrees $\geq m$. Next, suppose that $H^i(P) = 0$ for $i > a$. Then $ P \otimes^\mathbf{L} \sigma_{\geq m - N - 1 - a}\mathcal{G}^\bullet \longrightarrow P \otimes^\mathbf{L} \mathcal{G}^\bullet $ is an isomorphism on cohomology sheaves in degrees $\geq m - N - 1$. Thus again we find that $$ Rf_*(P \otimes^\mathbf{L} \sigma_{\geq m - N - 1 - a}\mathcal{G}^\bullet) \to Rf_*(P \otimes^\mathbf{L} \mathcal{G}^\bullet) $$ is an isomorphism on cohomology sheaves in degrees $\geq m$. By Lemma \ref{lemma-base-change-tensor-perfect} the source is a perfect complex. We conclude that $K$ is $m$-pseudo-coherent as desired.
The statement means that for any object $E$ of $D_\QCoh(\mathcal{O}_X)$ the induced map $$ \Hom(E, \mathcal{O}_X \otimes_R (\lim M_n)) \longrightarrow \Hom(E, \lim \mathcal{O}_X \otimes_R M_n) $$ is an isomorphism. Since $D_\QCoh(\mathcal{O}_X)$ has a perfect generator (Theorem \ref{theorem-bondal-van-den-Bergh}) it suffices to check this for perfect $E$. By Lemma \ref{lemma-Rlim-quasi-coherent} we have $\lim \mathcal{O}_X \otimes_R M_n = R\lim \mathcal{O}_X \otimes_R M_n$. The exact functor $R\Hom_X(E, -) : D_\QCoh(\mathcal{O}_X) \to D(R)$ of Cohomology on Sites, Section \ref{sites-cohomology-section-global-RHom} commutes with products and hence with derived limits, whence $$ R\Hom_X(E, \lim \mathcal{O}_X \otimes_R M_n) = R\lim R\Hom_X(E, \mathcal{O}_X \otimes_R M_n) $$ Let $E^\vee$ be the dual perfect complex, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-dual-perfect-complex}. We have $$ R\Hom_X(E, \mathcal{O}_X \otimes_R M_n) = R\Gamma(X, E^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*M_n) = R\Gamma(X, E^\vee) \otimes_R^\mathbf{L} M_n $$ by Lemma \ref{lemma-cohomology-base-change}. From Lemma \ref{lemma-flat-proper-perfect-direct-image-general} we see $R\Gamma(X, E^\vee)$ is a perfect complex of $R$-modules. In particular it is a pseudo-coherent complex and by More on Algebra, Lemma \ref{more-algebra-lemma-pseudo-coherent-tensor-limit} we obtain $$ R\lim R\Gamma(X, E^\vee) \otimes_R^\mathbf{L} M_n = R\Gamma(X, E^\vee) \otimes_R^\mathbf{L} \lim M_n $$ as desired.
There exists an integer $N$ such that $R\Gamma(X, -) : D_\QCoh(\mathcal{O}_X) \to D(A)$ has cohomological dimension $N$ as explained in Lemma \ref{lemma-quasi-coherence-direct-image}. Let $b \in \mathbf{Z}$ be such that $H^i(K) = 0$ for $i > b$. Let $E$ be pseudo-coherent on $X$. It suffices to show that $R\Gamma(X, E \otimes^\mathbf{L} K)$ is $m$-pseudo-coherent for every $m$. Choose an approximation $P \to E$ by a perfect complex $P$ of $(X, E, m - N - 1 - b)$. This is possible by Theorem \ref{theorem-approximation}. Choose a distinguished triangle $$ P \to E \to C \to P[1] $$ in $D_\QCoh(\mathcal{O}_X)$. The cohomology sheaves of $C$ are zero in degrees $\geq m - N - 1 - b$. Hence the cohomology sheaves of $C \otimes^\mathbf{L} K$ are zero in degrees $\geq m - N - 1$. Thus the cohomology of $R\Gamma(X, C \otimes^\mathbf{L} K)$ are zero in degrees $\geq m - 1$. Hence $$ R\Gamma(X, P \otimes^\mathbf{L} K) \to R\Gamma(X, E \otimes^\mathbf{L} K) $$ is an isomorphism on cohomology in degrees $\geq m$. By assumption the source is pseudo-coherent. We conclude that $R\Gamma(X, E \otimes^\mathbf{L} K)$ is $m$-pseudo-coherent as desired.
The statement on base change is Lemma \ref{lemma-base-change-RHom}. Thus it suffices to show that $K$ is a perfect object. If $Y$ is Noetherian, then this follows from Lemma \ref{lemma-ext-perfect}. We will reduce to this case by Noetherian approximation. We encourage the reader to skip the rest of this proof. \medskip\noindent The question is local on $Y$, hence we may assume $Y$ is affine. Say $Y = \Spec(R)$. We write $R = \colim R_i$ as a filtered colimit of Noetherian rings $R_i$. By Limits of Spaces, Lemma \ref{spaces-limits-lemma-descend-finite-presentation} there exists an $i$ and an algebraic space $X_i$ of finite presentation over $R_i$ whose base change to $R$ is $X$. By Limits of Spaces, Lemma \ref{spaces-limits-lemma-descend-modules-finite-presentation} we may assume after increasing $i$, that there exists a bounded complex of finitely presented $\mathcal{O}_{X_i}$-module $\mathcal{G}_i^\bullet$ whose pullback to $X$ is $\mathcal{G}$. After increasing $i$ we may assume $\mathcal{G}_i^n$ is flat over $R_i$, see Limits of Spaces, Lemma \ref{spaces-limits-lemma-descend-flat}. After increasing $i$ we may assume the support of $\mathcal{G}_i^n$ is proper over $R_i$, see Limits of Spaces, Lemma \ref{spaces-limits-lemma-eventually-proper-support}. Finally, by Lemma \ref{lemma-descend-perfect} we may, after increasing $i$, assume there exists a perfect object $E_i$ of $D(\mathcal{O}_{X_i})$ whose pullback to $X$ is $E$. Applying Lemma \ref{lemma-compute-ext-perfect} to $X_i \to \Spec(R_i)$, $E_i$, $\mathcal{G}_i^\bullet$ and using the base change property already shown we obtain the result.
Choose a scheme $U$ and a surjective \'etale morphism $\varphi : U \to X$. Then $L\varphi^*E$ is a pseudo-coherent complex on the scheme $U$ (use Lemma \ref{lemma-descend-pseudo-coherent}) and we can apply the result for schemes, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-jump-loci}. The meaning of part (3) is that the inverse image of the level sets to $U$ are locally constructible, see Properties of Spaces, Definition \ref{spaces-properties-definition-locally-constructible}.
By cohomology and base change (more precisely by Lemma \ref{lemma-flat-proper-perfect-direct-image-general}) the object $K = Rf_*\mathcal{F}$ is a perfect object of the derived category of $Y$ whose formation commutes with arbitrary base change. In particular we have $$ H^i(X_y, \mathcal{F}_y) = H^i(K \otimes_{\mathcal{O}_Y}^\mathbf{L} \kappa(y)) $$ Thus the lemma follows from Lemma \ref{lemma-jump-loci}.
By cohomology and base change ( Lemma \ref{lemma-flat-proper-perfect-direct-image-general}) the object $K = Rf_*\mathcal{F}$ is a perfect object of the derived category of $Y$ whose formation commutes with arbitrary base change. Thus this lemma follows immediately from Lemma \ref{lemma-open-where-cohomology-in-degree-i-rank-r}.
It suffices to prove (a) and (b) \'etale locally on $Y$, thus we may and do assume $Y$ is an affine scheme. By cohomology and base change (Lemma \ref{lemma-flat-proper-perfect-direct-image-general}) the complex $E = Rf_*\mathcal{O}_X$ is perfect and its formation commutes with arbitrary base change. In particular, for $y \in Y$ we see that $H^0(E \otimes^\mathbf{L} \kappa(y)) = H^0(X_y, \mathcal{O}_{X_y}) = \kappa(y)$. Thus $\beta_0(y) \leq 1$ for all $y \in Y$ with notation as in Lemma \ref{lemma-jump-loci}. Apply Lemma \ref{lemma-locally-closed-where-H0-locally-free} with $a = 0$ and $r = 1$. We obtain a universal closed subscheme $j : Z \to Y$ with $H^0(Lj^*E)$ invertible characterized by the equivalence of (4)(a), (b), and (c) of the lemma. Since formation of $E$ commutes with base change, we have $$ Lf^*E = R\text{pr}_{1, *}\mathcal{O}_{X \times_Y X} $$ The morphism $\text{pr}_1 : X \times_Y X$ has a section namely the diagonal morphism $\Delta$ for $X$ over $Y$. We obtain maps $$ \mathcal{O}_X \longrightarrow R\text{pr}_{1, *}\mathcal{O}_{X \times_Y X} \longrightarrow \mathcal{O}_X $$ in $D(\mathcal{O}_X)$ whose composition is the identity. Thus $R\text{pr}_{1, *}\mathcal{O}_{X \times_Y X} = \mathcal{O}_X \oplus E'$ in $D(\mathcal{O}_X)$. Thus $\mathcal{O}_X$ is a direct summand of $H^0(Lf^*E)$ and we conclude that $X \to Y$ factors through $Z$ by the equivalence of (4)(c) and (4)(a) of the lemma cited above. Since $\{X \to Y\}$ is an fppf covering, we have $Z = Y$. Thus $f_*\mathcal{O}_X$ is an invertible $\mathcal{O}_Y$-module. We conclude $\mathcal{O}_Y \to f_*\mathcal{O}_X$ is an isomorphism because a ring map $A \to B$ such that $B$ is invertible as an $A$-module is an isomorphism. Since the assumptions are preserved under base change, we see that (a) is true. \medskip\noindent Proof of (b). Above we have seen that for every $y \in Y$ the map $\mathcal{O}_Y \to H^0(E \otimes^\mathbf{L} \kappa(y))$ is surjective. Thus we may apply More on Algebra, Lemma \ref{more-algebra-lemma-better-cut-complex-in-two} to see that in an open neighbourhood of $y$ we have a decomposition $Rf_*\mathcal{O}_X = \mathcal{O}_Y \oplus P$
By Lemma \ref{lemma-proper-flat-h0} it suffices to show that $k = H^0(X_k, \mathcal{O}_{X_k})$ for all morphisms $\Spec(k) \to Y$ where $k$ is a field. This follows from Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-proper-geometrically-reduced-global-sections} and the fact that $X_k$ is geometrically connected and geometrically reduced.
Using Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-dual-perfect-complex} we see that it suffices to prove the result for the groups $H^i(U, K \otimes^\mathbf{L} E)$. We will use the induction principle to prove the lemma, see Lemma \ref{lemma-induction-principle}. \medskip\noindent When $U = \Spec(A)$ is affine the result follows from the case of schemes, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-countable-cohomology}. \medskip\noindent To finish the proof it suffices to show: if $(U \subset W, V \to W)$ is an elementary distinguished triangle and the result holds for $U$, $V$, and $U \times_W V$, then the result holds for $W$. This is an immediate consquence of the Mayer-Vietoris sequence, see Lemma \ref{lemma-unbounded-mayer-vietoris}.
If (1) is true, then (2) is true because homotopy colimits commutes with taking cohomology sheaves (by Derived Categories, Lemma \ref{derived-lemma-cohomology-of-hocolim}) and because a perfect complex is locally isomorphic to a finite complex of finite free $\mathcal{O}_X$-modules and therefore satisfies (2) by assumption on $X$. \medskip\noindent Assume (2). Choose a K-injective complex $\mathcal{K}^\bullet$ representing $K$. Choose a perfect generator $E$ of $D_\QCoh(\mathcal{O}_X)$ and represent it by a K-injective complex $\mathcal{I}^\bullet$. According to Theorem \ref{theorem-DQCoh-is-Ddga} and its proof there is an equivalence of triangulated categories $F : D_\QCoh(\mathcal{O}_X) \to D(A, \text{d})$ where $(A, \text{d})$ is the differential graded algebra $$ (A, \text{d}) = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)} (\mathcal{I}^\bullet, \mathcal{I}^\bullet) $$ which maps $K$ to the differential graded module $$ M = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)} (\mathcal{I}^\bullet, \mathcal{K}^\bullet) $$ Note that $H^i(A) = \Ext^i(E, E)$ and $H^i(M) = \Ext^i(E, K)$. Moreover, since $F$ is an equivalence it and its quasi-inverse commute with homotopy colimits. Therefore, it suffices to write $M$ as a homotopy colimit of compact objects of $D(A, \text{d})$. By Differential Graded Algebra, Lemma \ref{dga-lemma-countable} it suffices show that $\Ext^i(E, E)$ and $\Ext^i(E, K)$ are countable for each $i$. This follows from Lemma \ref{lemma-countable-cohomology}.
By Lemma \ref{lemma-cohomology-base-change} we have $$ R\Gamma(U, Lf^*K) = R\Gamma(X, Rf_*\mathcal{O}_U \otimes^\mathbf{L} K) $$ functorially in $K$. Observe that $R\Gamma(X, -)$ commutes with homotopy colimits because it commutes with direct sums by Lemma \ref{lemma-quasi-coherence-pushforward-direct-sums}. Similarly, $- \otimes^\mathbf{L} K$ commutes with derived colimits because $- \otimes^\mathbf{L} K$ commutes with direct sums (because direct sums in $D(\mathcal{O}_X)$ are given by direct sums of representing complexes). Hence to prove (2) it suffices to write $Rf_*\mathcal{O}_U = \text{hocolim} E_n$ for a system of perfect objects $E_n$ of $D(\mathcal{O}_X)$. Once this is done we obtain (1) by setting $L_n = E_n^\vee$, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-dual-perfect-complex}. \medskip\noindent Write $A = \colim A_i$ with $A_i$ of finite type over $\mathbf{Z}$. By Limits of Spaces, Lemma \ref{spaces-limits-lemma-descend-finite-presentation} we can find an $i$ and morphisms $U_i \to X_i \to \Spec(A_i)$ of finite presentation whose base change to $\Spec(A)$ recovers $U \to X \to \Spec(A)$. After increasing $i$ we may assume that $f_i : U_i \to X_i$ is flat, see Limits of Spaces, Lemma \ref{spaces-limits-lemma-descend-flat}. By Lemma \ref{lemma-compare-base-change} the derived pullback of $Rf_{i, *}\mathcal{O}_{U_i}$ by $g : X \to X_i$ is equal to $Rf_*\mathcal{O}_U$. Since $Lg^*$ commutes with derived colimits, it suffices to prove what we want for $f_i$. Hence we may assume that $U$ and $X$ are of finite type over $\mathbf{Z}$. \medskip\noindent Assume $f : U \to X$ is a morphism of algebraic spaces of finite type over $\mathbf{Z}$. To finish the proof we will show that $Rf_*\mathcal{O}_U$ is a homotopy colimit of perfect complexes. To see this we apply Lemma \ref{lemma-countable}. Thus it suffices to show that $R^if_*\mathcal{O}_U$ has countable sets of sections over affines \'etale over $X$. This follows from Lemma \ref{lemma-countable-cohomology} applied to the structure sheaf.
Using the triangle $ \tau_{< a} E \to E \to \tau_{\geq a} E \to$ we see that the equivalence follows if we can show $$ \Hom_{D(\mathcal{O}_X)}(P[-i], \tau_{< a} E) = \Hom_{D(\mathcal{O}_X)}(P, (\tau_{< a} E)[i]) = 0 $$ for $i \gg 0$. As $P$ is perfect this is true by Lemma \ref{lemma-ext-from-perfect-into-bounded-QCoh}.
Using the triangle $ \tau_{\leq a} E \to E \to \tau_{> a} E \to$ we see that the equivalence follows if we can show $$ \Hom_{D(\mathcal{O}_X)}(P[-i], \tau_{> a} E) = \Hom_{D(\mathcal{O}_X)}(P, (\tau_{> a} E)[i]) = 0 $$ for $i \ll 0$. As $P$ is perfect this is true by Lemma \ref{lemma-ext-from-perfect-into-bounded-QCoh}.
Let $V \to W$ be an \'etale morphism with $V$ affine and $W$ a quasi-compact open subspace of $X$. Then  the morphism $V \to W$ is affine as $V$ is affine and $W$ has affine diagonal (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-affine-permanence}). Lemma \ref{lemma-affine-pushforward} then guarantees that the assumption of Lemma \ref{lemma-argument-proves} holds. Hence we conclude.
If $K$ is a perfect object of $D(\mathcal{O}_X)$ with dual $K^\vee$ (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-dual-perfect-complex}) we have $$ \Hom_{D(\mathcal{O}_X)}(K, M) = H^0(X, K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} M) $$ functorially in $M$. Since $K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} -$ commutes with direct sums and since $H^0(X, -)$ commutes with direct sums on $D_\QCoh(\mathcal{O}_X)$ by Lemma \ref{lemma-quasi-coherence-pushforward-direct-sums} we conclude that $K$ is compact in $D_\QCoh(\mathcal{O}_X)$. \medskip\noindent Conversely, let $K$ be a compact object of $D_\QCoh(\mathcal{O}_X)$. To show that $K$ is perfect, it suffices to show that $K|_U$ is perfect for every affine scheme $U$ \'etale over $X$, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-perfect-independent-representative}. Observe that $j : U \to X$ is a quasi-compact and separated morphism. Hence $Rj_* : D_\QCoh(\mathcal{O}_U) \to D_\QCoh(\mathcal{O}_X)$ commutes with direct sums, see Lemma \ref{lemma-quasi-coherence-pushforward-direct-sums}. Thus the adjointness of restriction to $U$ and $Rj_*$ implies that $K|_U$ is a perfect object of $D_\QCoh(\mathcal{O}_U)$. Hence we reduce to the case that $X$ is affine, in particular a quasi-compact and quasi-separated scheme. Via Lemma \ref{lemma-derived-quasi-coherent-small-etale-site} and \ref{lemma-descend-perfect} we reduce to the case of schemes, i.e., to Derived Categories of Schemes, Proposition \ref{perfect-proposition-compact-is-perfect}.
Observe that $E[0] \cong R\Gamma_Z(\omega_A^\bullet)$ by Lemma \ref{lemma-local-cohomology-of-dualizing}. By More on Algebra, Lemma \ref{more-algebra-lemma-completion-RHom} completion on the left hand side goes inside. Thus we have to prove $$ R\Hom_A(K^\wedge, (\omega_A^\bullet)^\wedge) = R\Hom_A(R\Gamma_Z(K), R\Gamma_Z(\omega_A^\bullet)) $$ This follows from the equivalence between $D_{comp}(A, \mathfrak m)$ and $D_{\mathfrak m^\infty\text{-torsion}}(A)$ given in Proposition \ref{proposition-torsion-complete}. More precisely, it is a special case of Lemma \ref{lemma-compare-RHom}.
Let $E_i$ be a family of injective modules parametrized by a set $I$. Set $E = \bigcup E_i$. To show that $E$ is injective we use Injectives, Lemma \ref{injectives-lemma-criterion-baer}. Thus let $\varphi : I \to E$ be a module map from an ideal of $R$ into $E$. As $I$ is a finite $R$-module (because $R$ is Noetherian) we can find finitely many elements $i_1, \ldots, i_r \in I$ such that $\varphi$ maps into $\bigcup_{j = 1, \ldots, r} E_{i_j}$. Then we can extend $\varphi$ into $\bigcup_{j = 1, \ldots, r} E_{i_j}$ using the injectivity of the modules $E_{i_j}$.
Since $R \to S^{-1}R$ is an epimorphism of rings, it suffices to show that $S^{-1}E$ is injective as an $R$-module, see Lemma \ref{lemma-injective-epimorphism}. To show this we use Injectives, Lemma \ref{injectives-lemma-criterion-baer}. Thus let $I \subset R$ be an ideal and let $\varphi : I \to S^{-1} E$ be an $R$-module map. As $I$ is a finitely presented $R$-module (because $R$ is Noetherian) we can find an $f \in S$ and an $R$-module map $I \to E$ such that $f\varphi$ is the composition $I \to E \to S^{-1}E$ (Algebra, Lemma \ref{algebra-lemma-hom-from-finitely-presented}). Then we can extend $I \to E$ to a homomorphism $R \to E$. Then the composition $$ R \to E \to S^{-1}E \xrightarrow{f^{-1}} S^{-1}E $$ is the desired extension of $\varphi$ to $R$.
We will use Lemma \ref{lemma-essential-extensions-in-injective} to prove (1). Suppose that $E \subset E' \subset I$ and that $E'$ is an essential extension of $E$. We will show that $E' = E$. If not, then we can find $x \in E'$ and $x \not \in E$. Let $J = \{ a \in R \mid ax \in E\}$. Since $R$ is Noetherian, we may choose $x$ so that $J$ is maximal among ideals of this form. Again since $R$ is Noetherian, we may write $J = (g_1, \ldots, g_t)$ for some $g_i \in R$. By definition $E$ is the set of elements of $I$ annihilated by powers of $f$, so we may choose integers $n_i$ so that $f^{n_i}g_ix = 0$. Set $n = \mathrm{max}\{ n_i \}$.  Then $x' = f^n x$ is an element of $E'$ not in $E$ and is annihilated by $J$. Then by maximality of $J$ we have $J = \{ a \in R \mid ax' \in E \} = \text{Ann}(x')$, so $Rx' \cap E  = 0$. Hence $E'$ is not an essential extension of $E$, a contradiction. \medskip\noindent To prove (2) write $J = (f_1, \ldots, f_t)$. Then $I[J^\infty]$ is equal to $$ (\ldots((I[f_1^\infty])[f_2^\infty])\ldots)[f_t^\infty] $$ and the result follows from (1) and induction.
Let us write $E[x] = E \otimes_A A[x]$. Consider the short exact sequence of $A[x]$-modules $$ 0 \to E[x] \to \Hom_A(A[x], E[x]) \to \Hom_A(A[x], E[x]) \to 0 $$ where the first map sends $p \in E[x]$ to $f \mapsto fp$ and the second map sends $\varphi$ to $f \mapsto \varphi(xf) - x\varphi(f)$. The second map is surjective because $\Hom_A(A[x], E[x]) = \prod_{n \geq 0} E[x]$ as an abelian group and the map sends $(e_n)$ to $(e_{n + 1} - xe_n)$ which is surjective. As an $A$-module we have $E[x] \cong \bigoplus_{n \geq 0} E$ which is injective by Lemma \ref{lemma-sum-injective-modules}. Hence the $A[x]$-module $\Hom_A(A[x], E[x])$ is injective by Lemma \ref{lemma-hom-injective} and the proof is complete.
Let $M$ be a finite $R$-module. Let $r = \dim_\kappa(M/\mathfrak m M)$. Choose $x_1, \ldots, x_r \in M$ mapping to a basis of $M/\mathfrak m M$. Consider the map $f : R^{\oplus r} \to M$. By Nakayama's lemma this is a surjection (Algebra, Lemma \ref{algebra-lemma-NAK}). If $N \subset R^{\oplus r}$ is a proper submodule, then $N/\mathfrak m N \to \kappa^{\oplus r}$ is not surjective (by Nakayama's lemma again) hence $N/\mathfrak m N \to M/\mathfrak m M$ is not surjective. Thus $f$ is an essential surjection.
Let $M$ be an $R$-module. By More on Algebra, Section \ref{more-algebra-section-injectives-modules} the category of $R$-modules has enough injectives. Choose an injection $M \to I$ with $I$ an injective $R$-module. Consider the set $\mathcal{S}$ of submodules $M \subset E \subset I$ such that $E$ is an essential extension of $M$. We order $\mathcal{S}$ by inclusion. If $\{E_\alpha\}$ is a totally ordered subset of $\mathcal{S}$, then $\bigcup E_\alpha$ is an essential extension of $M$ too (Lemma \ref{lemma-union-essential-extensions}). Thus we can apply Zorn's lemma and find a maximal element $E \in \mathcal{S}$. We claim $M \subset E$ is an injective hull, i.e., $E$ is an injective $R$-module. This follows from Lemma \ref{lemma-essential-extensions-in-injective}.
Part (1) follows from the fact that $E'$ is an injective $R$-module. Part (2) follows as $\Ker(\psi) \cap M = 0$ and $E$ is an essential extension of $M$. Assume $\varphi$ is an essential injection. Then $E \cong \psi(E) \subset E'$ by (2) which implies $E' = \psi(E) \oplus E''$ because $E$ is injective. Since $E'$ is an essential extension of $M$ (Lemma \ref{lemma-essential}) we get $E'' = 0$. Part (4) is a special case of (3). Assume $M \to I$ as in (5). Choose a map $\alpha : E \to I$ extending the map $M \to I$. Arguing as before we see that $\alpha$ is injective. Thus as before $\alpha(E)$ splits off from $I$. This proves (5).
Part (1) follows from Lemma \ref{lemma-injective-hull-unique}. Part (2) follows from part (1) and the definition of injective hulls. \medskip\noindent Proof of (3). Set $A = \text{End}_R(E, E)$ and $I = \{\varphi \in A \mid \Ker(f) \not = 0\}$. The statement means that $I$ is a two sided ideal and that any $\varphi \in A$, $\varphi \not \in I$ is invertible. Suppose $\varphi$ and $\psi$ are not injective. Then $\Ker(\varphi) \cap \Ker(\psi)$ is nonzero by (2). Hence $\varphi + \psi \in I$. It follows that $I$ is a two sided ideal. If $\varphi \in A$, $\varphi \not \in I$, then $E \cong \varphi(E) \subset E$ is an injective submodule, hence $E = \varphi(E)$ because $E$ is indecomposable. \medskip\noindent Proof of (4). Consider the ring map $R \to A$ and let $\mathfrak p \subset R$ be the inverse image of the maximal ideal $I$. Then it is clear that $\mathfrak p$ is a prime ideal and that $R \to A$ extends to $R_\mathfrak p \to A$. Thus $E$ is an $R_\mathfrak p$-module. It follows from Lemma \ref{lemma-injective-epimorphism} that $E$ is injective as an $R_\mathfrak p$-module.