hcju/stacksqa huggingface dataset:
We strongly urge the reader to find their own proof of the existence of $U$. Let $\eta \in X$ be the generic point. The restriction of $E$ to $\eta$ is isomorphic in $D(\kappa(\eta))$ to a finite complex $V^\bullet$ of finite dimensional vector spaces with zero differentials. Set $r_i = \dim_{\kappa(\eta)} V^i$. Then the perfect object $E'$ in $D(\mathcal{O}_X)$ represented by the complex with terms $\mathcal{O}_X^{\oplus r_i}$ and zero differentials becomes isomorphic to $E$ after pulling back to $\eta$. Hence by Derived Categories of Schemes, Lemma \ref{perfect-lemma-descend-relatively-perfect} there is an open neighbourhood $U$ of $\eta$ such that $E|_U$ and $E'|_U$ are isomorphic. This proves the first assertion. The second follows from the first and Lemma \ref{lemma-blowup-complex} as any nonempty open is scheme theoretically dense in the integral scheme $X$.
The assumptions tell us that the object $(\mathcal{F} \to \mathcal{G})$ of $D(\mathcal{O}_X)$ is perfect. Thus we get a $U$-admissible blowup that works for the cokernel and kernel by Lemmas \ref{lemma-blowup-complex} and \ref{lemma-blowup-pd1-derived} (to see what the complex looks like after pullback). The image is the kernel of the cokernel and hence is going to be perfect of tor dimension $\leq 1$ as well.
Recall that $D'$ is an effective Cartier divisor (Divisors, Lemma \ref{divisors-lemma-blow-up-pullback-effective-Cartier}). Observe that $\mathcal{J}_i$ pulls back to an invertible ideal sheaf on $X'$ as $X'$ dominates the blowing up in $\mathcal{J}_i$, see Divisors, Lemma \ref{divisors-lemma-blowing-up-two-ideals}. By Remark \ref{remark-complex-and-divisor-ideal} we may replace $X$ by $X'$ and assume $\mathcal{J}_i$ is invertible for all $i$. Via Lemma \ref{lemma-eta-stalks} we obtain the result from More on Algebra, Lemma \ref{more-algebra-lemma-eta-locally-free}.
In Remark \ref{remark-complex-and-divisor-ideal} we have seen that $\mathcal{J}_i$ pulls back to the corresponding ideal on $Y$. Hence $Y'$ is the strict transform of $Y$ by Divisors, Lemma \ref{divisors-lemma-strict-transform}. The final statement follows from Lemma \ref{lemma-complex-and-divisor-eta-pull} applied to $Y' \to X'$.
Over $U$ all of the modules $\Im(d^i)$ and $\Ker(d^i)$ are finite locally free, see for example the discussion in Remark \ref{remark-when-you-have-a-complex}. Let $x \in U$. Choose an open neighbourhood $x \in V \subset U$ such that $\mathcal{I}|_V$, $\mathcal{E}^i$, $\Ker(d^i)$, $\Im(d^i)$, and $H^i(\mathcal{E}^\bullet)$ are free and choose splittings for the short exact sequences $$ 0 \to \Im(d^i) \to \Ker(d^{i + 1}) \to H^{i + 1}(\mathcal{E}^\bullet) \to 0 \quad\text{and}\quad 0 \to \Ker(d^i) \to \mathcal{E}^i \to \Im(d^i) \to 0 $$ Then we see that our complex looks like $$ \ldots \to \mathcal{O}_V^{\oplus n_{i - 1} + m_i + n_i} \to \mathcal{O}_V^{\oplus n_i + m_{i + 1} + n_{i + 1}} \to \ldots $$ where the map identifies the last $n_i$ summands with the first $n_i$ summands. Thus $\mathcal{J}_i|V$ is the ideal generated by $f^{n_{i - 1} + m_i}$ where $f \in \mathcal{O}_X(V)$ is a generator for $\mathcal{I}|_V$. Thus over $V$ we are blowing up an invertible ideal, which produces the identity morphism (Divisors, Lemma \ref{divisors-lemma-blow-up-effective-Cartier-divisor}).
Arguing exactly as in the proof of Lemma \ref{lemma-complex-and-divisor-blowup} we may replace $X$ by $X'$ and $U$ by $b^{-1}(U)$ and assume that the ideals $\mathcal{J}_i$ are invertible for all $i$. \medskip\noindent Assume $\mathcal{J}_i$ invertible for all $i$ so that $b = \text{id}_X$ and $\mathcal{Q}^\bullet = \eta_{\mathcal{I}}\mathcal{E}^\bullet$. Let $x \in D \cap U$ and choose a generator $f \in \mathcal{I}_x$. Since $H^i(\mathcal{E}^\bullet)_x$ is a finite free $\mathcal{O}_{X, x}$-module for all $i$ (by choice of $U$), we see that $$ \Ker(d^i : \mathcal{E}^i_x/f^2\mathcal{E}^i_x \to \mathcal{E}^{i + 1}_x/f^2\mathcal{E}^{i + 1}_x) \to \Ker(d^i : \mathcal{E}^i_x/f\mathcal{E}^i_x \to \mathcal{E}^{i + 1}_x/f\mathcal{E}^{i + 1}_x) $$ is surjective, see More on Algebra, Lemma \ref{more-algebra-lemma-vanishing-beta}. This means that if $X = \Spec(A)$ is affine, then via Lemma \ref{lemma-eta-stalks} we may apply More on Algebra, Lemma \ref{more-algebra-lemma-eta-vanishing-beta} to get a closed subscheme $T \subset D$ with all the desired properties (some details omitted). \medskip\noindent To glue this affine local construction, we remark that in the proof of  More on Algebra, Lemma \ref{more-algebra-lemma-eta-vanishing-beta-plus} the ideal cutting out $T$ is constructed with a certain universal property. Namely, the result of More on Algebra, Lemma \ref{more-algebra-lemma-eta-locally-free} tells us that the canonical maps $$ c^i : \mathcal{Q}^i \to \mathcal{I}^i\mathcal{E}^i \oplus \mathcal{I}^{i + 1}\mathcal{E}^{i + 1} $$ are locally split. The closed subscheme $T \subset D$ is characterized by the property that a morphism of schemes $g : W \to D$ factors through $T$ if and only if $g^*\mathcal{Q}^i$ is a direct sum of sheaves compatible with the map $g^*c^i$ for all $i$. Hence it is clear that the affine locally constructed closed subschemes glue.
By openness of flatness, see More on Morphisms, Theorem \ref{more-morphisms-theorem-openness-flatness} we may replace $X$ by an open neighbourhood of $x$ and assume that $\mathcal{F}$ is flat over $S$. Next, we apply Proposition \ref{proposition-existence-complete-at-x} to find a diagram as in the statement of the proposition such that $g^*\mathcal{F}/X'/S'$ has a complete d\'evissage over $s'$. (In particular $S'$ and $X'$ are affine.) By Morphisms, Lemma \ref{morphisms-lemma-flat-permanence} we see that $g^*\mathcal{F}$ is flat over $S$ and by Lemma \ref{lemma-etale-flat-up-down} we see that it is flat over $S'$. Via Remark \ref{remark-same-notion} we deduce that $$ \Gamma(X', g^*\mathcal{F})/ \Gamma(X', \mathcal{O}_{X'})/ \Gamma(S', \mathcal{O}_{S'}) $$ has a complete d\'evissage over the prime of $\Gamma(S', \mathcal{O}_{S'})$ corresponding to $s'$. Thus Lemma \ref{lemma-complete-devissage-flat-finitely-presented-module} implies that the result of the proposition holds after replacing $S'$ by a standard open neighbourhood of $s'$.
Part (1) is a special case of Lemma \ref{lemma-finite-type-flat-over-integral}. For Part (2) choose a surjection $R[x_1, \ldots, x_n] \to S$. By Lemma \ref{lemma-finite-type-flat-algebra} we find that $M$ is finitely presented as an $R[x_1, \ldots, x_n]$-module. We conclude by Algebra, Lemma \ref{algebra-lemma-finitely-presented-over-subring}.
The implication (1) $\Rightarrow$ (2) is Lemma \ref{lemma-affine-locally-projective-pure}. Assume $\mathcal{F}$ is pure relative to $S$. Note that by Lemma \ref{lemma-finite-type-flat-pure-along-fibre-is-universal} this implies $\mathcal{F}$ remains pure after any base change. By Descent, Lemma \ref{descent-lemma-locally-projective-descends} it suffices to prove $f_*\mathcal{F}$ is fpqc locally projective on $S$. Pick $s \in S$. We will prove that the restriction of $f_*\mathcal{F}$ to an \'etale neighbourhood of $s$ is locally projective. Namely, by Lemma \ref{lemma-finite-presentation-flat-along-fibre}, after replacing $S$ by an affine elementary \'etale neighbourhood of $s$, we may assume there exists a diagram $$ \xymatrix{ X \ar[dr] & & X' \ar[ll]^g \ar[ld] \\ & S & } $$ of schemes affine and of finite presentation over $S$, where $g$ is \'etale, $X_s \subset g(X')$, and with $\Gamma(X', g^*\mathcal{F})$ a projective $\Gamma(S, \mathcal{O}_S)$-module. Note that in this case $g^*\mathcal{F}$ is universally pure over $S$, see Lemma \ref{lemma-affine-locally-projective-pure}. Hence by Lemma \ref{lemma-criterion} we see that the open $g(X')$ contains the points of $\text{Ass}_{X/S}(\mathcal{F})$ lying over $\Spec(\mathcal{O}_{S, s})$. Set $$ E = \{t \in S \mid \text{Ass}_{X_t}(\mathcal{F}_t) \subset g(X') \}. $$ By More on Morphisms, Lemma \ref{more-morphisms-lemma-relative-assassin-constructible} $E$ is a constructible subset of $S$. We have seen that $\Spec(\mathcal{O}_{S, s}) \subset E$. By Morphisms, Lemma \ref{morphisms-lemma-constructible-containing-open} we see that $E$ contains an open neighbourhood of $s$. Hence after replacing $S$ by an affine neighbourhood of $s$ we may assume that $\text{Ass}_{X/S}(\mathcal{F}) \subset g(X')$. By Lemma \ref{lemma-base-change-universally-flat} this means that $$ \Gamma(X, \mathcal{F}) \longrightarrow \Gamma(X', g^*\mathcal{F}) $$ is $\Gamma(S, \mathcal{O}_S)$-universally injective. By Algebra, Lemma \ref{algebra-lemma-pure-submodule-ML} we conclude that $\Gamma(X, \mathcal{F})$ is Mittag-Leffler as an $\Gamma(S, \mathcal{O}_S)$-module. Since $\Gamma(X, \mathcal{F})$ is countably generated and flat as a $\Gamma(S, \mathcal{O}_S)$-module, we conclude it is projective by Algebra, Lemma \ref{algebra-lemma-countgen-projective}.
The final assertion is the content of Lemma \ref{lemma-colim-F-Vect}. To prove the proposition we will check conditions (1), (2), and (3) of Lemma \ref{lemma-refine-check-h-stack}. \medskip\noindent Condition (1) holds because by definition we have glueing for the Zariski topology. \medskip\noindent To see condition (2), suppose that $f : X \to Y$ is a surjective, flat, proper morphism of finite presentation over $S$ with $Y$ affine. Since $Y, X, X \times_Y X$ are quasi-compact and quasi-separated, we can use the description of fibre categories given in the statement of the proposition. Then it is clearly enough to show that $$ \textit{Vect}(Y) \longrightarrow \textit{Vect}(X) \times_{\textit{Vect}(X \times_Y X)} \textit{Vect}(X) $$ is an equivalence (as this will imply the same for the colimits). This follows immediately from fppf descent of finite locally free modules, see Descent, Proposition \ref{descent-proposition-fpqc-descent-quasi-coherent} and Lemma \ref{descent-lemma-finite-locally-free-descends}. \medskip\noindent Condition (3) is the content of Lemmas \ref{lemma-vector-bundle-I} and \ref{lemma-vector-bundle-II}.
The equivalence between categories (1) and (2) is the restriction of the equivalence of Varieties, Theorem \ref{varieties-theorem-varieties-rational-maps}. Namely, a variety is a curve if and only if its function field has transcendence degree $1$, see for example Varieties, Lemma \ref{varieties-lemma-dimension-locally-algebraic}. \medskip\noindent The categories in (3), (4), (5), and (6) are the same. First of all, the terms ``regular'' and ``nonsingular'' are synonyms, see Properties, Definition \ref{properties-definition-regular}. Being normal and regular are the same thing for Noetherian $1$-dimensional schemes (Properties, Lemmas \ref{properties-lemma-regular-normal} and \ref{properties-lemma-normal-dimension-1-regular}). See Varieties, Lemma \ref{varieties-lemma-regular-point-on-curve} for the case of curves. Thus (3) is the same as (5). Finally, (6) is the same as (3) by Varieties, Lemma \ref{varieties-lemma-dim-1-proper-projective}. \medskip\noindent If $f : X \to Y$ is a nonconstant morphism of nonsingular projective curves, then $f$ sends the generic point $\eta$ of $X$ to the generic point $\xi$ of $Y$. Hence we obtain a morphism $k(Y) = \mathcal{O}_{Y, \xi} \to \mathcal{O}_{X, \eta} = k(X)$ in the category (1). If two morphisms $f,g: X \to Y$ gives the same morphism $k(Y) \to k(X)$, then by the equivalence between (1) and (2), $f$ and $g$ are equivalent as rational maps, so $f=g$ by Lemma \ref{lemma-extend-over-normal-curve}. Conversely, suppose that we have a map $k(Y) \to k(X)$ in the category (1). Then we obtain a morphism $U \to Y$ for some nonempty open $U \subset X$. By Lemma \ref{lemma-extend-over-dvr} this extends to all of $X$ and we obtain a morphism in the category (5). Thus we see that there is a fully faithful functor (5)$\to$(1). \medskip\noindent To finish the proof we have to show that every $K/k$ in (1) is the function field of a normal projective curve. We already know that $K = k(X)$ for some curve $X$. After replacing $X$ by its normalization (which is a variety birational to $X$) we may assume $X$ is normal (Varieties, Lemma \ref{varieties-lemma-normalization-locally-algebraic}). Then we choose $X \to \overline{X}$ with $\overline{X} \setminus X = \{x_1, \ldots, x_n\}$ as in Varieties, Lemma \ref{varieties-lemma-reduced-dim-1-projective-completion}. Since $X$ is normal and since each of the local rings $\mathcal{O}_{\overline{X}, x_i}$ is normal we conclude that $\overline{X}$ is a normal projective curve as desired. (Remark: We can also first compactify using Varieties, Lemma \ref{varieties-lemma-dim-1-projective-completion} and then normalize using Varieties, Lemma \ref{varieties-lemma-normalization-locally-algebraic}. Doing it this way we avoid using the somewhat tricky Morphisms, Lemma \ref{morphisms-lemma-relative-normalization-normal-codim-1}.)
A rational map from $X$ to $Y$ can be extended to a morphism $X \to Y$ by Lemma \ref{lemma-extend-over-dvr} as every local ring is a discrete valuation ring (for example by Varieties, Lemma \ref{varieties-lemma-regular-point-on-curve}). Conversely, if two morphisms $f,g: X \to Y$ are equivalent as rational maps, then  $f = g$ by Morphisms, Lemma \ref{morphisms-lemma-equality-of-morphisms}.
Pick $x \in X$ mapping to $y \in Y$. Then $\mathcal{O}_{Y, y}$ is either a field or a discrete valuation ring (Varieties, Lemma \ref{varieties-lemma-regular-point-on-curve}). Since $f$ is nonconstant it is dominant (as it must map the generic point of $X$ to the generic point of $Y$). This implies that $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$ is injective (Morphisms, Lemma \ref{morphisms-lemma-dominant-between-integral}). Hence $\mathcal{O}_{X, x}$ is torsion free as a $\mathcal{O}_{Y, y}$-module and therefore $\mathcal{O}_{X, x}$ is flat as a $\mathcal{O}_{Y, y}$-module by More on Algebra, Lemma \ref{more-algebra-lemma-valuation-ring-torsion-free-flat}.
The morphism $f$ is proper by Morphisms, Lemma \ref{morphisms-lemma-image-proper-scheme-closed}. Thus $f(X)$ is closed and images of closed points are closed. Let $y \in Y$ be the image of a closed point in $X$. Then $f^{-1}(\{y\})$ is a closed subset of $X$ not containing any of the generic points of irreducible components of dimension $1$ by condition (3). It follows that $f^{-1}(\{y\})$ is finite. Hence $f$ is finite over an open neighbourhood of $y$ by More on Morphisms, Lemma \ref{more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood} (if $Y$ is Noetherian, then you can use the easier Cohomology of Schemes, Lemma \ref{coherent-lemma-proper-finite-fibre-finite-in-neighbourhood}). Since we've seen above that there are enough of these points $y$, the proof is complete.
Let $X'$ be a nonsingular projective model of $X$. Then $X'$ and $X$ have isomorphic nonempty open subschemes. In particular $X'$ is geometrically irreducible as $X$ is (some details omitted). Thus we may assume that $X$ is projective. \medskip\noindent Assume $X$ is proper. Then $X_K$ is proper and hence the normalization $(X_K)^\nu$ is proper as a scheme finite over a proper scheme (Varieties, Lemma \ref{varieties-lemma-normalization-locally-algebraic} and Morphisms, Lemmas \ref{morphisms-lemma-finite-proper} and \ref{morphisms-lemma-composition-proper}). On the other hand, $X_K$ is irreducible as $X$ is geometrically irreducible. Hence $X_K^\nu$ is proper, normal, irreducible, and birational to $(X_K)_{red}$. This proves (1) because a proper curve is projective (Varieties, Lemma \ref{varieties-lemma-dim-1-proper-projective}). \medskip\noindent Proof of (2). As $X$ is proper and we have (1), we can apply Varieties, Lemma \ref{varieties-lemma-finite-extension-geometrically-normal} to find $K/k$ finite purely inseparable such that $Y_K$ is geometrically normal. Then $Y_K$ is geometrically regular as normal and regular are the same for curves (Properties, Lemma \ref{properties-lemma-normal-dimension-1-regular}). Then $Y$ is a smooth variety by Varieties, Lemma \ref{varieties-lemma-geometrically-regular-smooth}. \medskip\noindent If $Y_K$ is geometrically reduced, then $Y_K$ is geometrically integral (Varieties, Lemma \ref{varieties-lemma-geometrically-integral}) and we see that $H^0(Y_K, \mathcal{O}_{Y_K}) = K$ by Varieties, Lemma \ref{varieties-lemma-regular-functions-proper-variety}. This proves (3) because a smooth variety is geometrically reduced (even geometrically regular, see Varieties, Lemma \ref{varieties-lemma-geometrically-regular-smooth}). \medskip\noindent If $Y_K$ is smooth, then for every extension $\Omega/K$ the base change $(Y_K)_\Omega$ is smooth over $\Omega$ (Morphisms, Lemma \ref{morphisms-lemma-base-change-smooth}). Hence it is clear that $Y_\Omega = (Y_K)_\Omega$. This proves (4).
Let $s_0, \ldots, s_r \in V$ be a $k$-basis. Since $X$ is nonsingular the image $\mathcal{L}' \subset \mathcal{L}$ of the map $s_0, \ldots, s_r : \mathcal{O}_X^{\oplus r} \to \mathcal{L}$ is an invertible $\mathcal{O}_X$-module for example by Divisors, Lemma \ref{divisors-lemma-torsion-free-over-regular-dim-1}. Then we use Constructions, Lemma \ref{constructions-lemma-projective-space} to get a morphism $$ \varphi = \varphi_{(\mathcal{L}', (s_0, \ldots, s_r))} : X \longrightarrow \mathbf{P}^r_k $$ as in the statement of the lemma.
By Lemma \ref{lemma-linear-series} we see that $\mathcal{L}' = \varphi^*\mathcal{O}_{\mathbf{P}^1_k}(1)$ has a nonzero map $\mathcal{L}' \to \mathcal{L}$. Hence by Varieties, Lemma \ref{varieties-lemma-check-invertible-sheaf-trivial} we see that $0 \leq \deg(\mathcal{L}') \leq d$. If $\deg(\mathcal{L}') = 0$, then the same lemma tells us $\mathcal{L}' \cong \mathcal{O}_X$ and since we have two linearly independent sections we find we are in case (2). If $\deg(\mathcal{L}') > 0$ then $\varphi$ is nonconstant (since the pullback of an invertible module by a constant morphism is trivial). Hence $$ \deg(\mathcal{L}') = \deg(X/\mathbf{P}^1_k) \deg(\mathcal{O}_{\mathbf{P}^1_k}(1)) $$ by Varieties, Lemma \ref{varieties-lemma-degree-pullback-map-proper-curves}. This finishes the proof as the degree of $\mathcal{O}_{\mathbf{P}^1_k}(1)$ is $1$.
Let $(\mathcal{L}, V)$ be a $\mathfrak g^r_d$. Since this will only increase $r$, we may assume $V = H^0(X, \mathcal{L})$. Choose a nonzero element $s \in V$.  Then the zero scheme of $s$ is an effective Cartier divisor $D \subset X$, we have $\mathcal{L} = \mathcal{O}_X(D)$, and we have a short exact sequence $$ 0 \to \mathcal{O}_X \to \mathcal{L} \to \mathcal{L}|_D \to 0 $$ see Divisors, Lemma \ref{divisors-lemma-characterize-OD} and Remark \ref{divisors-remark-ses-regular-section}. By Varieties, Lemma \ref{varieties-lemma-degree-effective-Cartier-divisor} we have $\deg(D) = \deg(\mathcal{L}) = d$. Since $D$ is an Artinian scheme we have $\mathcal{L}|_D \cong \mathcal{O}_D$\footnote{In our case this follows from Divisors, Lemma \ref{divisors-lemma-finite-trivialize-invertible-upstairs} as $D \to \Spec(k)$ is finite.}. Thus $$ \dim_k H^0(D, \mathcal{L}|_D) = \dim_k H^0(D, \mathcal{O}_D) = \deg(D) = d $$ On the other hand, by assumption $\dim_k H^0(X, \mathcal{O}_X) = 1$ and $\dim H^0(X, \mathcal{L}) = r + 1$. We conclude that $r + 1 \leq 1 + d$, i.e., $r \leq d$ as in the lemma. \medskip\noindent Assume equality holds. Then $H^0(X, \mathcal{L}) \to H^0(X, \mathcal{L}|_D)$ is surjective. If we knew that $H^1(X, \mathcal{L})$ was zero, then we would conclude that $H^1(X, \mathcal{O}_X)$ is zero by the long exact cohomology sequence and the proof would be complete. Our strategy will be to replace $\mathcal{L}$ by a large power which has vanishing. As $\mathcal{L}|_D$ is the trivial invertible module (see above), we can find a section $t$ of $\mathcal{L}$ whose restriction of $D$ generates $\mathcal{L}|_D$. Consider the multiplication map $$ \mu : H^0(X, \mathcal{L}) \otimes_k H^0(X, \mathcal{L}) \longrightarrow H^0(X, \mathcal{L}^{\otimes 2}) $$ and consider the short exact sequence $$ 0 \to \mathcal{L} \xrightarrow{s} \mathcal{L}^{\otimes 2} \to \mathcal{L}^{\otimes 2}|_D \to 0 $$ Since $H^0(\mathcal{L}) \to H^0(\mathcal{L}|_D)$ is surjective and since $t$ maps to a trivialization of $\mathcal{L}|_D$ we see that $\mu(H^0(X, \mathcal{L}) \otimes t)$ gives a subspace of $H^0(X, \mathcal{L}^{\otimes 2})$ surjecting onto the global sections of $\mathcal{L}^{\otimes 2}|_D$. Thus we see that $$ \dim H^0(X, \mathcal{L}^{\otimes 2}) = r + 1 + d = 2r + 1 = \deg(\mathcal{L}^{\otimes 2}) + 1 $$ Ok, so $\mathcal{L}^{\otimes 2}$ has the same property as $\mathcal{L}$, i.e., that the dimension of the space of global sections is equal to the degree plus one. Since $\mathcal{L}$ is ample (Varieties, Lemma \ref{varieties-lemma-ample-curve}) there exists some $n_0$ such that $\mathcal{L}^{\otimes n}$ has vanishing $H^1$ for all $n \geq n_0$ (Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-proper-ample}). Thus applying the argument above to $\mathcal{L}^{\otimes n}$ with $n = 2^m$ for some sufficiently large $m$ we conclude the lemma is true.
Denote $f : X \to \Spec(k)$ the structure morphism. We start with the relative dualizing complex $$ \omega_X^\bullet = \omega_{X/k}^\bullet = a(\mathcal{O}_{\Spec(k)}) $$ as described in Duality for Schemes, Remark \ref{duality-remark-relative-dualizing-complex}. Then property (4) holds by construction as $a$ is the right adjoint for $f_* : D_\QCoh(\mathcal{O}_X) \to D(\mathcal{O}_{\Spec(k)})$. Since $f$ is proper we have $f^!(\mathcal{O}_{\Spec(k)}) = a(\mathcal{O}_{\Spec(k)})$ by definition, see Duality for Schemes, Section \ref{duality-section-upper-shriek}. Hence $\omega_X^\bullet$ and $\omega_X$ are as in Duality for Schemes, Example \ref{duality-example-proper-over-local} and as in Duality for Schemes, Example \ref{duality-example-equidimensional-over-field}. Parts (1) and (2) follow from Duality for Schemes, Lemma \ref{duality-lemma-vanishing-good-dualizing}. For a closed point $x \in X$ we see that $\omega_{X, x}^\bullet$ is a normalized dualizing complex over $\mathcal{O}_{X, x}$, see Duality for Schemes, Lemma \ref{duality-lemma-good-dualizing-normalized}. Assertion (3) then follows from Dualizing Complexes, Lemma \ref{dualizing-lemma-apply-CM}. Assertion (5) follows from Duality for Schemes, Lemma \ref{duality-lemma-dualizing-module-proper-over-A} for coherent $\mathcal{F}$ and in general by unwinding (4) for $K = \mathcal{F}[0]$ and $i = -1$. Assertion (6) follows from Duality for Schemes, Lemma \ref{duality-lemma-smooth-proper}.
Recall from the proof of Lemma \ref{lemma-duality-dim-1} that $\omega_X$ is as in Duality for Schemes, Example \ref{duality-example-proper-over-local} and hence is a dualizing module. The other statements follow from Lemma \ref{lemma-duality-dim-1} and the fact that $\omega_X^\bullet = \omega_X[1]$ as $X$ is Cohen-Macualay (Duality for Schemes, Lemma \ref{duality-lemma-dualizing-module-CM-scheme}).
Denote $f : X \to \Spec(k)$ the structure morphism. Assertion (1) really means that $\omega_X^\bullet$ and $\omega_X$ are as in Lemma \ref{lemma-duality-dim-1} for the morphism $f' : X \to \Spec(k')$. In the proof of Lemma \ref{lemma-duality-dim-1} we took $\omega_X^\bullet = a(\mathcal{O}_{\Spec(k)})$ where $a$ be is the right adjoint of Duality for Schemes, Lemma \ref{duality-lemma-twisted-inverse-image} for $f$. Thus we have to show $a(\mathcal{O}_{\Spec(k)}) \cong a'(\mathcal{O}_{\Spec(k)})$ where $a'$ be is the right adjoint of Duality for Schemes, Lemma \ref{duality-lemma-twisted-inverse-image} for $f'$. Since $k' \subset H^0(X, \mathcal{O}_X)$ we see that $k'/k$ is a finite extension (Cohomology of Schemes, Lemma \ref{coherent-lemma-proper-over-affine-cohomology-finite}). By uniqueness of adjoints we have $a = a' \circ b$ where $b$ is the right adjoint of Duality for Schemes, Lemma \ref{duality-lemma-twisted-inverse-image} for $g : \Spec(k') \to \Spec(k)$. Another way to say this: we have $f^! = (f')^! \circ g^!$. Thus it suffices to show that $\Hom_k(k', k) \cong k'$ as $k'$-modules, see Duality for Schemes, Example \ref{duality-example-affine-twisted-inverse-image}. This holds because these are $k'$-vector spaces of the same dimension (namely dimension $1$). \medskip\noindent Proof of (2). This holds because we have base change for $a$ by Duality for Schemes, Lemma \ref{duality-lemma-more-base-change}. See discussion in Duality for Schemes, Remark \ref{duality-remark-relative-dualizing-complex}.