hcju/stacksqa huggingface dataset:
Part (1) is clear since $\mathcal{O}_U$ is just the restriction of $\mathcal{O}$, so Lemmas \ref{lemma-locally-ringed-intrinsic} and \ref{lemma-locally-ringed-intrinsic-morphism} apply. Part (2) is clear as the morphism $(j, j^\sharp)$ is actually a localization of a locally ringed site so (1) applies. Part (3) is clear also since $(f')^\sharp$ is just the restriction of $f^\sharp$ to the topos $\Sh(\mathcal{C})/\mathcal{F}$, see proof of Lemma \ref{lemma-localize-morphism-ringed-topoi} (hence the diagram of Definition \ref{definition-morphism-locally-ringed-topoi} for the morphism $f'$ is just the restriction of the corresponding diagram for $f$, and restriction is an exact functor). Part (4) follows formally on combining (2) and (3). Parts (5), (6), (7), and (8) follow from their counterparts (1), (2), (3), and (4) by enlarging the sites as in Lemma \ref{lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites} and translating everything in terms of sites and morphisms of sites using the comparisons of Lemmas \ref{lemma-localize-compare}, \ref{lemma-relocalize-compare}, \ref{lemma-localize-morphism-compare}, and \ref{lemma-relocalize-morphism-compare}. (Alternatively one could use the same arguments as in the proofs of (1), (2), (3), and (4) to prove (5), (6), (7), and (8) directly.)
Let $U$ be an object of $\mathcal{C}$. For any $\mathcal{O}_\mathcal{D}$-module $\mathcal{G}$ we have \begin{align*} \Hom_{\mathcal{O}_\mathcal{C}}(j_{U!}\mathcal{O}_U, g^{-1}\mathcal{G}) & = g^{-1}\mathcal{G}(U) \\ & = \mathcal{G}(u(U)) \\ & = \Hom_{\mathcal{O}_\mathcal{D}}(j_{u(U)!}\mathcal{O}_{u(U)}, \mathcal{G}) \end{align*} because $g^{-1}$ is described by restriction, see Sites, Lemma \ref{sites-lemma-when-shriek}. Of course a similar formula holds a direct sum of modules of the form $j_{U!}\mathcal{O}_U$. By Homology, Lemma \ref{homology-lemma-partially-defined-adjoint} and Lemma \ref{lemma-module-quotient-flat} we see that $g_!$ exists.
We have $(g')^*\mathcal{F} = (g')^{-1}\mathcal{F}$ and $g^*\mathcal{G} = g^{-1}\mathcal{G}$ because of condition (5). Thus the first equality follows immediately from the corresponding equality in Sites, Lemma \ref{sites-lemma-special-square-cocontinuous}. Since the left adjoint functors $g_!$ and $g'_!$ to $g^*$ and $(g')^*$ exist by Lemma \ref{lemma-lower-shriek-modules} we see that the second equality follows by uniqueness of adjoint functors.
We have $(g')^*\mathcal{F} = (g')^{-1}\mathcal{F}$ and $g^*\mathcal{G} = g^{-1}\mathcal{G}$ because of condition (5). Thus the first equality follows immediately from the corresponding equality in Sites, Lemma \ref{sites-lemma-special-square-continuous}. Since the left adjoint functors $g_!$ and $g'_!$ to $g^*$ and $(g')^*$ exist by Lemma \ref{lemma-lower-shriek-modules} we see that the second equality follows by uniqueness of adjoint functors.
Choose a presentation $\Lambda^{\oplus m} \to \Lambda^{\oplus n} \to Q \to 0$. This gives an exact sequence $M^{\oplus m} \to M^{\oplus n} \to M \otimes Q \to 0$. By Lemma \ref{lemma-constant-exact} we obtain an exact sequence $$ \underline{M}(U)^{\oplus m} \to \underline{M}(U)^{\oplus n} \to \underline{M \otimes Q}(U) \to 0 $$ which proves the lemma. (Note that taking sections over $U$ always commutes with finite direct sums, but not arbitrary direct sums.)
Let $I \subset \Lambda$ be a finitely generated ideal. By Algebra, Lemma \ref{algebra-lemma-flat} it suffices to show that $\underline{M}(U) \otimes_\Lambda I \to \underline{M}(U)$ is injective. As $\Lambda$ is coherent $I$ is finitely presented as a $\Lambda$-module. By Lemma \ref{lemma-tensor-with-finitely-presented} we see that $\underline{M}(U) \otimes I = \underline{M \otimes I}$. Since $M$ is flat the map $M \otimes I \to M$ is injective, whence $\underline{M \otimes I} \to \underline{M}$ is injective.
To prove $\underline{\Lambda}^\wedge$ is flat, it suffices to show that $\underline{\Lambda}^\wedge(U)$ is flat as a $\Lambda$-module for each $U \in \Ob(\mathcal{C})$, see Lemmas \ref{lemma-flatness-presheaves} and \ref{lemma-flatness-sheafification}. By Lemma \ref{lemma-flat-sections} we see that $$ \underline{\Lambda}^\wedge(U) = \lim \underline{\Lambda/I^n}(U) $$ is a limit of a system of flat $\Lambda/I^n$-modules. By Lemma \ref{lemma-constant-exact} we see that the transition maps are surjective. We conclude by More on Algebra, Lemma \ref{more-algebra-lemma-limit-flat}. \medskip\noindent To see the equalities, note that $\underline{\Lambda}(U)/I\underline{\Lambda}(U) = \underline{\Lambda/I}(U)$ by Lemma \ref{lemma-tensor-with-finitely-presented}. It follows that $\underline{\Lambda}/I\underline{\Lambda} = \underline{\Lambda}/\underline{I} = \underline{\Lambda/I}$. The system of short exact sequences $$ 0 \to \underline{I/I^n}(U) \to \underline{\Lambda/I^n}(U) \to \underline{\Lambda/I}(U) \to 0 $$ has surjective transition maps, hence gives a short exact sequence $$ 0 \to \lim \underline{I/I^n}(U) \to \lim \underline{\Lambda/I^n}(U) \to \lim \underline{\Lambda/I}(U) \to 0 $$ see Homology, Lemma \ref{homology-lemma-Mittag-Leffler}. Thus we see that $\underline{\Lambda}^\wedge/\underline{I}^\wedge = \underline{\Lambda/I}$. Since $$ I \underline{\Lambda}^\wedge \subset \underline{I} \cdot \underline{\Lambda}^\wedge \subset \underline{I}^\wedge $$ it suffices to show that $I \underline{\Lambda}^\wedge(U) = \underline{I}^\wedge(U)$ for all $U$. Choose generators $I = (f_1, \ldots, f_r)$. This gives a short exact sequence $0 \to K \to \Lambda^{\oplus r} \to I \to 0$. We obtain short exact sequences $$ 0 \to \underline{(K \cap I^n)/I^nK}(U) \to \underline{(\Lambda/I^n)^{\oplus r}}(U) \to \underline{I/I^n}(U) \to 0 $$ By Artin-Rees (Algebra, Lemma \ref{algebra-lemma-Artin-Rees}) the system of modules on the left hand side has ML. (It is zero as a pro-object.) Thus we see that $(\underline{\Lambda}^\wedge)^{\oplus r}(U) \to \underline{I}^\wedge(U)$ is surjective by Homology, Lemma \ref{homology-lemma-Mittag-Leffler} which is what we wanted to show.
Proof of (1). If $M$ is generated by $x_1, \ldots, x_r$ then $x_1, \ldots, x_r$ define global sections of $\underline{M}$ which generate it, hence $\underline{M}$ is of finite type. Conversely, assume $\underline{M}$ is of finite type. Let $U \in \mathcal{C}$ be an object which is not sheaf theoretically empty (Sites, Definition \ref{sites-definition-empty}). Such an object exists as we assumed $\Sh(\mathcal{C})$ is not the empty topos. Then there exists a covering $\{U_i \to U\}$ and finitely many sections $s_{ij} \in \underline{M}(U_i)$ generating $\underline{M}|_{U_i}$. After refining the covering we may assume that $s_{ij}$ come from elements $x_{ij}$ of $M$. Then $x_{ij}$ define global sections of $\underline{M}$ whose restriction to $U$ generate $\underline{M}$. \medskip\noindent Assume there exist elements $x_1, \ldots, x_r$ of $M$ which define global sections of $\underline{M}$ generating $\underline{M}$ as a sheaf of $\underline{\Lambda}$-modules. We will show that $x_1, \ldots, x_r$ generate $M$ as a $\Lambda$-module. Let $x \in M$. We can find a covering $\{U_i \to U\}_{i \in I}$ and $f_{i, j} \in \underline{\Lambda}(U_i)$ such that $x|_{U_i} = \sum f_{i, j} x_j|_{U_i}$. After refining the covering we may assume $f_{i, j} \in \Lambda$. Since $U$ is not sheaf theoretically empty we see that $I \not = \emptyset$. Thus we can pick $i \in I$ and we see that $x = \sum f_{i, j}x_j$ in $M$ as desired. \medskip\noindent Proof of (2). Assume $\underline{M}$ is a $\underline{\Lambda}$-module of finite presentation. By (1) we see that $M$ is of finite type. Choose generators $x_1, \ldots, x_r$ of $M$ as a $\Lambda$-module. This determines a short exact sequence $0 \to K \to \Lambda^{\oplus r} \to M \to 0$ which turns into a short exact sequence $$ 0 \to \underline{K} \to \underline{\Lambda}^{\oplus r} \to \underline{M} \to 0 $$ by Lemma \ref{lemma-constant-exact}. By Lemma \ref{lemma-kernel-surjection-finite-onto-finite-presentation} we see that $\underline{K}$ is of finite type. Hence $K$ is a finite $\Lambda$-module by (1). Thus $M$ is a $\Lambda$-module of finite presentation.
Proof of (1). Set $E = \Hom_\Lambda(M, N)$. We want to show the canonical map $$ \underline{E} \longrightarrow \SheafHom_{\underline{\Lambda}}(\underline{M}, \underline{N}) $$ is an isomorphism. The module $M$ has a presentation $\Lambda^{\oplus s} \to \Lambda^{\oplus t} \to M \to 0$. Then $E$ sits in an exact sequence $$ 0 \to E \to \Hom_\Lambda(\Lambda^{\oplus t}, N) \to \Hom_\Lambda(\Lambda^{\oplus s}, N) $$ and we have similarly $$ 0 \to \SheafHom_{\underline{\Lambda}}(\underline{M}, \underline{N}) \to \SheafHom_{\underline{\Lambda}}(\underline{\Lambda^{\oplus t}}, \underline{N}) \to \SheafHom_{\underline{\Lambda}}(\underline{\Lambda^{\oplus s}}, \underline{N}) $$ This reduces the question to the case where $M$ is a finite free module where the result is clear. \medskip\noindent Proof of (3). The question is local on $\mathcal{C}$, hence we may assume $\mathcal{F} = \underline{M}$ and $\mathcal{G} = \underline{N}$ for some $\Lambda$-modules $M$ and $N$. By Lemma \ref{lemma-locally-constant-finite-type} the module $M$ is of finite presentation. Thus the result follows from (1). \medskip\noindent Parts (2) and (4) follow from parts (1) and (3) and the fact that $\mathit{Isom}$ can be viewed as the subsheaf of sections of $\SheafHom_{\underline{\Lambda}}(\mathcal{F}, \mathcal{G})$ which have an inverse in $\SheafHom_{\underline{\Lambda}}(\mathcal{G}, \mathcal{F})$.
Proof of (1). We may work locally on $\mathcal{C}$. Hence by Lemma \ref{lemma-morphism-locally-constant} we may assume we are given a finite diagram of finite sets such that our diagram of sheaves is the associated diagram of constant sheaves. Then we just take the limit or colimit in the category of sets and take the associated constant sheaf. Some details omitted. \medskip\noindent To prove (2) and (3) we use the criterion of Homology, Lemma \ref{homology-lemma-characterize-weak-serre-subcategory}. Existence of kernels and cokernels is argued in the same way as above. Of course, the reason for using a Noetherian ring in (3) is to assure us that the kernel of a map of finite $\Lambda$-modules is a finite $\Lambda$-module. To see that the category is closed under extensions (in the case of sheaves $\Lambda$-modules), assume given an extension of sheaves of $\Lambda$-modules $$ 0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{G} \to 0 $$ on $\mathcal{C}$ with $\mathcal{F}$, $\mathcal{G}$ finite type and locally constant. Localizing on $\mathcal{C}$ we may assume $\mathcal{F}$ and $\mathcal{G}$ are constant, i.e., we get $$ 0 \to \underline{M} \to \mathcal{E} \to \underline{N} \to 0 $$ for some $\Lambda$-modules $M, N$. Choose generators $y_1, \ldots, y_m$ of $N$, so that we get a short exact sequence $0 \to K \to \Lambda^{\oplus m} \to N \to 0$ of $\Lambda$-modules. Localizing further we may assume $y_j$ lifts to a section $s_j$ of $\mathcal{E}$. Thus we see that $\mathcal{E}$ is a pushout as in the following diagram $$ \xymatrix{ 0 \ar[r] & \underline{K} \ar[d] \ar[r] & \underline{\Lambda^{\oplus m}} \ar[d] \ar[r] & \underline{N} \ar[d] \ar[r] & 0 \\ 0 \ar[r] & \underline{M} \ar[r] & \mathcal{E} \ar[r] & \underline{N} \ar[r] & 0 } $$ By Lemma \ref{lemma-morphism-locally-constant} again (and the fact that $K$ is a finite $\Lambda$-module as $\Lambda$ is Noetherian) we see that the map $\underline{K} \to \underline{M}$ is locally constant, hence we conclude.
We have $\Omega_{S'/S} = 0$ and $\Omega_{X'/X} = 0$, see for example Morphisms, Lemma \ref{morphisms-lemma-etale-at-point}. Then by the short exact sequences of Morphisms, Lemmas \ref{morphisms-lemma-triangle-differentials} and \ref{morphisms-lemma-triangle-differentials-smooth} we see that $\Omega_{X'/S'} = \Omega_{X'/S} = f^*\Omega_{X/S}$. Taking exterior powers we conclude.
There is a spectral sequence with first page $E_1^{a, b} = R^ap_*\Omega^q_{X/S}$ converging to $Rp_*\Omega^\bullet_{X/S}$ (see Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}). Hence by Homology, Lemma \ref{homology-lemma-first-quadrant-ss} it suffices to show that $R^ap_*\Omega^q_{X/S}$ is quasi-coherent. This follows from Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherence-higher-direct-images}.
In this case by Morphisms, Lemma \ref{morphisms-lemma-finite-type-differentials} the modules $\Omega^i_{X/S}$ are coherent. Hence we can use exactly the same argument as in the proof of Lemma \ref{lemma-quasi-coherence-relative} using Cohomology of Schemes, Proposition \ref{coherent-proposition-proper-pushforward-coherent}.
Since $f$ is smooth the modules $\Omega^p_{X/S}$ are finite locally free $\mathcal{O}_X$-modules, see Morphisms, Lemma \ref{morphisms-lemma-smooth-omega-finite-locally-free}. Their formation commutes with arbitrary change of base by Lemma \ref{lemma-base-change-de-rham}. Hence $Rf_*\Omega^p_{X/S}$ is a perfect object of $D(\mathcal{O}_S)$ whose formation commutes with abitrary base change, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-flat-proper-perfect-direct-image-general}. This proves the first assertion of the lemma. \medskip\noindent To prove that $Rf_*\Omega^\bullet_{X/S}$ is perfect on $S$ we may work locally on $S$. Thus we may assume $S$ is quasi-compact. This means we may assume that $\Omega^n_{X/S}$ is zero for $n$ large enough. For every $p \geq 0$ we claim that $Rf_*\sigma_{\geq p}\Omega^\bullet_{X/S}$ is a perfect object of $D(\mathcal{O}_S)$ whose formation commutes with arbitrary change of base. By the above we see that this is true for $p \gg 0$. Suppose the claim holds for $p$ and consider the distinguished triangle $$ \sigma_{\geq p}\Omega^\bullet_{X/S} \to \sigma_{\geq p - 1}\Omega^\bullet_{X/S} \to \Omega^{p - 1}_{X/S}[-(p - 1)] \to (\sigma_{\geq p}\Omega^\bullet_{X/S})[1] $$ in $D(f^{-1}\mathcal{O}_S)$. Applying the exact functor $Rf_*$ we obtain a distinguished triangle in $D(\mathcal{O}_S)$. Since we have the 2-out-of-3 property for being perfect (Cohomology, Lemma \ref{cohomology-lemma-two-out-of-three-perfect}) we conclude $Rf_*\sigma_{\geq p - 1}\Omega^\bullet_{X/S}$ is a perfect object of $D(\mathcal{O}_S)$. Similarly for the commutation with arbitrary base change.
We know that $ \Omega_{X \times_S Y/S} = p^*\Omega_{X/S} \oplus q^*\Omega_{Y/S} $ by Morphisms, Lemma \ref{morphisms-lemma-differential-product}. Taking exterior powers we obtain $$ \Omega^n_{X \times_S Y/S} = \bigoplus\nolimits_{i + j = n} p^*\Omega^i_{X/S} \otimes_{\mathcal{O}_{X \times_S Y}} q^*\Omega^j_{Y/S} = \bigoplus\nolimits_{i + j = n} \Omega^i_{X/S} \boxtimes \Omega^j_{Y/S} $$ by elementary properties of exterior powers. These identifications determine isomorphisms between the terms of the complexes on the left and the right of the arrow in the lemma. We omit the verification that these maps are compatible with differentials.
By Morphisms, Lemma \ref{morphisms-lemma-smooth-omega-finite-locally-free} the sheaves $\Omega^n_{X/S}$ and $\Omega^m_{Y/S}$ are finite locally free $\mathcal{O}_X$ and $\mathcal{O}_Y$-modules. On the other hand, $X$ and $Y$ are flat over $S$ (Morphisms, Lemma \ref{morphisms-lemma-smooth-flat}) and hence we find that $\Omega^n_{X/S}$ and $\Omega^m_{Y/S}$ are flat over $S$. Also, observe that $\Omega^\bullet_{X/S}$ is a locally bounded. Thus the result by Lemma \ref{lemma-de-rham-complex-product} and Derived Categories of Schemes, Lemma \ref{perfect-lemma-kunneth-special}.
We have constructed the map $\Omega^\bullet_{X/S} \to \Omega^\bullet_{L^\star/S, 0}$ above. \medskip\noindent Construction of $\text{Res} : \Omega^\bullet_{L^\star/S, 0} \to \Omega^\bullet_{X/S}[-1]$. Let $U \subset X$ be an open and let $s \in \mathcal{L}(U)$ and $s' \in \mathcal{L}^{\otimes -1}(U)$ be sections such that $s' s = 1$. Then $s$ gives an invertible section of the sheaf of algebras $(L^\star \to X)_*\mathcal{O}_{L^\star}$ over $U$ with inverse $s' = s^{-1}$. Then we can consider the $1$-form $\text{d}\log(s) = s' \text{d}(s)$ which is an element of $\Omega^1_{L^\star/S, 0}(U)$ by our construction of the grading on $\Omega^1_{L^\star/S}$. Our computations on affines given below will show that $1$ and $\text{d}\log(s)$ freely generate $\Omega^\bullet_{L^\star/S, 0}|_U$ as a right module over $\Omega^\bullet_{X/S}|_U$. Thus we can define $\text{Res}$ over $U$ by the rule $$ \text{Res}(\omega' + \text{d}\log(s) \wedge \omega) = \omega $$ for all $\omega', \omega \in \Omega^\bullet_{X/S}(U)$. This map is independent of the choice of local generator $s$ and hence glues to give a global map. Namely, another choice of $s$ would be of the form $gs$ for some invertible $g \in \mathcal{O}_X(U)$ and we would get $\text{d}\log(gs) = g^{-1}\text{d}(g) + \text{d}\log(s)$ from which the independence easily follows. Finally, observe that our rule for $\text{Res}$ is compatible with differentials as $\text{d}(\omega' + \text{d}\log(s) \wedge \omega) = \text{d}(\omega') - \text{d}\log(s) \wedge \text{d}(\omega)$ and because the differential on $\Omega^\bullet_{X/S}[-1]$ sends $\omega'$ to $-\text{d}(\omega')$ by our sign convention in Homology, Definition \ref{homology-definition-shift-cochain}. \medskip\noindent Local computation. We can cover $X$ by affine opens $U \subset X$ such that $\mathcal{L}|_U \cong \mathcal{O}_U$ which moreover map into an affine open $V \subset S$. Write $U = \Spec(A)$, $V = \Spec(R)$ and choose a generator $s$ of $\mathcal{L}$. We find that we have $$ L^\star \times_X U = \Spec(A[s, s^{-1}]) $$ Computing differentials we see that $$ \Omega^1_{A[s, s^{-1}]/R} = A[s, s^{-1}] \otimes_A \Omega^1_{A/R} \oplus A[s, s^{-1}] \text{d}\log(s) $$ and therefore taking exterior powers we obtain $$ \Omega^p_{A[s, s^{-1}]/R} = A[s, s^{-1}] \otimes_A \Omega^p_{A/R} \oplus A[s, s^{-1}] \text{d}\log(s) \otimes_A \Omega^{p - 1}_{A/R} $$ Taking degree $0$ parts we find $$ \Omega^p_{A[s, s^{-1}]/R, 0} = \Omega^p_{A/R} \oplus \text{d}\log(s) \otimes_A \Omega^{p - 1}_{A/R} $$ and the proof of the lemma is complete.
To be precise we consider the shift $$ 0 \to \Omega^\bullet_{X/S}[1] \to \Omega^\bullet_{L^\star/S, 0}[1] \to \Omega^\bullet_{X/S} \to 0 $$ of the short exact sequence of Lemma \ref{lemma-the-complex-for-L-star}. As the degree zero part of a grading on $(L^\star \to X)_*\Omega^\bullet_{L^\star/S}$ we see that $\Omega^\bullet_{L^\star/S, 0}$ is a differential graded $\mathcal{O}_X$-algebra and that the map $\Omega^\bullet_{X/S} \to \Omega^\bullet_{L^\star/S, 0}$ is a homomorphism of differential graded $\mathcal{O}_X$-algebras. Hence we may view $\Omega^\bullet_{X/S}[1] \to \Omega^\bullet_{L^\star/S, 0}[1]$ as a map of right differential graded $\Omega^\bullet_{X/S}$-modules on $X$. The map $\text{Res} : \Omega^\bullet_{L^\star/S, 0}[1] \to \Omega^\bullet_{X/S}$ is a map of right differential graded $\Omega^\bullet_{X/S}$-modules since it is locally defined by the rule $\text{Res}(\omega' + \text{d}\log(s) \wedge \omega) = \omega$, see proof of Lemma \ref{lemma-the-complex-for-L-star}. Thus by the discussion in Differential Graded Sheaves, Section \ref{sdga-section-misc} we see that $\delta$ comes from a map $\delta' : \Omega^\bullet_{X/S} \to \Omega^\bullet_{X/S}[2]$ in the derived category $D(\Omega^\bullet_{X/S}, \text{d})$ of right differential graded modules over the de Rham complex. The uniqueness averted in Remark \ref{remark-cup-product-as-a-map} shows it suffices to prove that $\delta(1) = c_1^{dR}(\mathcal{L})$. \medskip\noindent We claim that there is a commutative diagram $$ \xymatrix{ 0 \ar[r] & \mathcal{O}_X^* \ar[r] \ar[d]_{\text{d}\log} & E \ar[r] \ar[d] & \underline{\mathbf{Z}} \ar[d] \ar[r] & 0 \\ 0 \ar[r] & \Omega^\bullet_{X/S}[1] \ar[r] & \Omega^\bullet_{L^\star/S, 0}[1] \ar[r] & \Omega^\bullet_{X/S} \ar[r] & 0 } $$ where the top row is a short exact sequence of abelian sheaves whose boundary map sends $1$ to the class of $\mathcal{L}$ in $H^1(X, \mathcal{O}_X^*)$. It suffices to prove the claim by the compatibility of boundary maps with maps between short exact sequences. We define $E$ as the sheafification of the rule $$ U \longmapsto \{(s, n) \mid n \in \mathbf{Z},\ s \in \mathcal{L}^{\otimes n}(U)\text{ generator}\} $$ with group structure given by $(s, n) \cdot (t, m) = (s \otimes t, n + m)$. The middle vertical map sends $(s, n)$ to $\text{d}\log(s)$. This produces a map of short exact sequences because the map $Res : \Omega^1_{L^\star/S, 0} \to \mathcal{O}_X$ constructed in the proof of Lemma \ref{lemma-the-complex-for-L-star} sends $\text{d}\log(s)$ to $1$ if $s$ is a local generator of $\mathcal{L}$. To calculate the boundary of $1$ in the top row, choose local trivializations $s_i$ of $\mathcal{L}$ over opens $U_i$ as in Section \ref{section-first-chern-class}. On the overlaps $U_{i_0i_1} = U_{i_0} \cap U_{i_1}$ we have an invertible function $f_{i_0i_1}$ such that $f_{i_0i_1} = s_{i_1}|_{U_{i_0i_1}} s_{i_0}|_{U_{i_0i_1}}^{-1}$ and the cohomology class of $\mathcal{L}$ is given by the {\v C}ech cocycle $\{f_{i_0i_1}\}$. Then of course we have $$ (f_{i_0i_1}, 0) = (s_{i_1}, 1)|_{U_{i_0i_1}} \cdot (s_{i_0}, 1)|_{U_{i_0i_1}}^{-1} $$ as sections of $E$ which finishes the proof.
We are going to use the results of Cohomology of Schemes, Lemma \ref{coherent-lemma-cohomology-projective-space-over-ring} without further mention. In particular, the statements are true for $H^q(\mathbf{P}^n_A, \mathcal{O}(k))$. \medskip\noindent Proof for $p = 1$. Consider the short exact sequence $$ 0 \to \Omega \to \mathcal{O}(-1)^{\oplus n + 1} \to \mathcal{O} \to 0 $$ of Lemma \ref{lemma-euler-sequence}. Since $\mathcal{O}(-1)$ has vanishing cohomology in all degrees, this gives that $H^q(\mathbf{P}^n_A, \Omega)$ is zero except in degree $1$ where it is freely generated by the boundary of $1$ in $H^0(\mathbf{P}^n_A, \mathcal{O})$. \medskip\noindent Assume $p > 1$. Let us think of the short exact sequence above as defining a $2$ step filtration on $\mathcal{O}(-1)^{\oplus n + 1}$. The induced filtration on $\wedge^p\mathcal{O}(-1)^{\oplus n + 1}$ looks like this $$ 0 \to \Omega^p \to \wedge^p\left(\mathcal{O}(-1)^{\oplus n + 1}\right) \to \Omega^{p - 1} \to 0 $$ Observe that $\wedge^p\mathcal{O}(-1)^{\oplus n + 1}$ is isomorphic to a direct sum of $n + 1$ choose $p$ copies of $\mathcal{O}(-p)$ and hence has vanishing cohomology in all degrees. By induction hypothesis, this shows that $H^q(\mathbf{P}^n_A, \Omega^p)$ is zero unless $q = p$ and $H^p(\mathbf{P}^n_A, \Omega^p)$ is free of rank $1$ with generator the boundary of the generator in $H^{p - 1}(\mathbf{P}^n_A, \Omega^{p - 1})$. \medskip\noindent Let $k > 0$. Observe that $\Omega^n = \mathcal{O}(-n - 1)$ for example by the short exact sequence above for $p = n + 1$. Hence $\Omega^n(k)$ has vanishing cohomology in positive degrees. Using the short exact sequences $$ 0 \to \Omega^p(k) \to \wedge^p\left(\mathcal{O}(-1)^{\oplus n + 1}\right)(k) \to \Omega^{p - 1}(k) \to 0 $$ and {\it descending} induction on $p$ we get the vanishing of cohomology of $\Omega^p(k)$ in positive degrees for all $p$.