hcju/stacksqa huggingface dataset:
Consider the Hodge-to-de Rham spectral sequence of Section \ref{section-hodge-to-de-rham}. By the computation of the Hodge cohomology of $\mathbf{P}^n_A$ over $A$ done in Lemma \ref{lemma-hodge-cohomology-projective-space} we see that the spectral sequence degenerates on the $E_1$ page. In this way we see that $H^{2i}_{dR}(\mathbf{P}^n_A/A)$ is a free $A$-module of rank $1$ for $0 \leq i \leq n$ and zero else. Observe that $c_1^{dR}(\mathcal{O}(1))^i \in H^{2i}_{dR}(\mathbf{P}^n_A/A)$ for $i = 0, \ldots, n$ and that for $i = n$ this element is the image of $c_1^{Hodge}(\mathcal{L})^n$ by the map of complexes $$ \Omega^n_{\mathbf{P}^n_A/A}[-n] \longrightarrow \Omega^\bullet_{\mathbf{P}^n_A/A} $$ This follows for example from the discussion in Remark \ref{remark-truncations} or from the explicit description of cocycles representing these classes in Section \ref{section-first-chern-class}. The spectral sequence shows that the induced map $$ H^n(\mathbf{P}^n_A, \Omega^n_{\mathbf{P}^n_A/A}) \longrightarrow H^{2n}_{dR}(\mathbf{P}^n_A/A) $$ is an isomorphism and since $c_1^{Hodge}(\mathcal{L})^n$ is a generator of of the source (Lemma \ref{lemma-hodge-cohomology-projective-space}), we conclude that $c_1^{dR}(\mathcal{L})^n$ is a generator of the target. By the $A$-bilinearity of the cup products, it follows that also $c_1^{dR}(\mathcal{L})^i$ is a generator of $H^{2i}_{dR}(\mathbf{P}^n_A/A)$ for $0 \leq i \leq n$.
Consider $\Omega^\bullet_{X/S}$ as a filtered complex with the filtration introduced above. The spectral sequence is the spectral sequence of Cohomology, Lemma \ref{cohomology-lemma-relative-spectral-sequence-filtered-object}. By Derived Categories of Schemes, Lemma \ref{perfect-lemma-cohomology-de-rham-base-change} we have $$ Rf_*\text{gr}^k\Omega^\bullet_{X/S} = \Omega^k_{Y/S}[-k] \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*\Omega^\bullet_{X/Y} $$ and thus we conclude.
By Lemma \ref{lemma-proper-smooth-de-Rham} $Rf_*\Omega^\bullet_{X/Y}$ is a perfect object of $D(\mathcal{O}_Y)$ whose formation commutes with arbitrary base change. Thus the map of the lemma is a map $a : K \to L$ between perfect objects of $D(\mathcal{O}_Y)$ whose derived restriction to any point is an isomorphism by our assumption on fibres. Then the cone $C$ on $a$ is a perfect object of $D(\mathcal{O}_Y)$ (Cohomology, Lemma \ref{cohomology-lemma-two-out-of-three-perfect}) whose derived restriction to any point is zero. It follows that $C$ is zero by More on Algebra, Lemma \ref{more-algebra-lemma-lift-perfect-from-residue-field} and $a$ is an isomorphism. (This also uses Derived Categories of Schemes, Lemmas \ref{perfect-lemma-affine-compare-bounded} and \ref{perfect-lemma-perfect-affine} to translate into algebra.)
Our assumption is that for every $y \in Y$ and local equation $f \in \mathcal{O}_{X, y}$ of $Y$ we have $$ \Omega_{X/S, y} = \mathcal{O}_{X, y}\text{d}f \oplus M \quad\text{and}\quad \Omega^p_{X/S, y} = \wedge^{p - 1}(M)\text{d}f \oplus \wedge^p(M) $$ for some module $M$ with $f$-torsion free exterior powers $\wedge^p(M)$. It follows that $$ \Omega^p_{Y/S, y} = \wedge^p(M/fM) = \wedge^p(M)/f\wedge^p(M) $$ Below we will tacitly use these facts. In particular the sheaves $\Omega^p_{X/S}$ have no nonzero local sections supported on $Y$ and we have a canonical inclusion $$ \Omega^p_{X/S} \subset \Omega^p_{X/S}(Y) $$ see More on Flatness, Section \ref{flat-section-eta}. Let $U = \Spec(A)$ be an affine open subscheme such that $Y \cap U = V(f)$ for some nonzerodivisor $f \in A$. Let us consider the $\mathcal{O}_U$-submodule of $\Omega^p_{X/S}(Y)|_U$ generated by $\Omega^p_{X/S}|_U$ and $\text{d}\log(f) \wedge \Omega^{p - 1}_{X/S}$ where $\text{d}\log(f) = f^{-1}\text{d}(f)$. This is independent of the choice of $f$ as another generator of the ideal of $Y$ on $U$ is equal to $uf$ for a unit $u \in A$ and we get $$ \text{d}\log(uf) - \text{d}\log(f) = \text{d}\log(u) = u^{-1}\text{d}u $$ which is a section of $\Omega_{X/S}$ over $U$. These local sheaves glue to give a quasi-coherent submodule $$ \Omega^p_{X/S} \subset \Omega^p_{X/S}(\log Y) \subset \Omega^p_{X/S}(Y) $$ Let us agree to think of $\Omega^p_{Y/S}$ as a quasi-coherent $\mathcal{O}_X$-module. There is a unique surjective $\mathcal{O}_X$-linear map $$ \text{Res} : \Omega^p_{X/S}(\log Y) \to \Omega^{p - 1}_Y $$ defined by the rule $$ \text{Res}(\eta' + \text{d}\log(f) \wedge \eta) = \eta|_{Y \cap U} $$ for all opens $U$ as above and all $\eta' \in \Omega^p_{X/S}(U)$ and $\eta \in \Omega^{p - 1}_{X/S}(U)$. If a form $\eta$ over $U$ restricts to zero on $Y \cap U$, then $\eta = \text{d}f \wedge \eta' + f\eta''$ for some forms $\eta'$ and $\eta''$ over $U$. We conclude that we have a short exact sequence $$ 0 \to \Omega^p_{X/S} \to \Omega^p_{X/S}(\log Y) \to \Omega^{p - 1}_{Y/S} \to 0 $$ for all $p$. We still have to define the differentials $\Omega^p_{X/S}(\log Y) \to \Omega^{p + 1}_{X/S}(\log Y)$. On the subsheaf $\Omega^p_{X/S}$ we use the differential of the de Rham complex of $X$ over $S$. Finally, we define $\text{d}(\text{d}\log(f) \wedge \eta) = -\text{d}\log(f) \wedge \text{d}\eta$. The sign is forced on us by the Leibniz rule (on $\Omega^\bullet_{X/S}$) and it is compatible with the differential on $\Omega^\bullet_{Y/S}[-1]$ which is after all $-\text{d}_{Y/S}$ by our sign convention in Homology, Definition \ref{homology-definition-shift-cochain}. In this way we obtain a short exact sequence of complexes as stated in the lemma.
More precisely, we define $\delta$ as the boundary map corresponding to the shifted short exact sequence $$ 0 \to \Omega^\bullet_{X/S}[1] \to \Omega^\bullet_{X/S}(\log Y)[1] \to \Omega^\bullet_{Y/S} \to 0 $$ It suffices to prove each triangle commutes. Set $\mathcal{L} = \mathcal{O}_X(-Y)$. Denote $\pi : L \to X$ the line bundle with $\pi_*\mathcal{O}_L = \bigoplus_{n \geq 0} \mathcal{L}^{\otimes n}$. \medskip\noindent Commutativity of the upper left triangle. By Lemma \ref{lemma-the-complex-for-L-star-gives-chern-class} the map $\xi'$ is the boundary map of the triangle given in Lemma \ref{lemma-the-complex-for-L-star}. By functoriality it suffices to prove there exists a morphism of short exact sequences $$ \xymatrix{ 0 \ar[r] & \Omega^\bullet_{X/S}[1] \ar[r] \ar[d] & \Omega^\bullet_{L^\star/S, 0}[1] \ar[r] \ar[d] & \Omega^\bullet_{X/S} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \Omega^\bullet_{X/S}[1] \ar[r] & \Omega^\bullet_{X/S}(\log Y)[1] \ar[r] & \Omega^\bullet_{Y/S} \ar[r] & 0 } $$ where the left and right vertical arrows are the obvious ones. We can define the middle vertical arrow by the rule $$ \omega' + \text{d}\log(s) \wedge \omega \longmapsto \omega' + \text{d}\log(f) \wedge \omega $$ where $\omega', \omega$ are local sections of $\Omega^\bullet_{X/S}$ and where $s$ is a local generator of $\mathcal{L}$ and $f \in \mathcal{O}_X(-Y)$ is the corresponding section of the ideal sheaf of $Y$ in $X$. Since the constructions of the maps in Lemmas \ref{lemma-the-complex-for-L-star} and \ref{lemma-log-complex} match exactly, this works. \medskip\noindent Commutativity of the lower right triangle. Denote $\overline{L}$ the restriction of $L$ to $Y$. By Lemma \ref{lemma-the-complex-for-L-star-gives-chern-class} the map $\zeta'$ is the boundary map of the triangle given in Lemma \ref{lemma-the-complex-for-L-star} using the line bundle $\overline{L}$ on $Y$. By functoriality it suffices to prove there exists a morphism of short exact sequences $$ \xymatrix{ 0 \ar[r] & \Omega^\bullet_{X/S}[1] \ar[r] \ar[d] & \Omega^\bullet_{X/S}(\log Y)[1] \ar[r] \ar[d] & \Omega^\bullet_{Y/S} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \Omega^\bullet_{Y/S}[1] \ar[r] & \Omega^\bullet_{\overline{L}^\star/S, 0}[1] \ar[r] & \Omega^\bullet_{Y/S} \ar[r] & 0 \\ } $$ where the left and right vertical arrows are the obvious ones. We can define the middle vertical arrow by the rule $$ \omega' + \text{d}\log(f) \wedge \omega \longmapsto \omega'|_Y + \text{d}\log(\overline{s}) \wedge \omega|_Y $$ where $\omega', \omega$ are local sections of $\Omega^\bullet_{X/S}$ and where $f$ is a local generator of $\mathcal{O}_X(-Y)$ viewed as a function on $X$ and where $\overline{s}$ is $f|_Y$ viewed as a section of $\mathcal{L}|_Y = \mathcal{O}_X(-Y)|_Y$. Since the constructions of the maps in Lemmas \ref{lemma-the-complex-for-L-star} and \ref{lemma-log-complex} match exactly, this works.
This follows immediately from Lemma \ref{lemma-gysin-via-log-complex}. Namely, we have $$ c_1^{dR}(\mathcal{O}_X(Y)) \cup b = -c_1^{dR}(\mathcal{O}_X(-Y)) \cup b = -\xi'(b) = -\delta(b|_Y) = 0 $$ as desired. For the second equality, see Remark \ref{remark-cup-product-as-a-map}.
Let $y \in Y$ be a point. By More on Morphisms, Lemma \ref{more-morphisms-lemma-etale-local-structure} there exists an integer $0 \geq m$ and a commutative diagram $$ \xymatrix{ Y \ar[d] & V \ar[l] \ar[d] \ar[r] & \mathbf{A}^m_T \ar[d]^{(a_1, \ldots, a_m) \mapsto (a_1, \ldots, a_m, 0)} \\ X & U \ar[l] \ar[r]^-\pi & \mathbf{A}^{m + 1}_T } $$ where $U \subset X$ is open, $V = Y \cap U$, $\pi$ is \'etale, $V = \pi^{-1}(\mathbf{A}^m_T)$, and $y \in V$. Denote $z \in \mathbf{A}^m_T$ the image of $y$. Then we have $$ \Omega^p_{X/S, y} = \Omega^p_{\mathbf{A}^{m + 1}_T/S, z} \otimes_{\mathcal{O}_{\mathbf{A}^{m + 1}_T, z}} \mathcal{O}_{X, x} $$ by Lemma \ref{lemma-etale}. Denote $x_1, \ldots, x_{m + 1}$ the coordinate functions on $\mathbf{A}^{m + 1}_T$. Since the conditions (1) and (2) in Definition \ref{definition-local-product} do not depend on the choice of the local coordinate, it suffices to check the conditions (1) and (2) when $f$ is the image of $x_{m + 1}$ by the flat local ring homomorphism $\mathcal{O}_{\mathbf{A}^{m + 1}_T, z} \to \mathcal{O}_{X, x}$. In this way we see that it suffices to check conditions (1) and (2) for $\mathbf{A}^m_T \subset \mathbf{A}^{m + 1}_T$ and the point $z$. To prove this case we may assume $S = \Spec(A)$ and $T = \Spec(B)$ are affine. Let $A \to B$ be the ring map corresponding to the morphism $T \to S$ and set $P = B[x_1, \ldots, x_{m + 1}]$ so that $\mathbf{A}^{m + 1}_T = \Spec(B)$. We have $$ \Omega_{P/A} = \Omega_{B/A} \otimes_B P \oplus \bigoplus\nolimits_{j = 1, \ldots, m} P \text{d}x_j \oplus P \text{d}x_{m + 1} $$ Hence the map $P \to \Omega_{P/A}$, $g \mapsto g \text{d}x_{m + 1}$ is a split injection and $x_{m + 1}$ is a nonzerodivisor on $\Omega^p_{P/A}$ for all $p \geq 0$. Localizing at the prime ideal corresponding to $z$ finishes the proof.
Let us first prove part (2). Since $P = Z \times_S \mathbf{P}^{n - 1}_S$ we see that $$ \Omega^a_{P/S} = \bigoplus\nolimits_{a = r + s} \text{pr}_1^*\Omega^r_{Z/S} \otimes \text{pr}_2^*\Omega^s_{\mathbf{P}^{n - 1}_S/S} $$ Recalling that $p = \text{pr}_1$ by the projection formula (Cohomology, Lemma \ref{cohomology-lemma-projection-formula}) we obtain $$ p_*\Omega^a_{P/S} = \bigoplus\nolimits_{a = r + s} \Omega^r_{Z/S} \otimes \text{pr}_{1, *}\text{pr}_2^*\Omega^s_{\mathbf{P}^{n - 1}_S/S} $$ By the calculations in Section \ref{section-projective-space} and in particular in the proof of Lemma \ref{lemma-hodge-cohomology-projective-space} we have $\text{pr}_{1, *}\text{pr}_2^*\Omega^s_{\mathbf{P}^{n - 1}_S/S} = 0$ except if $s = 0$ in which case we get $\text{pr}_{1, *}\mathcal{O}_P = \mathcal{O}_Z$. This proves (2). \medskip\noindent By the material in Section \ref{section-line-bundle} and in particular Lemma \ref{lemma-push-omega-a} we have $\pi_*\Omega^a_{L/S} = \Omega^a_{P/S} \oplus \bigoplus_{k \geq 1} \Omega^a_{L/S, k}$. Since the composition $\pi \circ 0$ in the diagram above is the identity morphism on $P$ to prove part (3) it suffices to show that $\Omega^a_{L/S, k}$ has vanishing higher cohomology for $k > 0$. By Lemmas \ref{lemma-the-complex-for-L-star} and \ref{lemma-push-omega-a} there are short exact sequences $$ 0 \to \Omega^a_{P/S} \otimes \mathcal{O}_P(k) \to \Omega^a_{L/S, k} \to \Omega^{a - 1}_{P/S} \otimes \mathcal{O}_P(k) \to 0 $$ where $\Omega^{a - 1}_{P/S} = 0$ if $a = 0$. Since $P = Z \times_S \mathbf{P}^{n - 1}_S$ we have $$ \Omega^a_{P/S} = \bigoplus\nolimits_{i + j = a} \Omega^i_{Z/S} \boxtimes \Omega^j_{\mathbf{P}^{n - 1}_S/S} $$ by Lemma \ref{lemma-de-rham-complex-product}. Since $\Omega^i_{Z/S}$ is free of finite rank we see that it suffices to show that the higher cohomology of $\mathcal{O}_Z \boxtimes \Omega^j_{\mathbf{P}^{n - 1}_S/S}(k)$ is zero for $k > 0$. This follows from Lemma \ref{lemma-twisted-hodge-cohomology-projective-space} applied to $P = Z \times_S \mathbf{P}^{n - 1}_S = \mathbf{P}^{n - 1}_Z$ and the proof of (3) is complete. \medskip\noindent We still have to prove (1). If $n = 1$, then we are blowing up an effective Cartier divisor and $b$ is an isomorphism and we have (1). If $n > 1$, then the composition $$ \Gamma(X, \Omega^a_{X/S}) \to \Gamma(L, \Omega^a_{L/S}) \to \Gamma(L \setminus E, \Omega^a_{L/S}) = \Gamma(X \setminus Z, \Omega^a_{X/S}) $$ is an isomorphism as $\Omega^a_{X/S}$ is finite free (small detail omitted). Thus the only way (1) can fail is if there are nonzero elements of $\Gamma(L, \Omega^a_{L/S})$ which vanish outside of $E = 0(P)$. Since $L$ is a line bundle over $P$ with zero section $0 : P \to L$, it suffices to show that on a line bundle there are no nonzero sections of a sheaf of differentials which vanish identically outside the zero section. The reader sees this is true either (preferably) by a local caculation or by using that $\Omega_{L/S, k} \subset \Omega_{L^\star/S, k}$ (see references above).
The first arrow in the displayed formula is discussed in Section \ref{section-de-rham-complex}. To get the second arrow we have to show that if we view a local section of $\Omega^a_{L/S}$ as a ``meromorphic section'' of $b^*\Omega^a_{X/S}$, then it has a pole of order at most $n - 1$ along $E$. To see this we work on affine local charts on $L$. Namely, recall that $L$ is covered by the spectra of the affine blowup algebras $A[\frac{I}{y_i}]$ where $I = A_{+}$ is the ideal generated by $y_1, \ldots, y_n$. See Algebra, Section \ref{algebra-section-blow-up} and Divisors, Lemma \ref{divisors-lemma-blowing-up-affine}. By symmetry it is enough to work on the chart corresponding to $i = 1$. Then $$ A[\frac{I}{y_1}] = R[x_1, \ldots, x_m, y_1, t_2, \ldots, t_n] $$ where $t_i = y_i/y_1$, see More on Algebra, Lemma \ref{more-algebra-lemma-blowup-regular-sequence}. Thus the module $\Omega^1_{L/S}$ is over the corresponding affine open freely generated by $\text{d}x_1, \ldots, \text{d}x_m$, $\text{d}y_1$, and $\text{d}t_1, \ldots, \text{d}t_n$. Of course, the first $m + 1$ of these generators come from $b^*\Omega^1_{X/S}$ and for the remaining $n - 1$ we have $$ \text{d}t_j = \text{d}\frac{y_j}{y_1} = \frac{1}{y_1}\text{d}y_j - \frac{y_j}{y_1^2}\text{d}y_1 = \frac{\text{d}y_j - t_j \text{d}y_1}{y_1} $$ which has a pole of order $1$ along $E$ since $E$ is cut out by $y_1$ on this chart. Since the wedges of $a$ of these elements give a basis of $\Omega^a_{L/S}$ over this chart, and since there are at most $n - 1$ of the $\text{d}t_j$ involved this finishes the proof.
By the projection formula it is enough to show this for $\mathcal{E} = \mathcal{O}_X$, see Cohomology, Lemma \ref{cohomology-lemma-projection-formula}. Since $X$ is affine it suffices to show that the maps $$ H^0(X, \mathcal{O}_X) \to H^0(L, \mathcal{O}_L) \to H^0(L, \mathcal{O}_L(iE)) $$ are isomorphisms and that $H^j(X, \mathcal{O}_L(iE)) = 0$ for $j > 0$ and $0 \leq i \leq n - 1$, see Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherence-higher-direct-images-application}. Since $\pi$ is affine, we can compute global sections and cohomology after taking $\pi_*$, see Cohomology of Schemes, Lemma \ref{coherent-lemma-relative-affine-cohomology}. If $n = 1$, then $L \to X$ is an isomorphism and $i = 0$ hence the first statement holds. If $n > 1$, then we consider the composition $$ H^0(X, \mathcal{O}_X) \to H^0(L, \mathcal{O}_L) \to H^0(L, \mathcal{O}_L(iE)) \to H^0(L \setminus E, \mathcal{O}_L) = H^0(X \setminus Z, \mathcal{O}_X) $$ Since $H^0(X \setminus Z, \mathcal{O}_X) = H^0(X, \mathcal{O}_X)$ in this case as $Z$ has codimension $n \geq 2$ in $X$ (details omitted) we conclude the first statement holds. For the second, recall that $\mathcal{O}_L(E) = \mathcal{O}_L(-1)$, see Divisors, Lemma \ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor}. Hence we have $$ \pi_*\mathcal{O}_L(iE) = \pi_*\mathcal{O}_L(-i) = \bigoplus\nolimits_{k \geq -i} \mathcal{O}_P(k) $$ as discussed in More on Morphisms, Section \ref{more-morphisms-section-proj-spec}. Thus we conclude by the vanishing of the cohomology of twists of the structure sheaf on $P = \mathbf{P}^{n - 1}_Z$ shown in Cohomology of Schemes, Lemma \ref{coherent-lemma-cohomology-projective-space-over-ring}.
By Divisors, Lemma \ref{divisors-lemma-immersion-smooth-into-smooth-regular-immersion} the immersion $Z \to X$ is a regular immmersion, hence the ideal sheaf $\mathcal{I}$ is of finite type, hence $b$ is a projective morphism with relatively ample invertible sheaf $\mathcal{O}_{X'}(1) = \mathcal{O}_{X'}(-E)$, see Divisors, Lemmas \ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor} and \ref{divisors-lemma-blowing-up-projective}. The canonical map $\mathcal{I} \to b_*\mathcal{O}_{X'}(1)$ gives a closed immersion $$ X' \longrightarrow \mathbf{P}\left(\bigoplus\nolimits_{n \geq 0} \text{Sym}^n_{\mathcal{O}_X}(\mathcal{I})\right) $$ by the very construction of the blowup. The restriction of this morphism to $E$ gives a canonical map $$ E \longrightarrow \mathbf{P}\left(\bigoplus\nolimits_{n \geq 0} \text{Sym}^n_{\mathcal{O}_Z}(\mathcal{I}/\mathcal{I}^2)\right) $$ over $Z$. Since $\mathcal{I}/\mathcal{I}^2$ is finite locally free if this canonical map is an isomorphism, then the final part of the lemma holds. Having said all of this, now the question is \'etale local on $X$. Namely, blowing up commutes with flat base change by Divisors, Lemma \ref{divisors-lemma-flat-base-change-blowing-up} and we can check smoothness after precomposing with a surjective \'etale morphism. Thus by the \'etale local structure of a closed immersion of schemes over $S$ given in More on Morphisms, Lemma \ref{more-morphisms-lemma-etale-local-structure} this reduces to the situation discussed in Section \ref{section-calculations}.
Let $\epsilon : X_1 \to X$ be a surjective \'etale morphism. Denote $i_1 : Z_1 \to X_1$, $b_1 : X'_1 \to X_1$, $E_1 \subset X'_1$, and $p_1 : E_1 \to Z_1$ the base changes of the objects considered in Lemma \ref{lemma-blowup}. Observe that $i_1$ is a closed immersion of schemes smooth over $S$ and that $b_1$ is the blowing up with center $Z_1$ by Divisors, Lemma \ref{divisors-lemma-flat-base-change-blowing-up}. Suppose that we can prove (1), (2), and (3) for the morphisms $b_1$, $p_1$, and $i_1$. Then by Lemma \ref{lemma-etale} we obtain that the pullback by $\epsilon$ of the maps in (1), (2), and (3) are isomorphisms. As $\epsilon$ is a surjective flat morphism we conclude. Thus working \'etale locally, by More on Morphisms, Lemma \ref{more-morphisms-lemma-etale-local-structure}, we may assume we are in the situation discussed in Section \ref{section-calculations}. In this case the lemma is the same as Lemma \ref{lemma-comparison}.
Choose a distinguished triangle $$ C \to Rb_*\Omega^\bullet_{X'/S} \oplus i_*\Omega^\bullet_{Z/S} \to i_*Rp_*\Omega^\bullet_{E/S} \to C[1] $$ in $D(X, f^{-1}\mathcal{O}_S)$. It suffices to show that $\Omega^\bullet_{X/S}$ is isomorphic to $C$ in a manner compatible with the canonical maps. By the axioms of triangulated categories there exists a map of distinguished triangles $$ \xymatrix{ C' \ar[r] \ar[d] & b_*\Omega^\bullet_{X'/S} \oplus i_*\Omega^\bullet_{Z/S} \ar[r] \ar[d] & i_*p_*\Omega^\bullet_{E/S} \ar[r] \ar[d] & C'[1] \ar[d] \\ C \ar[r] & Rb_*\Omega^\bullet_{X'/S} \oplus i_*\Omega^\bullet_{Z/S} \ar[r] & i_*Rp_*\Omega^\bullet_{E/S} \ar[r] & C[1] } $$ By Lemma \ref{lemma-comparison-general} part (3) and Derived Categories, Proposition \ref{derived-proposition-9} we conclude that $C' \to C$ is an isomorphism. By Lemma \ref{lemma-comparison-general} part (2) the map $i_*\Omega^\bullet_{Z/S} \to i_*p_*\Omega^\bullet_{E/S}$ is an isomorphism. Thus $C' = b_*\Omega^\bullet_{X'/S}$ in the derived category. Finally we use Lemma \ref{lemma-comparison-general} part (1) tells us this is equal to $\Omega^\bullet_{X/S}$. We omit the verification this is compatible with the canonical maps.
We may decompose $X$ into open and closed subschemes having fixed relative dimension to $S$, see Morphisms, Lemma \ref{morphisms-lemma-smooth-omega-finite-locally-free}. Since the derived category $D(X, f^{-1}\mathcal{O})_S)$ correspondingly decomposes as a product of categories, we may assume $X$ has fixed relative dimension $N$ over $S$. We may decompose $Z = \coprod Z_m$ into open and closed subschemes of relative dimension $m \geq 0$ over $S$. The restriction $i_m : Z_m \to X$ of $i$ to $Z_m$ is a regular immersion of codimension $N - m$, see Divisors, Lemma \ref{divisors-lemma-immersion-smooth-into-smooth-regular-immersion}. Let $E = \coprod E_m$ be the corresponding decomposition, i.e., we set $E_m = p^{-1}(Z_m)$. We claim that there are natural maps $$ b^*\Omega^a_{X/S} \to \Omega^a_{X'/S} \to b^*\Omega^a_{X/S} \otimes_{\mathcal{O}_{X'}} \mathcal{O}_{X'}(\sum (N - m - 1)E_m) $$ whose composition is induced by the inclusion $\mathcal{O}_{X'} \to \mathcal{O}_{X'}(\sum (N - m - 1)E_m)$. Namely, in order to prove this, it suffices to show that the cokernel of the first arrow is locally on $X'$ annihilated by a local equation of the effective Cartier divisor $\sum (N - m - 1)E_m$. To see this in turn we can work \'etale locally on $X$ as in the proof of Lemma \ref{lemma-comparison-general} and apply Lemma \ref{lemma-comparison-bis}. Computing \'etale locally using Lemma \ref{lemma-blowup-twist-same-cohomology} we see that the induced composition $$ \Omega^a_{X/S} \to Rb_*\Omega^a_{X'/S} \to Rb_*\left(b^*\Omega^a_{X/S} \otimes_{\mathcal{O}_{X'}} \mathcal{O}_{X'}(\sum (N - m - 1)E_m)\right) $$ is an isomorphism in $D(\mathcal{O}_X)$ which is how we obtain the existence of the map in the lemma. \medskip\noindent For uniqueness, it suffices to show that there are no nonzero maps from $\tau_{\geq 1}Rb_*\Omega_{X'/S}$ to $\Omega^a_{X/S}$ in $D(\mathcal{O}_X)$. For this it suffices in turn to show that there are no nonzero maps from $R^qb_*\Omega_{X'/s}[-q]$ to $\Omega^a_{X/S}$ in $D(\mathcal{O}_X)$ for $q \geq 1$ (details omitted). By Lemma \ref{lemma-comparison-general} we see that $R^qb_*\Omega_{X'/s} \cong i_*R^qp_*\Omega^a_{E/S}$ is the pushforward of a module on $Z = \coprod Z_m$. Moreover, observe that the restriction of $R^qp_*\Omega^a_{E/S}$ to $Z_m$ is nonzero only for $q < N - m$. Namely, the fibres of $E_m \to Z_m$ have dimension $N - m - 1$ and we can apply Limits, Lemma \ref{limits-lemma-higher-direct-images-zero-above-dimension-fibre}. Thus the desired vanishing follows from Lemma \ref{lemma-ext-zero}.
By Duality for Schemes, Lemma \ref{duality-lemma-duality-proper-over-field} we have $\omega_X^\bullet = \Omega^d_{X/k}[d]$. Since $\Omega_{X/k}$ is locally free of rank $d$ (Morphisms, Lemma \ref{morphisms-lemma-smooth-omega-finite-locally-free}) we have $$ \Omega^d_{X/k} \otimes_{\mathcal{O}_X} (\Omega^p_{X/k})^\vee \cong \Omega^{d - p}_{X/k} $$ Thus we obtain a $k$-linear map $t : H^d(X, \Omega^d_{X/k}) \to k$ such that the statement is true by Duality for Schemes, Lemma \ref{duality-lemma-duality-proper-over-field-perfect}. In particular the pairing $H^0(X, \mathcal{O}_X) \times H^d(X, \Omega^d_{X/k}) \to k$ is perfect, which implies that any $k$-linear map $t' : H^d(X, \Omega^d_{X/k}) \to k$ is of the form $\xi \mapsto t(g\xi)$ for some $g \in H^0(X, \mathcal{O}_X)$. Of course, in order for $t'$ to still produce a duality between $H^0(X, \mathcal{O}_X)$ and $H^d(X, \Omega^d_{X/k})$ we need $g$ to be a unit. Denote $\langle -, - \rangle_{p, q}$ the pairing constructed using $t$ and denote $\langle -, - \rangle'_{p, q}$ the pairing constructed using $t'$. Clearly we have $$ \langle \xi, \xi' \rangle'_{p, q} = \langle g\xi, \xi' \rangle_{p, q} $$ for $\xi \in H^q(X, \Omega^p_{X/k})$ and $\xi' \in H^{d - q}(X, \Omega^{d - p}_{X/k})$. Since $g$ is a unit, i.e., invertible, we see that using $t'$ instead of $t$ we still get perfect pairings for all $p, q$.