hcju/stacksqa huggingface dataset:
As blowing up commutes with restrictions to open subschemes (Lemma \ref{lemma-flat-base-change-blowing-up}) the first statement just means that $X' = X$ if $Z = \emptyset$. In this case we are blowing up in the ideal sheaf $\mathcal{I} = \mathcal{O}_X$ and the result follows from Constructions, Example \ref{constructions-example-trivial-proj}. \medskip\noindent The second statement is local on $X$, hence we may assume $X$ affine. Say $X = \Spec(A)$ and $Z = \Spec(A/I)$. By Lemma \ref{lemma-blowing-up-affine} we see that $X'$ is covered by the spectra of the affine blowup algebras $A' = A[\frac{I}{a}]$. Then $IA' = aA'$ and $a$ maps to a nonzerodivisor in $A'$ according to Algebra, Lemma \ref{algebra-lemma-affine-blowup}. This proves the lemma as the inverse image of $Z$ in $\Spec(A')$ corresponds to $\Spec(A'/IA') \subset \Spec(A')$. \medskip\noindent Consider the canonical map $\psi_{univ, 1} : b^*\mathcal{I} \to \mathcal{O}_{X'}(1)$, see discussion following Constructions, Definition \ref{constructions-definition-relative-proj}. We claim that this factors through an isomorphism $\mathcal{I}_E \to \mathcal{O}_{X'}(1)$ (which proves the final assertion). Namely, on the affine open corresponding to the blowup algebra $A' = A[\frac{I}{a}]$ mentioned above $\psi_{univ, 1}$ corresponds to the $A'$-module map $$ I \otimes_A A' \longrightarrow \left(\Big(\bigoplus\nolimits_{d \geq 0} I^d\Big)_{a^{(1)}}\right)_1 $$ where $a^{(1)}$ is as in Algebra, Definition \ref{algebra-definition-blow-up}. We omit the verification that this is the map $I \otimes_A A' \to IA' = aA'$.
We see that $b : X' \to X$ is an object of $\mathcal{C}$ according to Lemma \ref{lemma-blowing-up-gives-effective-Cartier-divisor}. Let $f : Y \to X$ be an object of $\mathcal{C}$. We have to show there exists a unique morphism $Y \to X'$ over $X$. Let $D = f^{-1}(Z)$. Let $\mathcal{I} \subset \mathcal{O}_X$ be the ideal sheaf of $Z$ and let $\mathcal{I}_D$ be the ideal sheaf of $D$. Then $f^*\mathcal{I} \to \mathcal{I}_D$ is a surjection to an invertible $\mathcal{O}_Y$-module. This extends to a map $\psi : \bigoplus f^*\mathcal{I}^d \to \bigoplus \mathcal{I}_D^d$ of graded $\mathcal{O}_Y$-algebras. (We observe that $\mathcal{I}_D^d = \mathcal{I}_D^{\otimes d}$ as $D$ is an effective Cartier divisor.) By the material in Constructions, Section \ref{constructions-section-relative-proj} the triple $(1, f : Y \to X, \psi)$ defines a morphism $Y \to X'$ over $X$. The restriction $$ Y \setminus D \longrightarrow X' \setminus b^{-1}(Z) = X \setminus Z $$ is unique. The open $Y \setminus D$ is scheme theoretically dense in $Y$ according to Lemma \ref{lemma-complement-effective-Cartier-divisor}. Thus the morphism $Y \to X'$ is unique by Morphisms, Lemma \ref{morphisms-lemma-equality-of-morphisms} (also $b$ is separated by Constructions, Lemma \ref{constructions-lemma-relative-proj-separated}).
Since the exceptional divisor over $U'$ is cut out by the image of $a$ in $A' = A[\frac{I}{a}]$ one direction is clear. Conversely, assume that the image of $a$ in $\mathcal{O}_{X', x'}$ cuts out $E$. Since every element of $I$ maps to an element of the ideal defining $E$ over $b^{-1}(U)$ we see that elements of $I$ become divisible by $a$ in $\mathcal{O}_{X', x'}$. Thus for $f \in I^n$ we can write $f = \psi(f) a^n$ for some $\psi(f) \in \mathcal{O}_{X', x'}$. Observe that since $a$ maps to a nonzerodivisor of $\mathcal{O}_{X', x'}$ the element $\psi(f)$ is uniquely characterized by this. Then we define $$ A' \longrightarrow \mathcal{O}_{X', x'},\quad f/a^n \longmapsto \psi(f) $$ Here we use the description of blowup algebras given following Algebra, Definition \ref{definition-blow-up}. The uniqueness mentioned above shows that this is an $A$-algebra homomorphism. This gives a morphism $\Spec(\mathcal{O}_{X', x"}) \to \Spec(A') = U'$. By the universal property of blowing up (Lemma \ref{lemma-universal-property-blowing-up}) this is a morphism over $X'$, which of course implies that $x' \in U'$.
The exceptional divisor $E \subset X'$ is an effective Cartier divisor (Lemma \ref{lemma-blowing-up-gives-effective-Cartier-divisor}) hence is nowhere dense in $X'$ (Lemma \ref{lemma-complement-effective-Cartier-divisor}). On the other hand, $X' \setminus E \to X \setminus Z$ is an isomorphism. The lemma follows.
By Lemmas \ref{lemma-blowing-up-affine} and \ref{lemma-characterize-effective-Cartier-divisor} this reduces to the following algebra fact: Let $A$ be a ring, $I \subset A$ an ideal, $a \in I$, and $x \in A$ a nonzerodivisor. Then the image of $x$ in $A[\frac{I}{a}]$ is a nonzerodivisor. Namely, suppose that $x (y/a^n) = 0$ in $A[\frac{I}{a}]$. Then $a^mxy = 0$ in $A$ for some $m$. Hence $a^my = 0$ as $x$ is a nonzerodivisor. Whence $y/a^n$ is zero in $A[\frac{I}{a}]$ as desired.
Let $E \subset X'$ be the exceptional divisor of $b$ which is an effective Cartier divisor by Lemma \ref{lemma-blowing-up-gives-effective-Cartier-divisor}. Then $(b')^{-1}E$ is an effective Cartier divisor on $X''$ by Lemma \ref{lemma-blow-up-pullback-effective-Cartier}. Let $E' \subset X''$ be the exceptional divisor of $b'$ (also an effective Cartier divisor). Consider the effective Cartier divisor $E'' = E' + (b')^{-1}E$. By construction the ideal of $E''$ is $(b \circ b')^{-1}\mathcal{I} (b \circ b')^{-1}\mathcal{J} \mathcal{O}_{X''}$. Hence according to Lemma \ref{lemma-universal-property-blowing-up} there is a canonical morphism from $X''$ to the blowup $c : Y \to X$ of $X$ in $\mathcal{I}\mathcal{J}$. Conversely, as $\mathcal{I}\mathcal{J}$ pulls back to an invertible ideal we see that $c^{-1}\mathcal{I}\mathcal{O}_Y$ defines an effective Cartier divisor, see Lemma \ref{lemma-sum-closed-subschemes-effective-Cartier}. Thus a morphism $c' : Y \to X'$ over $X$ by Lemma \ref{lemma-universal-property-blowing-up}. Then $(c')^{-1}b^{-1}\mathcal{J}\mathcal{O}_Y = c^{-1}\mathcal{J}\mathcal{O}_Y$ which also defines an effective Cartier divisor. Thus a morphism $c'' : Y \to X''$ over $X'$. We omit the verification that this morphism is inverse to the morphism $X'' \to Y$ constructed earlier.
The surjection of graded $\mathcal{O}_X$-algebras $$ \text{Sym}_{\mathcal{O}_X}^*(\mathcal{I}) \longrightarrow \bigoplus\nolimits_{d \geq 0} \mathcal{I}^d $$ defines via Constructions, Lemma \ref{constructions-lemma-surjective-generated-degree-1-map-relative-proj} a closed immersion $$ X' = \underline{\text{Proj}}_X (\bigoplus\nolimits_{d \geq 0} \mathcal{I}^d) \longrightarrow \mathbf{P}(\mathcal{I}). $$ Hence $b$ is projective, see Morphisms, Definition \ref{morphisms-definition-projective}. The second statement follows for example from the characterization of relatively ample invertible sheaves in Morphisms, Lemma \ref{morphisms-lemma-characterize-relatively-ample}. Some details omitted.
Let $X'' \to X$ be the blowup of $X$ in $f^{-1}Z$. By the universal property of blowing up (Lemma \ref{lemma-universal-property-blowing-up}) there exists a commutative diagram $$ \xymatrix{ X'' \ar[r] \ar[d] & X \ar[d] \\ S' \ar[r] & S } $$ whence a morphism $X'' \to X \times_S S'$. Thus the first assertion is that this morphism is a closed immersion with image $X'$. The question is local on $X$. Thus we may assume $X$ and $S$ are affine. Say that $S = \Spec(A)$, $X = \Spec(B)$, and $Z$ is cut out by the ideal $I \subset A$. Set $J = IB$. The map $B \otimes_A \bigoplus_{n \geq 0} I^n \to \bigoplus_{n \geq 0} J^n$ defines a closed immersion $X'' \to X \times_S S'$, see Constructions, Lemmas \ref{constructions-lemma-base-change-map-proj} and \ref{constructions-lemma-surjective-graded-rings-generated-degree-1-map-proj}. We omit the verification that this morphism is the same as the one constructed above from the universal property. Pick $a \in I$ corresponding to the affine open $\Spec(A[\frac{I}{a}]) \subset S'$, see Lemma \ref{lemma-blowing-up-affine}. The inverse image of $\Spec(A[\frac{I}{a}])$ in the strict transform $X'$ of $X$ is the spectrum of $$ B' = (B \otimes_A A[\textstyle{\frac{I}{a}}])/a\text{-power-torsion} $$ see Properties, Lemma \ref{properties-lemma-sections-supported-on-closed-subset}. On the other hand, letting $b \in J$ be the image of $a$ we see that $\Spec(B[\frac{J}{b}])$ is the inverse image of $\Spec(A[\frac{I}{a}])$ in $X''$. By Algebra, Lemma \ref{algebra-lemma-blowup-base-change} the open $\Spec(B[\frac{J}{b}])$ maps isomorphically to the open subscheme $\text{pr}_{S'}^{-1}(\Spec(A[\frac{I}{a}]))$ of $X'$. Thus $X'' \to X'$ is an isomorphism. \medskip\noindent In the notation above, let $\mathcal{F}$ correspond to the $B$-module $N$. The strict transform of $\mathcal{F}$ corresponds to the $B \otimes_A A[\frac{I}{a}]$-module $$ N' = (N \otimes_A A[\textstyle{\frac{I}{a}}])/a\text{-power-torsion} $$ see Properties, Lemma \ref{properties-lemma-sections-supported-on-closed-subset}. The strict transform of $\mathcal{F}$ relative to the blowup of $X$ in $f^{-1}Z$ corresponds to the $B[\frac{J}{b}]$-module $N \otimes_B B[\frac{J}{b}]/b\text{-power-torsion}$. In exactly the same way as above one proves that these two modules are isomorphic. Details omitted.
We will prove part (2) as it implies part (1) by the definition of the strict transform of a scheme over $S$. The question is local on $X$. Thus we may assume that $S = \Spec(A)$, $X = \Spec(B)$, and that $\mathcal{F}$ corresponds to the $B$-module $N$. Then $\mathcal{F}'$ over the open $\Spec(B \otimes_A A[\frac{I}{a}])$ of $X \times_S S'$ corresponds to the module $$ N' = (N \otimes_A A[\textstyle{\frac{I}{a}}])/a\text{-power-torsion} $$ see Properties, Lemma \ref{properties-lemma-sections-supported-on-closed-subset}. Thus we have to show that the $a$-power-torsion of $N \otimes_A A[\frac{I}{a}]$ is zero. Let $y \in N \otimes_A A[\frac{I}{a}]$ with $a^n y = 0$. If $\mathfrak q \subset B$ is a prime and $a \not \in \mathfrak q$, then $y$ maps to zero in $(N \otimes_A A[\frac{I}{a}])_\mathfrak q$. on the other hand, if $a \in \mathfrak q$, then $N_\mathfrak q$ is a flat $A$-module and we see that $N_\mathfrak q \otimes_A A[\frac{I}{a}] =(N \otimes_A A[\frac{I}{a}])_\mathfrak q$ has no $a$-power torsion (as $A[\frac{I}{a}]$ doesn't). Hence $y$ maps to zero in this localization as well. We conclude that $y$ is zero by Algebra, Lemma \ref{algebra-lemma-characterize-zero-local}.
Observe that $g'_*\text{pr}_X^*\mathcal{F} = \text{pr}_Y^*g_*\mathcal{F}$ by Cohomology of Schemes, Lemma \ref{coherent-lemma-affine-base-change}. Let $\mathcal{K} \subset \text{pr}_X^*\mathcal{F}$ be the subsheaf of sections supported in the inverse image of $Z$ in $X \times_S S'$. By Properties, Lemma \ref{properties-lemma-push-sections-supported-on-closed-subset} the pushforward $g'_*\mathcal{K}$ is the subsheaf of sections of $\text{pr}_Y^*g_*\mathcal{F}$ supported in the inverse image of $Z$ in $Y \times_S S'$. As $g'$ is affine (Morphisms, Lemma \ref{morphisms-lemma-base-change-affine}) we see that $g'_*$ is exact, hence we conclude.
The first statement follows on combining Lemmas \ref{lemma-blowing-up-two-ideals} and \ref{lemma-blow-up-effective-Cartier-divisor}. Using Lemma \ref{lemma-blowing-up-affine} the second statement translates into the following algebra problem. Let $A$ be a ring, $I \subset A$ an ideal, $x \in A$ a nonzerodivisor, and $a \in I$. Let $M$ be an $A$-module whose $x$-torsion is zero. To show: the $a$-power torsion in $M \otimes_A A[\frac{I}{a}]$ is equal to the $xa$-power torsion. The reason for this is that the kernel and cokernel of the map $A \to A[\frac{I}{a}]$ is $a$-power torsion, so this map becomes an isomorphism after inverting $a$. Hence the kernel and cokernel of $M \to M \otimes_A A[\frac{I}{a}]$ are $a$-power torsion too. This implies the result.
Let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$ with respect to the blowup $S' \to S$ of $S$ in $Z$. Let $\mathcal{F}''$ be the strict transform of $\mathcal{F}'$ with respect to the blowup $S'' \to S'$ of $S'$ in $Z'$. Let $\mathcal{G}$ be the strict transform of $\mathcal{F}$ with respect to the blowup $S'' \to S$ of $S$ in $Y$. We also label the morphisms $$ \xymatrix{ X \times_S S'' \ar[r]_q \ar[d]^{f''} & X \times_S S' \ar[r]_p \ar[d]^{f'} & X \ar[d]^f \\ S'' \ar[r] & S' \ar[r] & S } $$ By definition there is a surjection $p^*\mathcal{F} \to \mathcal{F}'$ and a surjection $q^*\mathcal{F}' \to \mathcal{F}''$ which combine by right exactness of $q^*$ to a surjection $(p \circ q)^*\mathcal{F} \to \mathcal{F}''$. Also we have the surjection $(p \circ q)^*\mathcal{F} \to \mathcal{G}$. Thus it suffices to prove that these two surjections have the same kernel. \medskip\noindent The kernel of the surjection $p^*\mathcal{F} \to \mathcal{F}'$ is supported on $(f \circ p)^{-1}Z$, so this map is an isomorphism at points in the complement. Hence the kernel of $q^*p^*\mathcal{F} \to q^*\mathcal{F}'$ is supported on $(f \circ p \circ q)^{-1}Z$. The kernel of $q^*\mathcal{F}' \to \mathcal{F}''$ is supported on $(f' \circ q)^{-1}Z'$. Combined we see that the kernel of $(p \circ q)^*\mathcal{F} \to \mathcal{F}''$ is supported on $(f \circ p \circ q)^{-1}Z \cup (f' \circ q)^{-1}Z' = (f \circ p \circ q)^{-1}Y$. By construction of $\mathcal{G}$ we see that we obtain a factorization $(p \circ q)^*\mathcal{F} \to \mathcal{F}'' \to \mathcal{G}$. To finish the proof it suffices to show that $\mathcal{F}''$ has no nonzero (local) sections supported on $(f \circ p \circ q)^{-1}(Y) = (f \circ p \circ q)^{-1}Z \cup (f' \circ q)^{-1}Z'$. This follows from Lemma \ref{lemma-strict-transform-different-centers} applied to $\mathcal{F}'$ on $X \times_S S'$ over $S'$, the closed subscheme $Z'$ and the effective Cartier divisor $b^{-1}Z$.
We may localize on $S$ and $X$ and assume both are affine. Then we may push $\mathcal{F}_i$ to $S$, see Lemma \ref{lemma-strict-transform-affine}. We may assume that our blowup is the morphism $1 : S \to S$ associated to an effective Cartier divisor $D \subset S$. Then the translation into algebra is the following: Suppose that $A$ is a ring and $0 \to M_1 \to M_2 \to M_3 \to 0$ is a universally exact sequence of $A$-modules. Let $a\in A$. Then the sequence $$ 0 \to M_1/a\text{-power torsion} \to M_2/a\text{-power torsion} \to M_3/a\text{-power torsion} \to 0 $$ is exact too. Namely, surjectivity of the last map and injectivity of the first map are immediate. The problem is exactness in the middle. Suppose that $x \in M_2$ maps to zero in $M_3/a\text{-power torsion}$. Then $y = a^n x \in M_1$ for some $n$. Then $y$ maps to zero in $M_2/a^nM_2$. Since $M_1 \to M_2$ is universally injective we see that $y$ maps to zero in $M_1/a^nM_1$. Thus $y = a^n z$ for some $z \in M_1$. Thus $a^n(x - y) = 0$. Hence $y$ maps to the class of $x$ in $M_2/a\text{-power torsion}$ as desired.
Let $\mathcal{I} \subset \mathcal{O}_V$ be the finite type quasi-coherent sheaf of ideals such that $V(\mathcal{I})$ is disjoint from $U \cap V$ and such that $V'$ is isomorphic to the blowup of $V$ in $\mathcal{I}$. Let $\mathcal{I}' \subset \mathcal{O}_{U \cup V}$ be the quasi-coherent sheaf of ideals whose restriction to $U$ is $\mathcal{O}_U$ and whose restriction to $V$ is $\mathcal{I}$ (see Sheaves, Section \ref{sheaves-section-glueing-sheaves}). By Properties, Lemma \ref{properties-lemma-extend} there exists a finite type quasi-coherent sheaf of ideals $\mathcal{J} \subset \mathcal{O}_X$ whose restriction to $U \cup V$ is $\mathcal{I}'$. The lemma follows.
Choose a finite type quasi-coherent sheaf of ideals $\mathcal{I}$, resp.\ $\mathcal{J}$ such that $X \setminus U = V(\mathcal{I})$, resp.\ $X \setminus V = V(\mathcal{J})$, see Properties, Lemma \ref{properties-lemma-quasi-coherent-finite-type-ideals}. Then $V(\mathcal{I}\mathcal{J}) = X$ set theoretically, hence $\mathcal{I}\mathcal{J}$ is a locally nilpotent sheaf of ideals. Since $\mathcal{I}$ and $\mathcal{J}$ are of finite type and $X$ is quasi-compact there exists an $n > 0$ such that $\mathcal{I}^n \mathcal{J}^n = 0$. We may and do replace $\mathcal{I}$ by $\mathcal{I}^n$ and $\mathcal{J}$ by $\mathcal{J}^n$. Whence $\mathcal{I} \mathcal{J} = 0$. Let $b : X' \to X$ be the blowing up in $\mathcal{I} + \mathcal{J}$. This is $U \cup V$-admissible as $V(\mathcal{I} + \mathcal{J}) = X \setminus U \cup V$. We will show that $X'$ is a disjoint union of open subschemes $X' = X'_1 \amalg X'_2$ such that $b^{-1}\mathcal{I}|_{X'_2} = 0$ and $b^{-1}\mathcal{J}|_{X'_1} = 0$ which will prove the lemma. \medskip\noindent We will use the description of the blowing up in Lemma \ref{lemma-blowing-up-affine}. Suppose that $U = \Spec(A) \subset X$ is an affine open such that $\mathcal{I}|_U$, resp.\ $\mathcal{J}|_U$ corresponds to the finitely generated ideal $I \subset A$, resp.\ $J \subset A$. Then $$ b^{-1}(U) = \text{Proj}(A \oplus (I + J) \oplus (I + J)^2 \oplus \ldots) $$ This is covered by the affine open subsets $A[\frac{I + J}{x}]$ and $A[\frac{I + J}{y}]$ with $x \in I$ and $y \in J$. Since $x \in I$ is a nonzerodivisor in $A[\frac{I + J}{x}]$ and $IJ = 0$ we see that $J A[\frac{I + J}{x}] = 0$. Since $y \in J$ is a nonzerodivisor in $A[\frac{I + J}{y}]$ and $IJ = 0$ we see that $I A[\frac{I + J}{y}] = 0$. Moreover, $$ \Spec(A[\textstyle{\frac{I + J}{x}}]) \cap \Spec(A[\textstyle{\frac{I + J}{y}}]) = \Spec(A[\textstyle{\frac{I + J}{xy}}]) = \emptyset $$ because $xy$ is both a nonzerodivisor and zero. Thus $b^{-1}(U)$ is the disjoint union of the open subscheme $U_1$ defined as the union of the standard opens $\Spec(A[\frac{I + J}{x}])$ for $x \in I$ and the open subscheme $U_2$ which is the union of the affine opens $\Spec(A[\frac{I + J}{y}])$ for $y \in J$. We have seen that $b^{-1}\mathcal{I}\mathcal{O}_{X'}$ restricts to zero on $U_2$ and $b^{-1}\mathcal{I}\mathcal{O}_{X'}$ restricts to zero on $U_1$. We omit the verification that these open subschemes glue to global open subschemes $X'_1$ and $X'_2$.
From the definition of the ideal of denominators in Definition \ref{definition-regular-meromorphic-ideal-denominators} we immediately see that $b$ is a $U$-admissible blowup. For the notation $1_D$, $1_E$, and $\mathcal{O}_{X'}(D - E)$ please see Definition \ref{definition-invertible-sheaf-effective-Cartier-divisor}. The pullback $b^*s$ is defined by Lemmas \ref{lemma-blow-up-pullback-effective-Cartier} and \ref{lemma-meromorphic-sections-pullback}. Thus the statement of the lemma makes sense. We can reinterpret the final assertion as saying that $b^*s$ is a global regular section of $b^*\mathcal{L}(E)$ whose zero scheme is $D$. This uniquely defines $D$ hence to prove the lemma we may work affine locally on $X$ and $X'$. Assume $X = \Spec(A)$ is affine and $\mathcal{L} = \mathcal{O}_X$. Then $s$ is a regular meromorphic function and shrinking further we may assume $s = a'/a$ with $a', a \in A$ nonzerodivisors. Then the ideal of denominators of $s$ corresponds to the ideal $I = \{x \in A \mid xa' \in aA\}$. Recall that $X'$ is covered by spectra of affine blowup algebras $A' = A[\frac{I}{x}]$ with $x \in I$ (Lemma \ref{lemma-blowing-up-affine}). Fix $x \in I$ and write $xa' = a a''$ for some $a'' \in A$. The divisor $E \subset X'$ is cut out by $x \in A'$ over the spectrum of $A'$ and hence $1/x$ is a  generator of $\mathcal{O}_{X'}(E)$ over $\Spec(A')$. Finally, in the total quotient ring of $A'$ we have $a'/a = a''/x$. Hence $b^*s = a'/a$ restricts to a regular section of $\mathcal{O}_{X'}(E)$ which is over $\Spec(A')$ given by $a''/x$. This finishes the proof. (The divisor $D \cap \Spec(A')$ is cut out by the image of $a''$ in $A'$.)
Recall that $\mathcal{F}'$ can locally be generated by $\leq k$ sections if and only if $\text{Fit}_k(\mathcal{F}') = \mathcal{O}_{S'}$, see Lemma \ref{lemma-fitting-ideal-generate-locally}. Hence this lemma is a translation of More on Algebra, Lemma \ref{more-algebra-lemma-blowup-fitting-ideal}.
The ideal $\text{Fit}_r(\mathcal{F})$ is of finite type by Lemma \ref{lemma-fitting-ideal-of-finitely-presented} and its restriction to $U$ is equal to $\mathcal{O}_U$ by Lemma \ref{lemma-fitting-ideal-finite-locally-free}. Hence $b : X' \to X$ is $U$-admissible, see Definition \ref{definition-admissible-blowup}. \medskip\noindent By Lemma \ref{lemma-fitting-ideal-finite-locally-free} the restriction of $\text{Fit}_{r - 1}(\mathcal{F})$ to $U$ is zero, and since $U$ is scheme theoretically dense we conclude that $\text{Fit}_{r - 1}(\mathcal{F}) = 0$ on all of $X$. Thus it follows from Lemma \ref{lemma-fitting-ideal-finite-locally-free} that $\mathcal{F}$ is locally free of rank $r$ on the complement of subscheme cut out by the $r$th Fitting ideal of $\mathcal{F}$ (this complement may be bigger than $U$ which is why we had to do this step in the argument). Hence by Lemma \ref{lemma-strict-transform-blowup-fitting-ideal-locally-free} the strict transform $$ b^*\mathcal{F} \longrightarrow \mathcal{F}' $$ is locally free of rank $r$. The kernel $\mathcal{K}$ of this map is supported on the exceptional divisor of the blowup $b$ and hence $\mathcal{K}|_U = 0$. Finally, since $\mathcal{F}'$ is finite locally free and since the displayed arrow is surjective, we can locally on $X'$ write $b^*\mathcal{F}$ as the direct sum of $\mathcal{K}$ and $\mathcal{F}'$. Since $b^*\mathcal{F}'$ is finitely presented (Modules, Lemma \ref{modules-lemma-pullback-finite-presentation}) the same is true for $\mathcal{K}$. \medskip\noindent The statement on tor dimension follows from More on Algebra, Lemma \ref{more-algebra-lemma-fitting-ideals-and-pd1}.
We prove this by induction on the rank $r$ of $\mathcal{E}$. If $r = 1$ or $r = 0$ the lemma is obvious. Assume $r > 1$. Let $P = \mathbf{P}(\mathcal{E})$ with structure morphism $\pi : P \to X$, see Constructions, Section \ref{constructions-section-projective-bundle}. Then $\pi$ is proper (Lemma \ref{lemma-relative-proj-proper}). There is a canonical surjection $$ \pi^*\mathcal{E} \to \mathcal{O}_P(1) $$ whose kernel is finite locally free of rank $r - 1$. Choose a nonempty open subscheme $U \subset X$ such that $\mathcal{E}|_U \cong \mathcal{O}_U^{\oplus r}$. Then $P_U = \pi^{-1}(U)$ is isomorphic to $\mathbf{P}^{r - 1}_U$. In particular, there exists a section $s : U \to P_U$ of $\pi$. Let $X' \subset P$ be the scheme theoretic image of the morphism $U \to P_U \to P$. Then $X'$ is integral (Morphisms, Lemma \ref{morphisms-lemma-scheme-theoretic-image-reduced}), the morphism $f = \pi|_{X'} : X' \to X$ is proper (Morphisms, Lemmas \ref{morphisms-lemma-closed-immersion-proper} and \ref{morphisms-lemma-composition-proper}), and $f^{-1}(U) \to U$ is an isomorphism. Hence $f$ is a modification (Morphisms, Definition \ref{morphisms-definition-modification}). By construction the pullback $f^*\mathcal{E}$ has a two step filtration whose quotient is invertible because it is equal to $\mathcal{O}_P(1)|_{X'}$ and whose sub $\mathcal{E}'$ is locally free of rank $r - 1$. By induction we can find a modification $g : X'' \to X'$ such that $g^*\mathcal{E}'$ has a filtration as in the statement of the lemma. Thus $f \circ g : X'' \to X$ is the required modification.
Denote $j : U \to X$ the inclusion morphism. Let $X' \subset Y \times_S X$ be the scheme theoretic image of $(f, j) : U \to Y \times_S X$ (Morphisms, Definition \ref{morphisms-definition-scheme-theoretic-image}). The projection $g : Y \times_S X \to X$ is proper (Morphisms, Lemma \ref{morphisms-lemma-base-change-proper}). The composition $p : X' \to X$ of $X' \to Y \times_S X$ and $g$ is proper (Morphisms, Lemmas \ref{morphisms-lemma-closed-immersion-proper} and \ref{morphisms-lemma-composition-proper}). Since $g$ is separated and $U \subset X$ is retrocompact (as $X$ is Noetherian) we conclude that $p^{-1}(U) \to U$ is an isomorphism by Morphisms, Lemma \ref{morphisms-lemma-scheme-theoretic-image-of-partial-section}. On the other hand, the composition $f' : X' \to Y$ of $X' \to Y \times_S X$ and the projection $Y \times_S X \to Y$ agrees with $f$ on $p^{-1}(U)$.
Case (1) follows from a combination of Lemmas \ref{lemma-finite-locally-free-has-norm} and \ref{lemma-norm-ample}. Case (3) follows from a combination of Lemmas \ref{lemma-Frobenius-gives-norm-for-reduction} and \ref{lemma-norm-ample}. In case (2) we first replace $X$ by an irreducible component of $X$ which dominates $Y$ (viewed as a reduced closed subscheme of $X$). Then we can apply Lemma \ref{lemma-norm-in-normal-case}.