hcju/stacksqa huggingface dataset:
Since $Z \amalg X' \to X$ is a surjective proper morphism of finite presentation we see that $\{Z \amalg X' \to X\}$ is an h covering (Lemma \ref{lemma-surjective-proper-finite-presentation-h}). We have $$ (Z \amalg X') \times_X (Z \amalg X') = Z \amalg E \amalg E \amalg X' \times_X X' $$ Since $\mathcal{F}$ is a Zariski sheaf we see that $\mathcal{F}$ sends disjoint unions to products. Thus the sheaf condition for the covering $\{Z \amalg X' \to X\}$ says that $\mathcal{F}(X) \to \mathcal{F}(Z) \times \mathcal{F}(X')$ is injective with image the set of pairs $(t, s')$ such that (a) $t|_E = s'|_E$ and (b) $s'$ is in the equalizer of the two maps $\mathcal{F}(X') \to \mathcal{F}(X' \times_X X')$. Next, observe that the obvious morphism $$ E \times_Z E \amalg X' \longrightarrow X' \times_X X' $$ is a surjective proper morphism of finite presentation as $b$ induces an isomorphism $X' \setminus E \to X \setminus Z$. We conclude that $\mathcal{F}(X' \times_X X') \to \mathcal{F}(E \times_Z E) \times \mathcal{F}(X')$ is injective. It follows that (a) $\Rightarrow$ (b) which means that the lemma is true.
First proof. Observe that $X \to X'$ is a proper surjective morphism of finite presentation and $X \times_{X'} X = X$. By the sheaf property for the h covering $\{X \to X'\}$ (Lemma \ref{lemma-surjective-proper-finite-presentation-h}) we conclude. \medskip\noindent Second proof (silly). The blow up of $X'$ in $X$ is the empty scheme. The reason is that the affine blowup algebra $A[\frac{I}{a}]$ (Algebra, Section \ref{algebra-section-blow-up}) is zero if $a$ is a nilpotent element of $A$. Details omitted. Hence we get an almost blow up square of the form $$ \xymatrix{ \emptyset \ar[r] \ar[d] & \emptyset \ar[d] \\ X \ar[r] & X' } $$ Since $\mathcal{F}$ is a sheaf we have that $\mathcal{F}(\emptyset)$ is a singleton. Applying Lemma \ref{lemma-blow-up-square-h} we get the conclusion.
This lemma is a formal consequence of Lemma \ref{lemma-refine-check-h} and our defnition of stacks in groupoids. For example, assume (1), (2), (3). To show that $\mathcal{S}$ is a stack, we have to prove descent for morphisms and objects, see Stacks, Definition \ref{stacks-definition-stack-in-groupoids}. \medskip\noindent If $x, y$ are objects of $\mathcal{S}$ over an object $U$ of $(\Sch/S)_h$, then our assumptions imply $\mathit{Isom}(x, y)$ is a presheaf on $(\Sch/U)_h$ which satisfies (1), (2), (3), and (4) of Lemma \ref{lemma-refine-check-h} and therefore is a sheaf. Some details omitted. \medskip\noindent Let $\{U_i \to U\}_{i \in I}$ be a covering of $(\Sch/S)_h$. Let $(x_i, \varphi_{ij})$ be a descent datum in $\mathcal{S}$ relative to the family $\{U_i \to U\}_{i \in I}$, see Stacks, Definition \ref{stacks-definition-descent-data}. Consider the rule $F$ which to $V/U$ in $(\Sch/U)_h$ associates the set of pairs $(y, \psi_i)$ where $y$ is an object of $\mathcal{S}_V$ and $\psi_i : y|_{U_i \times_U V} \to x_i|_{U_i \times_U V}$ is a morphism of $\mathcal{S}$ over $U_i \times_U V$ such that $$ \varphi_{ij}|_{U_i \times_U U_j \times_U V} \circ \psi_i|_{U_i \times_U U_j \times_U V} = \psi_j|_{U_i \times_U U_j \times_U V} $$ up to isomorphism. Since we already have descent for morphisms, it is clear that $F(V/U)$ is either empty or a singleton set. On the other hand, we have $F(U_{i_0}/U)$ is nonempty because it contains $(x_{i_0}, \varphi_{i_0i})$. Since our goal is to prove that $F(U/U)$ is nonempty, it suffices to show that $F$ is a sheaf on $(\Sch/U)_h$. To do this we may use the criterion of Lemma \ref{lemma-refine-check-h}. However, our assumptions (1), (2), (3) imply (by drawing some commutative diagrams which we omit), that properties (1), (2), (3), and (4) of Lemma \ref{lemma-refine-check-h} hold for $F$. \medskip\noindent We omit the verification that if $\mathcal{S}$ is a stack in groupoids, then (1), (2), and (3) are satisfied.
First assume that $A = \mathbf{Z}[f_1, f_2]$ is the polynomial ring. In this case our almost blow up square is the blowing up of $X = \Spec(A)$ in the closed subscheme $Z$ and in fact $X' \subset \mathbf{P}^1_X$ is an effective Cartier divisor cut out by the global section $f_2T_0 - f_1 T_1$ of $\mathcal{O}_{\mathbf{P}^1_X}(1)$. Thus we have a resolution $$ 0 \to \mathcal{O}_{\mathbf{P}^1_X}(-1) \to \mathcal{O}_{\mathbf{P}^1_X} \to \mathcal{O}_{X'} \to 0 $$ Using the description of the cohomology given in Cohomology of Schemes, Section \ref{coherent-section-cohomology-projective-space} it follows that in this case $\Gamma(X, \mathcal{O}_X) \to \Gamma(X', \mathcal{O}_{X'})$ is an isomorphism and $H^1(X', \mathcal{O}_{X'}) = 0$. \medskip\noindent Next, we observe that any diagram as in Example \ref{example-two-generators} is the base change of the diagram in the previous paragraph by the ring map $\mathbf{Z}[f_1, f_2] \to A$. Hence by More on Morphisms, Lemmas \ref{more-morphisms-lemma-check-h1-fibre-zero}, \ref{more-morphisms-lemma-h1-fibre-zero}, and \ref{more-morphisms-lemma-h1-fibre-zero-check-h0-kappa} we conclude that $H^1(X', \mathcal{O}_{X'})$ is zero in general and the surjectivity of the map $H^0(X, \mathcal{O}_X) \to H^0(X', \mathcal{O}_{X'})$ in general. \medskip\noindent Next, in the general case, let us study the kernel. If $a \in A$ maps to zero, then looking on affine charts we see that $$ a = (f_1x - f_2)(a_0 + a_1x + \ldots + a_rx^r)\text{ in }A[x] $$ for some $r \geq 0$ and $a_0, \ldots, a_r \in A$ and similarly $$ a = (f_1 - f_2y)(b_0 + b_1y + \ldots + b_s y^s)\text{ in }A[y] $$ for some $s \geq 0$ and $b_0, \ldots, b_s \in A$. This means we have $$ a = f_2 a_0,\ f_1 a_0 = f_2 a_1,\ \ldots,\ f_1 a_r = 0, \ a = f_1 b_0,\ f_2 b_0 = f_1 b_1,\ \ldots,\ f_2 b_s = 0 $$ If $(a', r', a'_i, s', b'_j)$ is a second such system, then we have $$ aa' = f_1f_2a_0b'_0 = f_1f_2a_1b'_1 = f_1f_2a_2b'_2 = \ldots = 0 $$ as desired.
Denote $\mathcal{F}$ the Zariski sheafification of the functor $$ X \longrightarrow \colim_F \Gamma(X, \mathcal{O}_X) $$ For quasi-compact and quasi-separated schemes $X$ we have $\mathcal{F}(X) = \colim_F \Gamma(X, \mathcal{O}_X)$. by Sheaves, Lemma \ref{sheaves-lemma-directed-colimits-sections} and the fact that $\mathcal{O}$ is a sheaf for the Zariski topology. Thus it suffices to show that $\mathcal{F}$ is a h sheaf. To prove this we check conditions (1), (2), (3), and (4) of Lemma \ref{lemma-refine-check-h}. Condition (1) holds because we performed an (almost unnecessary) Zariski sheafification. Condition (2) holds because $\mathcal{O}$ is an fppf sheaf (Descent, Lemma \ref{descent-lemma-sheaf-condition-holds}) and if $A$ is the equalizer of two maps $B \to C$ of $\mathbf{F}_p$-algebras, then $\colim_F A$ is the equalizer of the two maps $\colim_F B \to \colim_F C$. \medskip\noindent We check condition (3). Let $A, f, J$ be as in Example \ref{example-one-generator}. We have to show that $$ \colim_F A = \colim_F A/J \times_{\colim_F A/fA + J} \colim_F A/fA $$ This reduces to the following algebra question: suppose $a', a'' \in A$ are such that $F^n(a' - a'') \in fA + J$. Find $a \in A$ and $m \geq 0$ such that $a - F^m(a') \in J$ and $a - F^m(a'') \in fA$ and show that the pair $(a, m)$ is uniquely determined up to a replacement of the form $(a, m) \mapsto (F(a), m + 1)$. To do this just write $F^n(a' - a'') = f h + g$ with $h \in A$ and $g \in J$ and set $a = F^n(a') - g = F^n(a'') + fh$ and set $m = n$. To see uniqueness, suppose $(a_1, m_1)$ is a second solution. By a replacement of the form given above we may assume $m = m_1$. Then we see that $a - a_1 \in J$ and $a - a_1 \in fA$. Since $J$ is annihilated by a power of $f$ we see that $a - a_1$ is a nilpotent element. Hence $F^k(a - a_1)$ is zero for some large $k$. Thus after doing more replacements we get $a = a_1$. \medskip\noindent We check condition (4). Let $X, X', Z, E$ be as in Example \ref{example-two-generators}. By Lemma \ref{lemma-funny-blow-up} we see that $$ \mathcal{F}(X) = \colim_F \Gamma(X, \mathcal{O}_X) \longrightarrow \colim_F \Gamma(X', \mathcal{O}_{X'}) = \mathcal{F}(X') $$ is bijective. Since $E = \mathbf{P}^1_Z$ in this case we also see that $\mathcal{F}(Z) \to \mathcal{F}(E)$ is bijective. Thus the conclusion holds in this case as well.
To prove $\mathcal{F}$ is a sheaf, let's check conditions (1), (2), (3), and (4) of Lemma \ref{lemma-refine-check-h}. Condition (1) holds because limits of sheaves are sheaves and $\mathcal{O}$ is a Zariski sheaf. Condition (2) holds because $\mathcal{O}$ is an fppf sheaf (Descent, Lemma \ref{descent-lemma-sheaf-condition-holds}) and if $A$ is the equalizer of two maps $B \to C$ of $\mathbf{F}_p$-algebras, then $\lim_F A$ is the equalizer of the two maps $\lim_F B \to \lim_F C$. \medskip\noindent We check condition (3). Let $A, f, J$ be as in Example \ref{example-one-generator}. We have to show that \begin{align*} \lim_F A & \to \lim_F A/J \times_{\lim_F A/fA + J} \lim_F A/fA \\ & = \lim_F (A/J \times_{A/fA + J} A/fA) \\ & = \lim_F A/(fA \cap J) \end{align*} is bijective. Since $J$ is annihilated by a power of $f$ we see that $\mathfrak a = fA \cap J$ is a nilpotent ideal, i.e., there exists an $n$ such that $\mathfrak a^n = 0$. It is straightforward to verify that in this case $\lim_F A \to \lim_F A/\mathfrak a$ is bijective. \medskip\noindent We check condition (4). Let $X, X', Z, E$ be as in Example \ref{example-two-generators}. By Lemma \ref{lemma-funny-blow-up} and the same argument as above we see that $$ \mathcal{F}(X) = \lim_F \Gamma(X, \mathcal{O}_X) \longrightarrow \lim_F \Gamma(X', \mathcal{O}_{X'}) = \mathcal{F}(X') $$ is bijective. Since $E = \mathbf{P}^1_Z$ in this case we also see that $\mathcal{F}(Z) \to \mathcal{F}(E)$ is bijective. Thus the conclusion holds in this case as well.
To prove $\mathcal{F}$ is a sheaf, let's check conditions (1) and (2) of Topologies, Lemma \ref{topologies-lemma-characterize-sheaf}. Condition (1) holds because formation of $X^{awn}$ commutes with open coverings, see Morphisms, Lemma \ref{morphisms-lemma-seminormalization} and its proof. \medskip\noindent Let $\pi : Y \to X$ be a surjective proper morphism. We have to show that the equalizer of the two maps $$ \Gamma(Y^{awn}, \mathcal{O}_{Y^{awn}}) \to \Gamma((Y \times_X Y)^{awn}, \mathcal{O}_{(Y \times_X Y)^{awn}}) $$ is equal to $\Gamma(X^{awn}, \mathcal{O}_{X^{awn}})$. Let $f$ be an element of this equalizer. Then we consider the morphism $$ f : Y^{awn} \longrightarrow \mathbf{A}^1_X $$ Since $Y^{awn} \to X$ is universally closed, the scheme theoretic image $Z$ of $f$ is a closed subscheme of $\mathbf{A}^1_X$ proper over $X$ and $f : Y^{awn} \to Z$ is surjective. See Morphisms, Lemma \ref{morphisms-lemma-scheme-theoretic-image-is-proper}. Thus $Z \to X$ is finite (Morphisms, Lemma \ref{morphisms-lemma-finite-proper}) and surjective. \medskip\noindent Let $k$ be a field and let $z_1, z_2 : \Spec(k) \to Z$ be two morphisms equalized by $Z \to X$. We claim that $z_1 = z_2$. It suffices to show the images $\lambda_i = z_i^*f \in k$ agree (as the structure sheaf of $Z$ is generated by $f$ over the structure sheaf of $X$). To see this we choose a field extension $K/k$ and morphisms $y_1, y_2 : \Spec(K) \to Y^{awn}$ such that $z_i \circ (\Spec(K) \to \Spec(k)) = f \circ y_i$. This is possible by the surjectivity of the map $Y^{awn} \to Z$. Choose an algebraically closed extension $\Omega/k$ of very large cardinality. For any $k$-algebra maps $\sigma_i : K \to \Omega$ we obtain $$ \Spec(\Omega) \xrightarrow{\sigma_1, \sigma_2} \Spec(K \otimes_k K) \xrightarrow{y_1, y_2} Y^{awn} \times_X Y^{awn} $$ Since the canonical morphism $(Y \times_X Y)^{awn} \to Y^{awn} \times_X Y^{awn}$ is a universal homeomorphism and since $\Omega$ is algebraically closed, we can lift the composition above uniquely to a morphism $\Spec(\Omega) \to (Y \times_X Y)^{awn}$. Since $f$ is in the equalizer above, this proves that $\sigma_1(\lambda_1) = \sigma_2(\lambda_2)$. An easy lemma about field extensions shows that this implies $\lambda_1 = \lambda_2$; details omitted. \medskip\noindent We conclude that $Z \to X$ is universally injective, i.e., $Z \to X$ is injective on points and induces purely inseparated residue field extensions (Morphisms, Lemma \ref{morphisms-lemma-universally-injective}). All in all we conclude that $Z \to X$ is a universal homeomorphism, see Morphisms, Lemma \ref{morphisms-lemma-universal-homeomorphism}. \medskip\noindent Let $g : X^{awn} \to Z$ be the map obtained from the universal property of $X^{awn}$. Then $Y^{awn} \to X^{awn} \to Z$ and $f : Y^{awn} \to Z$ are two morphisms over $X$. By the universal property of $Y^{awn} \to Y$ the two corresponding morphisms $Y^{awn} \to Y \times_X Z$ over $Y$ have to be equal. This implies that $g \circ \pi^{wan} = f$ as morphisms into $\mathbf{A}^1_X$ and we conclude that $g \in \Gamma(X^{awn}, \mathcal{O}_{X^{awn}})$ is the element we were looking for.
In Lemma \ref{lemma-weak-normalization-ph-sheaf} we have seen that the rule $\mathcal{F}$ of the lemma defines a sheaf in the ph topology and hence a fortiori a sheaf for the h topology. Clearly, there is a canonical map of presheaves of rings $\mathcal{O} \to \mathcal{F}$. To finish the proof, it suffices to show \begin{enumerate} \item if $f \in \mathcal{O}(X)$ maps to zero in $\mathcal{F}(X)$, then there is a h covering $\{X_i \to X\}$ such that $f|_{X_i} = 0$, and \item given $f \in \mathcal{F}(X)$ there is a h covering $\{X_i \to X\}$ such that $f|_{X_i}$ is the image of $f_i \in \mathcal{O}(X_i)$. \end{enumerate} Let $f$ be as in (1). Then $f|_{X^{awn}} = 0$. This means that $f$ is locally nilpotent. Thus if $X' \subset X$ is the closed subscheme cut out by $f$, then $X' \to X$ is a surjective closed immersion of finite presentation. Hence $\{X' \to X\}$ is the desired h covering. Let $f$ be as in (2). After replacing $X$ by the members of an affine open covering we may assume $X = \Spec(A)$ is affine. Then $f \in A^{awn}$, see Morphisms, Lemma \ref{morphisms-lemma-seminormalization-ring}. By Morphisms, Lemma \ref{morphisms-lemma-universal-homeo-limit} we can find a ring map $A \to B$ of finite presentation such that $\Spec(B) \to \Spec(A)$ is a universal homeomorphism and such that $f$ is the image of an element $b \in B$ under the canonical map $B \to A^{awn}$. Then $\{\Spec(B) \to \Spec(A)\}$ is an h covering and we conclude. The statement about the ph topology follows in the same manner (or it can be deduced from the statement for the h topology).
It is immediate from condition (2)(b) in Morphisms, Definition \ref{morphisms-definition-seminormal-ring} that if $A$ is absolutely weakly normal, then it is perfect. \medskip\noindent Assume $A$ is perfect. Suppose $x, y \in A$ with $x^3 = y^2$. If $p > 3$ then we can write $p = 2n + 3m$ for some $n, m > 0$. Choose $a, b \in A$ with $a^p = x$ and $b^p = y$. Setting $c = a^n b^m$ we have $$ c^{2p} = x^{2n} y^{2m} = x^{2n + 3m} = x^p $$ and hence $c^2 = x$. Similarly $c^3 = y$. If $p = 2$, then write $x = a^2$ to get $a^6 = y^2$ which implies $a^3 = y$. If $p = 3$, then write $y = a^3$ to get $x^3 = a^6$ which implies $x = a^2$. \medskip\noindent Suppose $x, y \in A$ with $\ell^\ell x = y^\ell$ for some prime number $\ell$. If $\ell \not = p$, then $a = y/\ell$ satsifies $a^\ell = x$ and $\ell a = y$. If $\ell = p$, then $y = 0$ and $x = a^p$ for some $a$.
Proof of (1). Observe that $A \to \colim_F A$ induces a universal homeomorphism on spectra by Algebra, Lemma \ref{algebra-lemma-p-ring-map}. Thus it suffices to show that $B = \colim_F A$ is absolutely weakly normal, see Morphisms, Lemma \ref{morphisms-lemma-seminormalization-ring}. Note that the ring map $F : B \to B$ is an automorphism, in other words, $B$ is a perfect ring. Hence Lemma \ref{lemma-perfect-weankly-normal} applies. \medskip\noindent Proof of (2). This follows from (1) and Lemmas \ref{lemma-h-sheaf-colim-F} and \ref{lemma-weak-normalization-h-sheaf} by looking affine locally.
Let $A, f, J$ be as in Example \ref{example-one-generator}. Since all our schemes are affine and since we have internal Hom in the category of vector bundles, the fully faithfulness of the functor follows if we can show that $$ \colim P \otimes_{A, F^N} A = \colim P \otimes_{A, F^N} A/J \times_{\colim P \otimes_{A, F^N} A/fA + J} \colim P \otimes_{A, F^N} A/fA $$ for a finite projective $A$-module $P$. After writing $P$ as a summand of a finite free module, this follows from the case where $P$ is finite free. This case immediately reduces to the case $P = A$. The case $P = A$ follows from Lemma \ref{lemma-h-sheaf-colim-F} (in fact we proved this case directly in the proof of this lemma). \medskip\noindent Essential surjectivity. Here we obtain the following algebra problem. Suppose $P_1$ is a finite projective $A/J$-module, $P_2$ is a finite projective $A/fA$-module, and $$ \varphi : P_1 \otimes_{A/J} A/fA + J \longrightarrow P_2 \otimes_{A/fA} A/fA + J $$ is an isomorphism. Goal: show that there exists an $N$, a finite projective $A$-module $P$, an isomorphism $\varphi_1 : P \otimes_A A/J \to P_1 \otimes_{A/J, F^N} A/J$, and an isomorphism $\varphi_2 : P \otimes_A A/fA \to P_2 \otimes_{A/fA, F^N} A/fA$ compatible with $\varphi$ in an obvious manner. This can be seen as follows. First, observe that $$ A/(J \cap fA) = A/J \times_{A/fA + J} A/fA $$ Hence by More on Algebra, Lemma \ref{more-algebra-lemma-finitely-presented-module-over-fibre-product} there is a finite projective module $P'$ over $A/(J \cap fA)$ which comes with isomorphisms $\varphi'_1 : P' \otimes_A A/J \to P_1$ and $\varphi_2 : P' \otimes_A A/fA \to P_2$ compatible with $\varphi$. Since $J$ is a finitely generated ideal and $f$-power torsion we see that $J \cap fA$ is a nilpotent ideal. Hence for some $N$ there is a factorization $$ A \xrightarrow{\alpha} A/(J \cap fA) \xrightarrow{\beta} A $$ of $F^N$. Setting $P = P' \otimes_{A/(J \cap fA), \beta} A$ we conclude.
We will prove the canonical map $$ \Phi : f^*f_*\mathcal{E} \longrightarrow \mathcal{E} $$ is an isomorphism. By flat base change (Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}) and assumptions (1) and (3) we see that the pullback of this to $X \times_S \Spec(\mathcal{O}_{S, s})$ is an isomorphism for every codimension $1$ point $s \in S$. By Divisors, Lemma \ref{divisors-lemma-check-isomorphism-via-depth-and-ass} it suffices to prove that $\text{depth}((f^*f_*\mathcal{E})_x) \geq 2$ for any point $x \in X$ mapping to a point $s \in S$ of codimension $\geq 2$. Since $f$ is flat and $(f^*f_*\mathcal{E})_x = (f_*\mathcal{E})_s \otimes_{\mathcal{O}_{S, s}} \mathcal{O}_{X, x}$, it suffices to prove that $\text{depth}((f_*\mathcal{E})_s) \geq 2$, see Algebra, Lemma \ref{algebra-lemma-apply-grothendieck}. Since $S$ is a normal Noetherian scheme and $\dim(\mathcal{O}_{S, s}) \geq 2$ we have $\text{depth}(\mathcal{O}_{S, s}) \geq 2$, see Properties, Lemma \ref{properties-lemma-criterion-normal}. Thus we get what we want from Divisors, Lemma \ref{divisors-lemma-depth-pushforward}.
Observe that $E$ locally on $X$ has the shape as stated in the lemma, see More on Algebra, Section \ref{more-algebra-section-pseudo-coherent}, Cohomology, Section \ref{cohomology-section-pseudo-coherent}, and Derived Categories of Schemes, Section \ref{perfect-section-spell-out}. Thus it suffices to prove that the ideal of minors is independent of the chosen representative. To do this, it suffices to check in local rings. Over a local ring $(R, \mathfrak m, \kappa)$ consider a bounded above complex $$ F^\bullet : \ldots \to R^{\oplus n_{b - 2}} \xrightarrow{d_{b - 2}} R^{\oplus n_{b - 1}} \xrightarrow{d_{b - 1}} R^{\oplus n_b} \to 0 \to \ldots $$ Denote $\text{Fit}_{k, p}(F^\bullet) \subset R$ the ideal generated by the minors of size $k - n_{p + 1} + n_{p + 2} - \ldots + (-1)^{b - p} n_b$ in the matrix of $d_p$. Suppose some matrix coefficient of some differential of $F^\bullet$ is invertible. Then we pick a largest integer $i$ such that $d_i$ has an invertible matrix coefficient. By Algebra, Lemma \ref{algebra-lemma-add-trivial-complex} the complex $F^\bullet$ is isomorphic to a direct sum of a trivial complex $\ldots \to 0 \to R \to R \to 0 \to \ldots$ with nonzero terms in degrees $i$ and $i + 1$ and a complex $(F')^\bullet$. We leave it to the reader to see that $\text{Fit}_{p, k}(F^\bullet) = \text{Fit}_{p, k}((F')^\bullet)$; this is where the formula for the size of the minors is used. If $(F')^\bullet$ has another differential with an invertible matrix coefficient, we do it again, etc. Continuing in this manner, we eventually reach a complex $(F^\infty)^\bullet$ all of whose differentials have matrices with coefficients in $\mathfrak m$. Here you may have to do an infinite number of steps, but for any cutoff only a finite number of these steps affect the complex in degrees $\geq $ the cutoff. Thus the ``limit'' $(F^\infty)^\bullet$ is a well defined bounded above complex of finite free modules, comes equipped with a quasi-isomorphism $(F^\infty)^\bullet \to F^\bullet$ into the complex we started with, and $\text{Fit}_{p, k}(F^\bullet) = \text{Fit}_{p, k}((F^\infty)^\bullet)$. Since the complex $(F^\infty)^\bullet$ is unique up to isomorphism by More on Algebra, Lemma \ref{more-algebra-lemma-lift-pseudo-coherent-from-residue-field} the proof is complete.
We will construct and study the blowup affine locally. Namely, suppose that $V \subset X$ is an affine open subscheme such that $E|_V$ can be represented by the complex $$ \mathcal{O}_V^{\oplus n_a} \xrightarrow{d_a} \ldots \xrightarrow{d_{b - 1}} \mathcal{O}_V^{\oplus n_b} $$ Set $k_ i = r_{i + 1} - r_{i + 2} + \ldots + (-1)^{b - i + 1}r_b$. A computation which we omit show that over $U \cap V$ the rank of $d_i$ is $$ \rho_i = - k_i + n_{i + 1} - n_{i + 2} + \ldots + (-1)^{b - i + 1}n_b $$ in the sense that the cokernel of $d_i$ is finite locally free of rank $n_{i + 1} - \rho_i$. Let $\mathcal{I}_i \subset \mathcal{O}_V$ be the ideal generated by the minors of size $\rho_i \times \rho_i$ in the matrix of $d_i$. \medskip\noindent On the one hand, comparing with Lemma \ref{lemma-fitting-ideals-complex} we see the ideal $\mathcal{I}_i$ corresponds to the global ideal $\text{Fit}_{i, k_i}(E)$ which was shown to be independent of the choice of the complex representing $E|_V$. On the other hand, $\mathcal{I}_i$ is the $(n_{i + 1} - \rho_i)$th Fitting ideal of $\Coker(d_i)$. Please keep this in mind. \medskip\noindent We let $b : X' \to X$ be the blowing up in the product of the ideals $\text{Fit}_{i, k_i}(E)$; this makes sense as locally on $X$ almost all of these ideals are equal to the unit ideal (see above). This blowup dominates the blowups $b_i : X'_i \to X$ in the ideals $\text{Fit}_{i, k_i}(E)$, see Divisors, Lemma \ref{divisors-lemma-blowing-up-two-ideals}. By Divisors, Lemma \ref{divisors-lemma-blowup-fitting-ideal} each $b_i$ is a $U$-admissible blowup. It follows that $b$ is a $U$-admissible blowup (tiny detail omitted; compare with the proof of Divisors, Lemma \ref{divisors-lemma-dominate-admissible-blowups}). Finally, $U$ is still a scheme theoretically dense open subscheme of $X'$. Thus after replacing $X$ by $X'$ we end up in the situation discussed in the next paragraph. \medskip\noindent Assume $\text{Fit}_{i, k_i}(E)$ is an invertible ideal for all $i$. Choose an affine open $V$ and a complex of finite free modules representing $E|_V$ as above. It follows from Divisors, Lemma \ref{divisors-lemma-blowup-fitting-ideal} that $\Coker(d_i)$ has tor dimension $\leq 1$. Thus $\Im(d_i)$ is finite locally free as the kernel of a map from a finite locally free module to a finitely presented module of tor dimension $\leq 1$. Hence $\Ker(d_i)$ is finite locally free as well (same argument). Thus the short exact sequence $$ 0 \to \Im(d_{i - 1}) \to \Ker(d_i) \to H^i(E)|_V \to 0 $$ shows what we want and the proof is complete.