hcju/stacksqa huggingface dataset:
We will prove this by verifying hypotheses (a), (b), and (c) of Lemma \ref{lemma-when-done}. For every $n$ we have a short exact sequence $$ 0 \to I^n\mathcal{F}_{n + 1} \to \mathcal{F}_{n + 1} \to \mathcal{F}_n \to 0 $$ Since $f_1, \ldots, f_d$ forms a regular sequence (and hence quasi-regular, see Algebra, Lemma \ref{algebra-lemma-regular-quasi-regular}) on each of the ``stalks'' $\mathcal{F}_y^\wedge$ and since we have $I\mathcal{F}_n = (f_1, \ldots, f_d)\mathcal{F}_n$ for all $n$, we find that $$ I^n\mathcal{F}_{n + 1} = \bigoplus\nolimits_{e_1 + \ldots + e_d = n} \mathcal{F}_1 \cdot f_1^{e_1} \ldots f_d^{e_d} $$ by checking on stalks. Using the assumption of finiteness of $H^0(U, \mathcal{F}_1)$ and induction, we first conclude that $M_n = H^0(U, \mathcal{F}_n)$ is a finite $A$-module for all $n$. In this way we see that condition (c) of Lemma \ref{lemma-when-done} holds. We also see that $$ \bigoplus\nolimits_{n \geq 0} H^1(U, I^n\mathcal{F}_{n + 1}) $$ is a finite graded $R = \bigoplus I^n/I^{n +1}$-module. By Lemma \ref{lemma-ML-general} we conclude that condition (a) of Lemma \ref{lemma-when-done} is satisfied. Finally, condition (b) of Lemma \ref{lemma-when-done} is satisfied because $\bigoplus H^0(U, I^n\mathcal{F}_{n + 1})$ is a finite graded $R$-module and we can apply Lemma \ref{lemma-topology-I-adic-general}.
The proof is exactly the same as the proof of Proposition \ref{proposition-algebraization-regular-sequence}. Namely, if $\kappa = R/\mathfrak m$ then for $n \geq 0$ there is an isomorphism $$ I^n \mathcal{F}_{n + 1} \cong \mathcal{F}_1 \otimes_\kappa \mathfrak m^n/\mathfrak m^{n + 1} $$ and the right hand side is a finite direct sum of copies of $\mathcal{F}_1$. This can be checked by looking at stalks. Everything else is exactly the same.
By Lemma \ref{lemma-local-pic-to-completion} we may assume that $A$ is a complete local ring. By assumption we can write $A = B/(f_1, \ldots, f_r)$ where $B$ is a complete regular local ring and $f_1, \ldots, f_r$ is a regular sequence. We will finish the proof by induction on $r$. The base case is $r = 0$ which follows from Lemma \ref{lemma-trivial-local-pic-regular}. \medskip\noindent Assume that $A = B/(f_1, \ldots, f_r)$ and that the proposition holds for $r - 1$. Set $A' = B/(f_1, \ldots, f_{r - 1})$ and apply Lemma \ref{lemma-surjective-Pic-first-better} to $f_r \in A'$. This is permissible: \begin{enumerate} \item condition (1) of Lemma \ref{lemma-surjective-Pic-first} holds because our local rings are complete, \item condition (2) of Lemma \ref{lemma-surjective-Pic-first} holds holds as $f_1, \ldots, f_r$ is a regular sequence, \item condition (3) and (4) of Lemma \ref{lemma-surjective-Pic-first} hold as $A = A'/f_r A'$ is Cohen-Macaulay of dimension $\dim(A) \geq 4$, \item condition (2) of Lemma \ref{lemma-surjective-Pic-first-better} holds by induction hypothesis as $\dim((A'_{f_r})_\mathfrak p) \geq 4$ for a maximal prime $\mathfrak p$ of $A'_{f_r}$ and as $(A'_{f_r})_\mathfrak p = B_\mathfrak q/(f_1, \ldots, f_{r - 1})$ for some prime ideal $\mathfrak q \subset B$ and $B_\mathfrak q$ is regular. \end{enumerate} This finishes the proof.
To prove fully faithfulness it suffices to prove that $$ \colim_\mathcal{V} \Gamma(V, \mathcal{L}^{\otimes m}) \longrightarrow \lim \Gamma(Y_n, \mathcal{L}^{\otimes m}|_{Y_n}) $$ is an isomorphism for all $m$, see Lemma \ref{lemma-completion-fully-faithful-general}. This follows from Lemma \ref{lemma-lefschetz-addendum}. \medskip\noindent Essential surjectivity. Let $(\mathcal{F}_n)$ be a finite locally free object of $\textit{Coh}(X, \mathcal{I})$. Then for $y \in Y$ we have $\mathcal{F}_y^\wedge = \lim \mathcal{F}_{n, y}$ is is a finite free $\mathcal{O}_{X, y}^\wedge$-module. Let $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge$ be a prime with $\mathfrak p \not \in V(\mathcal{I}_y^\wedge)$. Then $\mathfrak p$ lies over a prime $\mathfrak p_0 \subset \mathcal{O}_{X, y}$ which corresponds to a specialization $x \leadsto y$ with $x \not \in Y$. By Local Cohomology, Lemma \ref{local-cohomology-lemma-change-completion} and some dimension theory (see Varieties, Section \ref{varieties-section-algebraic-schemes}) we have $$ \text{depth}((\mathcal{O}_{X, y}^\wedge)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) = \text{depth}(\mathcal{O}_{X, x}) + \dim(\overline{\{x\}}) - \dim(\overline{\{y\}}) $$ Thus our assumptions imply the assumptions of Proposition \ref{proposition-lefschetz-existence} are satisfied and we find that $(\mathcal{F}_n)$ is the completion of a coherent $\mathcal{O}_X$-module $\mathcal{F}$. It then follows that $\mathcal{F}_y$ is finite free for all $y \in Y$ and hence $\mathcal{F}$ is finite locally free in an open neighbourhood $V$ of $Y$. This finishes the proof.
We want to reduce the proof to the case of schemes. To do this we choose a commutative diagram $$ \xymatrix{ U \ar[r] \ar[d]_g & X \ar[d]^f \\ V \ar[r] & Y } $$ where $U$, $V$ are schemes and where the horizontal arrows are \'etale and surjective. Pick $v \in V$ mapping to $y$. Observe that $V$ is locally Noetherian and that $\dim(\mathcal{O}_{V, v}) \leq 1$ (see Properties of Spaces, Definitions \ref{spaces-properties-definition-dimension-local-ring} and Remark \ref{spaces-properties-remark-list-properties-local-etale-topology}). The fibre $U_v$ of $U \to V$ over $v$ surjects onto $f^{-1}(\{y\}) \subset |X|$. The inverse image of $X^0$ in $U$ is exactly the set of generic points of irreducible components of $U$ (Properties of Spaces, Lemma \ref{spaces-properties-lemma-codimension-0-points}). If $\eta \in U$ is such a point with image $x \in X^0$, then the transcendence degree of $x / f(x)$ is the transcendence degree of $\kappa(\eta)$ over $\kappa(g(\eta))$ (Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-dimension-fibre}). Observe that $U \to V$ is quasi-finite at $u \in U$ if and only if $f$ is quasi-finite at the image of $u$ in $X$. \medskip\noindent Case (1). Here case (1) of Varieties, Lemma \ref{varieties-lemma-quasi-finite-in-codim-1} applies and we conclude that $U \to V$ is quasi-finite at all points of $U_v$. Hence $f$ is quasi-finite at every point lying over $y$. \medskip\noindent Case (2). Let $u \in U$ be a generic point of an irreducible component whose image in $V$ specializes to $v$. Then the image $x \in X^0$ of $u$ has the property that $f(x) \leadsto y$. Hence we see that case (2) of Varieties, Lemma \ref{varieties-lemma-quasi-finite-in-codim-1} applies and we conclude as before. \medskip\noindent Case (3) follows from case (3) of Varieties, Lemma \ref{varieties-lemma-quasi-finite-in-codim-1}. \medskip\noindent In case (4), since $|U| \to |X|$ is open, we see that the set of points where $U \to V$ is quasi-finite is dense as well. Hence case (4) of Varieties, Lemma \ref{varieties-lemma-quasi-finite-in-codim-1} applies.
By Lemma \ref{lemma-quasi-finite-in-codim-1} the morphism $f$ is quasi-finite at every point lying over $y$. Let $\overline{y} : \Spec(k) \to Y$ be a geometric point lying over $y$. Then $|X_{\overline{y}}|$ is a discrete space (Decent Spaces, Lemma \ref{decent-spaces-lemma-conditions-on-fibre-and-qf}). Since $X_{\overline{y}}$ is quasi-compact as $f$ is proper we conclude that $|X_{\overline{y}}|$ is finite. Thus we can apply Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-proper-finite-fibre-finite-in-neighbourhood} to conclude.
Part (1) follows from Decent Spaces, Lemma \ref{decent-spaces-lemma-birational-isomorphism-over-dense-open} (and the fact that the Noetherian algebraic spaces $X$ and $Y$ are quasi-separated and hence decent). Part (2) follows from part (3) and Lemma \ref{lemma-finite-in-codim-1} (and the fact that finite morphisms have finite fibres). Let $x \in |X|$ be as in (3). By Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-proper-finite-fibre-finite-in-neighbourhood} (which applies by Decent Spaces, Lemma \ref{decent-spaces-lemma-conditions-on-fibre-and-qf}) we may assume $f$ is finite. Choose an affine scheme $X'$ and an \'etale morphism $X' \to X$ and a point $x' \in X$ mapping to $x$. It suffices to show there exists an open neighbourhood $U'$ of $x' \in X'$ such that $Y \times_X X' \to X'$ is an isomorphism over $U'$ (namely, then $U$ contains the image of $U'$ in $X$, see Spaces, Lemma \ref{spaces-lemma-descent-representable-transformations-property}). Then $Y \times_X X' \to X$ is a finite birational (Decent Spaces, Lemma \ref{decent-spaces-lemma-birational-etale-localization}) morphism. Since a finite morphism is affine we reduce to the case of a finite birational morphism of Noetherian affine schemes $Y \to X$ and $x \in X$ such that $\mathcal{O}_{X, x}$ is a normal domain. This is treated in Varieties, Lemma \ref{varieties-lemma-modification-normal-iso-over-codimension-1}.
Set $R = \Gamma(X, \mathcal{O}_X)$. If $f, g \in R$ are nonzero and $fg = 0$ then $X = V(f) \cup V(g)$ where $V(f)$ denotes the closed subspace of $X$ cut out by $f$. Since $X$ is irreducible, we see that either $V(f) = X$ or $V(g) = X$. Then either $f = 0$ or $g = 0$ by Properties of Spaces, Lemma \ref{spaces-properties-lemma-reduced-space}.
Choose an affine scheme $U$ and an \'etale morphism $U \to X$ such that $x$ is in the image. Let $u_i$, $i \in I$ be the generic points of irreducible components of $U$. Then each $u_i$ maps to the generic point of $X$ (Decent Spaces, Lemma \ref{decent-spaces-lemma-decent-generic-points}). By  our definition of a decent space (Decent Spaces, Definition \ref{decent-spaces-definition-very-reasonable}), we see that $I$ is finite. Hence $U = \Spec(A)$ where $A$ is a normal ring with finitely many minimal primes. Thus $A = \prod_{i \in I} A_i$ is a product of normal domains by Algebra, Lemma \ref{algebra-lemma-characterize-reduced-ring-normal}. Then $U = \coprod U_i$ with $U_i = \Spec(A_i)$ and $x$ is in the image of $U_i \to X$ for some $i$. This proves the lemma.
Set $R = \Gamma(X, \mathcal{O}_X)$. Then $R$ is a domain by Lemma \ref{lemma-integral-sections}. Let $f = a/b$ be an element of the fraction field of $R$ which is integral over $R$. For any $U \to X$ \'etale with $U$ a scheme there is at most one $f_U \in \Gamma(U, \mathcal{O}_U)$ with $b|_U f_U = a|_U$. Namely, $U$ is reduced and the generic points of $U$ map to the generic point of $X$ which implies that $b|_U$ is a nonzerodivisor. For every $x \in |X|$ we choose $U \to X$ as in Lemma \ref{lemma-normal-integral-cover-by-affines}. Then there is a unique $f_U \in \Gamma(U, \mathcal{O}_U)$ with $b|_U f_U = a|_U$ because $\Gamma(U, \mathcal{O}_U)$ is a normal domain by Properties, Lemma \ref{properties-lemma-normal-integral-sections}. By the uniqueness mentioned above these $f_U$ glue and define a global section $f$ of the structure sheaf, i.e., of $R$.
Our map defines a bijection between (1) and (2) as $|X|$ is sober by  Decent Spaces, Proposition \ref{decent-spaces-proposition-reasonable-sober}. Given $T \subset |X|$ closed and irreducible, there is a unique reduced closed subspace $Z \subset X$ such that $|Z| = T$, namely, $Z$ is the reduced induced subspace structure on $T$, see Properties of Spaces, Definition \ref{spaces-properties-definition-reduced-induced-space}. This is an integral algebraic space because it is decent, reduced, and irreducible.
By elementary topology, we see that $f(x) = y$ as $f$ is dominant. Let $Y' \subset Y$ be the schematic locus of $Y$ and let $X' \subset f^{-1}(Y')$ be the schematic locus of $f^{-1}(Y')$. By the discussion above, using Decent Spaces, Proposition \ref{decent-spaces-proposition-reasonable-sober} and Theorem \ref{decent-spaces-theorem-decent-open-dense-scheme}, we see that $x \in |X'|$ and $y \in |Y'|$. Then $f|_{X'} : X' \to Y'$ is a morphism of integral schemes which is locally of finite type. Thus we see that (1), (2), (3) are equivalent by Morphisms, Lemma \ref{morphisms-lemma-finite-degree}. \medskip\noindent Condition (4) implies condition (1) by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-compare-tr-deg} applied to $X \to Y \to Y$. On the other hand, condition (3) implies condition (4) as a finite morphism is quasi-finite and as $x \in U$ because $x$ is the generic point. Thus (1) -- (4) are equivalent. \medskip\noindent Assume the equivalent conditions (1) -- (4). Suppose that $x' \mapsto y$. Then $x \leadsto x'$ is a specialization in the fibre of $|X| \to |Y|$ over $y$. If $x' \not = x$, then $f$ is not quasi-finite at $x$ by Decent Spaces, Lemma \ref{decent-spaces-lemma-conditions-on-point-in-fibre-and-qf}. Hence $x = x'$ and (5) holds. Conversely, if (5) holds, then (5) holds for the morphism of schemes $X' \to Y'$ (see above) and we can use Morphisms, Lemma \ref{morphisms-lemma-finite-degree} to see that (1) holds. \medskip\noindent Observe that (6) implies the equivalent conditions (1) -- (5) without any further assumptions on $f$. To finish the proof we have to show the equivalent conditions (1) -- (5) imply (6). This follows from Decent Spaces, Lemma \ref{decent-spaces-lemma-finite-over-dense-open}.
The topological space $|X|$ is locally Noetherian (Properties of Spaces, Lemma \ref{spaces-properties-lemma-Noetherian-topology}). A Noetherian topological space has a finite number of irreducible components and a subspace of a Noetherian space is Noetherian (Topology, Lemma \ref{topology-lemma-Noetherian}). Thus the lemma follows from the definition of locally finite (Topology, Definition \ref{topology-definition-locally-finite}).
We will use the results of Decent Spaces, Section \ref{decent-spaces-section-residue-fields-henselian-local-rings}. Let $(U, u) \to (X, \xi)$ be an elementary \'etale neighbourhood. Observe that $U$ is locally Noetherian and reduced. Thus $\mathcal{O}_{U, u}$ is a $1$-dimensional (by our definition of prime divisors) reduced Noetherian ring. After replacing $U$ by an affine open neighbourhood of $u$ we may assume $U$ is Noetherian and affine. After replacing $U$ by a smaller open, we may assume every irreducible component of $U$ passes through $u$. Since $U \to X$ is open and $X$ irreducible, $U \to X$ is dominant. Hence we obtain a ring map $R(X) \to R(U)$ by composing rational maps, see Morphisms of Spaces, Section \ref{spaces-morphisms-section-rational-maps}. Since $R(X)$ is a field, this map is injective. By our choice of $U$ we see that $R(U)$ is the total quotient ring $Q(\mathcal{O}_{U, u})$, see Morphisms, Lemma \ref{morphisms-lemma-integral-scheme-rational-functions} and Algebra, Lemma \ref{algebra-lemma-total-ring-fractions-no-embedded-points}. \medskip\noindent At this point we have proved all the statements in the lemma with $\mathcal{O}_{U, u}$ in stead of $\mathcal{O}_{X, \xi}^h$. However, $\mathcal{O}_{X, \xi}^h$ is the henselization of $\mathcal{O}_{U, u}$. Thus $\mathcal{O}_{X, \xi}^h$ is a $1$-dimensional reduced Noetherian ring, see More on Algebra, Lemmas \ref{more-algebra-lemma-henselization-reduced}, \ref{more-algebra-lemma-henselization-dimension}, and \ref{more-algebra-lemma-henselization-noetherian}. Since $\mathcal{O}_{U, u} \to \mathcal{O}_{X, \xi}^h$ is faithfully flat by More on Algebra, Lemma \ref{more-algebra-lemma-dumb-properties-henselization} it sends nonzerodivisors to nonzerodivisors. Therefore we obtain a canonical map $Q(\mathcal{O}_{U, u}) \to Q(\mathcal{O}_{X, \xi}^h)$ and we obtain our map. We omit the verification that the map is independent of the choice of $(U, u) \to (X, x)$; a slightly better approach would be to first observe that $\colim Q(\mathcal{O}_{U, u}) = Q(\mathcal{O}_{X, \xi}^h)$.
After shrinking $X$ we may assume $X$ is an integral Noetherian scheme. If $\xi \in Z$ denotes the generic point, then we find that $\mathcal{O}_{X, \xi}^h$ is the henselization of $\mathcal{O}_{X, \xi}$ (Decent Spaces, Lemma \ref{decent-spaces-lemma-describe-henselian-local-ring}). To prove the lemma it suffices and is necessary to show that $$ \text{length}_{\mathcal{O}_{X, \xi}} (\mathcal{O}_{X, \xi}/a \mathcal{O}_{X, \xi}) = \text{length}_{\mathcal{O}_{X, \xi}^h} (\mathcal{O}_{X, \xi}^h/a \mathcal{O}_{X, \xi}^h) $$ This follows immediately from Algebra, Lemma \ref{algebra-lemma-pullback-module} (and the fact that $\mathcal{O}_{X, \xi} \to \mathcal{O}_{X, \xi}^h$ is a flat local ring homomorphism of local Noetherian rings).
There exists a nonempty open subspace $U \subset X$ such that $f$ corresponds to a section of $\Gamma(U, \mathcal{O}_X^*)$. Hence the prime divisors which can occur in the sets of the lemma all correspond to irreducible components of $|X| \setminus |U|$. Hence Lemma \ref{lemma-components-locally-finite} gives the desired result.
There exists a nonempty open subspace $U \subset X$ such that $s$ corresponds to a section of $\Gamma(U, \mathcal{L})$ which generates $\mathcal{L}$ over $U$. Hence the prime divisors which can occur in the sets of the lemma all correspond to irreducible components of $|X| \setminus |U|$. Hence Lemma \ref{lemma-components-locally-finite}. gives the desired result.
Let $s$, resp.\ $t$ be a nonzero meromorphic section of $\mathcal{L}$, resp.\ $\mathcal{N}$. Then $st$ is a nonzero meromorphic section of $\mathcal{L} \otimes \mathcal{N}$. Let $Z \subset X$ be a prime divisor. Let $\xi \in |Z|$ be its generic point. Choose generators $s_\xi \in \mathcal{L}_\xi$, and $t_\xi \in \mathcal{N}_\xi$ with notation as described earlier in this section. Then $s_\xi \otimes t_\xi$ is a generator for $(\mathcal{L} \otimes \mathcal{N})_\xi$. So $st/(s_\xi t_\xi) = (s/s_\xi)(t/t_\xi)$ in $Q(\mathcal{O}_{X, \xi}^h)$. Applying the additivity of Algebra, Lemma \ref{algebra-lemma-ord-additive} we conclude that $$ \text{div}_{\mathcal{L} \otimes \mathcal{N}, Z}(st) = \text{div}_{\mathcal{L}, Z}(s) + \text{div}_{\mathcal{N}, Z}(t) $$ Some details omitted.
Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module whose associated Weil divisor class is trivial. Let $s$ be a regular meromorphic section of $\mathcal{L}$. The assumption means that $\text{div}_\mathcal{L}(s) = \text{div}(f)$ for some $f \in R(X)^*$. Then we see that $t = f^{-1}s$ is a regular meromorphic section of $\mathcal{L}$ with $\text{div}_\mathcal{L}(t) = 0$, see Lemma \ref{lemma-divisor-meromorphic-well-defined}. We claim that $t$ defines a trivialization of $\mathcal{L}$. The claim finishes the proof of the lemma. Our proof of the claim is a bit awkward as we don't yet have a lot of theory at our dispposal; we suggest the reader skip the proof. \medskip\noindent We may check our claim \'etale locally. Let $U \in X_\etale$ be affine such that $\mathcal{L}|_U$ is trivial. Say $s_U \in \Gamma(U, \mathcal{L}|_U)$ is a trivialization. By Properties, Lemma \ref{properties-lemma-normal-locally-finite-nr-irreducibles} we may also assume $U$ is integral. Write $U = \Spec(A)$ as the spectrum of a normal Noetherian domain $A$ with fraction field $K$. We may write $t|_U = f s_U$ for some element $f$ of $K$, see Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-meromorphic-quasi-coherent} for example. Let $\mathfrak p \subset A$ be a height one prime corresponding to a codimension $1$ point $u \in U$ which maps to a codimension $1$ point $\xi \in |X|$. Choose a trivialization $s_\xi$ of $\mathcal{L}_\xi$ as in the beginning of this section. Choose a geometric point $\overline{u}$ of $U$ lying over $u$. Then $$ (\mathcal{O}_{X, \xi}^h)^{sh} = \mathcal{O}_{X, \overline{u}} = \mathcal{O}_{U, u}^{sh} = (A_\mathfrak p)^{sh} $$ see Decent Spaces, Lemmas \ref{decent-spaces-lemma-henselian-local-ring-strict} and Properties of Spaces, Lemma \ref{spaces-properties-lemma-describe-etale-local-ring}. The normality of $X$ shows that all of these are discrete valuation rings. The trivializations $s_U$ and $s_\xi$ differ by a unit as sections of $\mathcal{L}$ pulled back to $\Spec(\mathcal{O}_{X, \overline{u}})$. Write $t = f_\xi s_\xi$ with $f_\xi \in Q(\mathcal{O}_{X, \xi}^h)$. We conclude that $f_\xi$ and $f$ differ by a unit in $Q(\mathcal{O}_{X, \overline{u}})$. If $Z \subset X$ denotes the prime divisor corresponding to $\xi$ (Lemma \ref{lemma-decent-irreducible-closed}), then $0 = \text{ord}_{Z, \mathcal{L}}(t) = \text{ord}_{\mathcal{O}_{X, \xi}^h}(f_\xi)$ and since $\mathcal{O}_{X, \xi}^h$ is a discrete valuation ring we see that $f_\xi$ is a unit. Thus $f$ is a unit in $\mathcal{O}_{X, \overline{u}}$ and hence in particular $f \in A_\mathfrak p$. This implies $f \in A$ by Algebra, Lemma \ref{algebra-lemma-normal-domain-intersection-localizations-height-1}. We conclude that $t \in \Gamma(X, \mathcal{L})$. Repeating the argument with $t^{-1}$ viewed as a meromorphic section of $\mathcal{L}^{\otimes -1}$ finishes the proof.
By Lemma \ref{lemma-finite-degree} there exists a nonempty $U \subset X$ such that $f^{-1}(U) \to U$ is finite. By generic flatness (Morphisms of Spaces, Proposition \ref{spaces-morphisms-proposition-generic-flatness-reduced}) we may assume $f^{-1}(U) \to U$ is flat and of finite presentation. So $f^{-1}(U) \to U$ is finite locally free (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-flat}). Since $f$ is birational, the degree of $X'$ over $X$ is $1$. Hence $f^{-1}(U) \to U$ is finite locally free of degree $1$, in other words it is an isomorphism.
Let $V \subset Y$ be a nonempty open affine (Decent Spaces, Theorem \ref{decent-spaces-theorem-decent-open-dense-scheme}). Let $\eta \in V$ be the generic point. Then $X_\eta$ is a nonempty proper algebraic space over $\eta$. Choose a closed point $x \in |X_\eta|$ (exists because $|X_\eta|$ is a quasi-compact, sober topological space, see Decent Spaces, Proposition \ref{decent-spaces-proposition-reasonable-sober} and Topology, Lemma \ref{topology-lemma-quasi-compact-closed-point}.) Let $X'$ be the reduced induced closed subspace structure on $\overline{\{x\}} \subset |X|$ (Properties of Spaces, Definition \ref{spaces-properties-definition-reduced-induced-space}. Then $f' : X' \to Y$ is surjective as the image contains $\eta$. Also $f'$ is proper as a composition of a closed immersion and a proper morphism. Finally, the fibre $X'_\eta$ has a single point; to see this use Decent Spaces, Lemma \ref{decent-spaces-lemma-topology-fibre} for both $X \to Y$ and $X' \to Y$ and the point $\eta$. Since $Y$ is decent and $X' \to Y$ is separated we see that $X'$ is decent (Decent Spaces, Lemmas \ref{decent-spaces-lemma-properties-trivial-implications} and \ref{decent-spaces-lemma-property-over-property}). Thus $f'$ is an alteration by Lemma \ref{lemma-alteration-generically-finite}.
The equivalence of (1), (2), and (3) follows from Decent Spaces, Lemma \ref{decent-spaces-lemma-decent-generic-points} and the fact that a quasi-separated algebraic space is decent (Decent Spaces, Section \ref{decent-spaces-section-reasonable-decent}). However in the next paragraph we will give a more elementary proof of the equivalence. \medskip\noindent Note that (1) and (2) are equivalent by definition (Properties of Spaces, Definition \ref{spaces-properties-definition-dimension-local-ring}). To prove the equivalence of (1) and (3) we may assume $X$ is quasi-compact. Choose $$ \emptyset = U_{n + 1} \subset U_n \subset U_{n - 1} \subset \ldots \subset U_1 = X $$ and $f_i : V_i \to U_i$ as in Decent Spaces, Lemma \ref{decent-spaces-lemma-filter-quasi-compact-quasi-separated}. Say $x \in U_i$, $x \not \in U_{i + 1}$. Then $x = f_i(y)$ for a unique $y \in V_i$. If (1) holds, then $y$ is a generic point of an irreducible component of $V_i$ (Properties of Spaces, Lemma \ref{spaces-properties-lemma-codimension-0-points}). Since $f_i^{-1}(U_{i + 1})$ is a quasi-compact open of $V_i$ not containing $y$, there is an open neighbourhood $W \subset V_i$ of $y$ disjoint from $f_i^{-1}(V_i)$ (see Properties, Lemma \ref{properties-lemma-generic-point-in-constructible} or more simply Algebra, Lemma \ref{algebra-lemma-standard-open-containing-maximal-point}). Then $f_i|_W : W \to X$ is an isomorphism onto its image and hence $x = f_i(y)$ is a generic point of $|X|$. Conversely, assume (3) holds. Then $f_i$ maps $\overline{\{y\}}$ onto the irreducible component $\overline{\{x\}}$ of $|U_i|$. Since $|f_i|$ is bijective over $\overline{\{x\}}$, it follows that $\overline{\{y\}}$ is an irreducible component of $U_i$. Thus $x$ is a point of codimension $0$. \medskip\noindent The final statement of the lemma is Properties of Spaces, Proposition \ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme}.