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\section{Algebraization of formal
schemes}\label{algebraization-of-formal-schemes}

\emph{Written by GPT-6.1 Sol (OpenAI), in Codex, at Ultra effort,
October 2026. Self-checked by the writing AI, GPT-6.1 Sol, at Ultra
effort. Public domain (CC0).}

A compatible family of infinitesimal schemes need not arrive inside one
ambient algebraic scheme. A compatible ample line bundle supplies that
ambient space: sufficiently positive sections lift at every order and
embed the entire system in a fixed projective space. Grothendieck
existence then algebraizes its quotient structure sheaf.

We use \href{grothendiecks-existence-theorem.md}{Grothendieck
existence}, \href{the-theorem-on-formal-functions.md}{formal functions},
and \href{cohomology-of-projective-space.md}{projective-space
cohomology}. The ample embedding and positive-power criteria are proved
in the Serre lesson, Section 1. The preceding formal-functions lesson,
Lemma 5.4, proves the cartesian nilpotent-thickening criteria for finite
type, closed immersion and properness, without a flatness assumption.
Relative Spec and the reconstruction of finite morphisms are proved in
the existing earlier \href{../AG-MO/AG-MO-05.html}{\emph{Affine
morphisms, relative Spec, and finite morphisms}, Theorem 1.1 and
Corollary 2.2}.

Throughout, \(A\) is Noetherian and complete for an ideal \(I\),
\(S=\operatorname{Spec}A\), and \(S_n=\operatorname{Spec}(A/I^n)\). Such
an ideal lies in the Jacobson radical. Indeed \(1-a\) has inverse
\(\sum_{j\geq0}a^j\) for \(a\in I\), so no maximal ideal can omit \(I\).

\subsection{1. Algebraizing closed
subschemes}\label{algebraizing-closed-subschemes}

Let \(X\) be proper over \(S\), and write \(X_n=X\times_S S_n\). Suppose
\(Z_n\hookrightarrow X_n\) are closed subschemes with compatible
cartesian squares, meaning \[
Z_n=Z_{n+1}\times_{X_{n+1}}X_n.
\] Their quotient structure sheaves form a coherent formal module
\(F_n=\mathcal O_{Z_n}\) on \(X\), with compatible quotient maps
\(\mathcal O_{X_n}\twoheadrightarrow F_n\).

\textbf{Theorem 1.1.} There is a unique closed subscheme
\(Z\hookrightarrow X\) whose base changes are the specified \(Z_n\),
compatibly as subschemes of \(X_n\).

\textbf{Proof.} Grothendieck existence gives a coherent \(F\) with
\(\widehat F=(F_n)\). Full faithfulness algebraizes the formal quotient
map to \(q:\mathcal O_X\to F\). Its coherent cokernel has zero
completion, by exactness. Faithfulness of completion makes that cokernel
zero: its identity has zero image, hence is zero. Thus \(q\) is
surjective, and its kernel \(K\) is a coherent ideal. Set \(Z=V(K)\);
reducing the quotient modulo \(I^n\) gives exactly the prescribed
quotient algebra on \(X_n\).

For uniqueness, the quotient modules of two algebraizations have a
specified formal isomorphism respecting the quotient maps. Full
faithfulness gives a unique algebraic isomorphism respecting those maps,
because their difference has zero completion. Their kernels are
therefore the same ideal. This proves uniqueness as an embedded closed
subscheme. \(\square\)

The kernels of the separate quotients \(\mathcal O_{X_n}\to F_n\) need
not themselves satisfy the quotient compatibility for a coherent formal
module. The argument uses the quotient systems and the actual
formal-category kernel, avoiding that mistake.

We also need the proper-support version. If \(W\) is separated of finite
type over \(S\), the same conclusion holds for closed \(Z_n\subset W_n\)
whose first member is proper over \(S_1\), and the resulting \(Z\) is
proper over \(S\). This is {[}Stacks, Tag 0899{]}. Here is why the
quotient construction still works. Existence with proper support
algebraizes \(F_n\) to a coherent \(F\) with proper scheme-theoretic
support. The formal units are compatible sections of \(F_n\). Formal
functions on that proper support identifies them with a unique section
of \(F\), giving \(\mathcal O_W\to F\). Its cokernel has proper support
and zero completion, so vanishes. Uniqueness is proved on the proper
closed union of two candidate supports, using full faithfulness there.
The quotient defines the required proper closed subscheme.

\subsection{2. Finite schemes and
morphisms}\label{finite-schemes-and-morphisms}

\textbf{Proposition 2.1.} Suppose \(X\) is proper over \(S\), and
\(Y_n\to X_n\) are compatible finite morphisms with cartesian squares.
They algebraize to a finite morphism \(Y\to X\), with \(Y\) proper over
\(S\).

\textbf{Proof.} The finite algebras
\(B_n=(Y_n\to X_n)_*\mathcal O_{Y_n}\) are a coherent formal module:
affine base change identifies their quotient transitions. Algebraize its
underlying module to \(B\). The formal multiplications algebraize to
\(B\otimes B\to B\), since completion commutes with tensor products of
finite modules. The unit algebraizes to \(\mathcal O_X\to B\).
Associativity, commutativity and the unit equations are equalities of
coherent-module maps; full faithfulness lifts them from every formal
level. Thus \(B\) is a coherent algebra. Take
\(Y=\underline{\operatorname{Spec}}_X B\). It is finite over \(X\),
hence proper over \(S\), and its base changes recover the finite
algebras \(B_n\) and the schemes \(Y_n\). \(\square\)

The same argument works when \(X\) is only separated of finite type and
\(Y_1\) is proper over \(S_1\), using proper-support existence. The
algebraized algebra is supported on a proper closed subscheme, and its
finite relative Spec is proper. This is the exact generality of
{[}Stacks, Tag 09ZT{]}.

\textbf{Theorem 2.2 (algebraization of morphisms).} If \(X\) is proper
and \(Y\) is separated of finite type over \(S\), any compatible system
\(g_n:X_n\to Y_n\) comes from a unique \(S\)-morphism \(g:X\to Y\).

\textbf{Proof.} Separatedness makes the graphs
\(\Gamma_n\subset X_n\times_{S_n}Y_n\) closed. They form a cartesian
system with proper first member. The proper-support closed-subscheme
result gives a proper \(Z\subset X\times_S Y\) algebraizing these
graphs. Its projection \(p:Z\to X\) is proper, by the closed-graph
factorization, and is an isomorphism modulo every \(I^n\).

The fibers of \(p\) over \(X_1\) have one point. The preceding
formal-functions lesson, Corollary 5.3, proves that a proper morphism
over a Noetherian scheme is finite on a neighborhood of a finite fiber.
Thus \(p\) is finite over an open neighborhood \(U\) of \(X_1\). The
closed complement \(X\setminus U\) has closed image in \(S\), because
\(X\) is proper. This image misses \(V(I)\). Every nonempty closed
subset of \(S\) meets \(V(I)\), since \(I\) lies in the Jacobson
radical. The complement is therefore empty, so \(p\) is finite
everywhere.

Finite pushforward commutes with all base changes, by the affine algebra
description. Hence the unit \(\mathcal O_X\to p_*\mathcal O_Z\) is an
isomorphism on every formal level. Full faithfulness for proper \(X\)
makes it an isomorphism algebraically. A finite map is recovered from
its algebra, so \(p\) is an isomorphism. Its inverse followed by the
other projection defines \(g\). Any other algebraization has a proper
closed graph with the same formal graph system; uniqueness of the
proper-support closed-subscheme result identifies the graphs and the
morphisms. \(\square\)

This proves morphism recovery with the target hypotheses of {[}Stacks,
Tag 0A42{]}; the target need not be proper.

\subsection{3. An abstract polarized
system}\label{an-abstract-polarized-system}

Now suppose schemes \(T_n\to S_n\) and transition morphisms are given
with cartesian squares, so \[
T_n=T_{n+1}\times_{S_{n+1}}S_n.
\] They share one underlying topological space, since \(I\) is nilpotent
on each finite level. Assume that \(T_1\to S_1\) is proper and that
there are invertible sheaves \(L_n\) with specified compatible
isomorphisms \[
L_{n+1}|_{T_n}\cong L_n,
\qquad L_1\text{ ample on }T_1.
\] The nilpotent-thickening criterion makes all \(T_n\) proper and of
finite type over their Noetherian bases. No flatness is assumed in this
section.

\textbf{Lemma 3.1 (uniform section lifting).} There is an integer
\(d_0\) such that, for \(d\geq d_0\), \[
\Gamma(T_{n+1},L_{n+1}^d)\longrightarrow\Gamma(T_n,L_n^d)
\] is onto for every \(n\geq1\).

\textbf{Proof.} On the common space set \(D_0=\mathcal O_{T_1}\) and \[
D_n=\ker(\mathcal O_{T_{n+1}}\to\mathcal O_{T_n})
=I^n\mathcal O_{T_{n+1}}\quad(n\geq1).
\] Each is annihilated by \(I\), so is coherent on \(T_1\). Computing at
a level larger than the indices defines products
\(D_a\otimes D_b\to D_{a+b}\). They are independent of that level, by
the cartesian quotient identities. With \(B=\bigoplus I^n/I^{n+1}\),
multiplication gives a graded surjection \[
\mathcal O_{T_1}\otimes_{A/I}B\twoheadrightarrow D=\bigoplus D_n.
\] The ring \(B\) is Noetherian and finitely generated. On the proper
scheme \(T_1\times_{A/I}\operatorname{Spec}B\), the graded module \(D\)
corresponds under affine pushforward to a coherent sheaf \(E\). Pullback
of \(L_1\) is relatively ample. Serre vanishing for \(E\), followed by
affine pushforward and direct-sum cohomology, supplies one bound for
\(H^1(T_1,D_n\otimes L_1^d)=0\), simultaneously in \(n\). The exact
sequences \[
0\to D_n\otimes L_1^d\to L_{n+1}^d\to L_n^d\to0
\] then give the asserted surjectivity. \(\square\)

The argument uses only a surjection of graded algebras, so survives
nonflatness. It is the same uniform-vanishing mechanism that algebraized
formal modules in the preceding lesson.

\subsection{4. Grothendieck's algebraization
theorem}\label{grothendiecks-algebraization-theorem}

\textbf{Theorem 4.1.} The polarized system of Section 3 comes from a
projective scheme \(T\to S\) and an ample invertible \(L\), with
compatible identifications \[
T_n\cong T\times_S S_n,
\qquad L_n\cong L|_{T_n}.
\]

\textbf{Proof.} Choose \(d\) large enough for Lemma 3.1 and for a finite
set of sections of \(L_1^d\) to give a closed immersion
\(T_1\hookrightarrow\mathbf P^N_{A/I}\). The ample embedding criterion
gives an immersion for a sufficiently large power, and properness makes
it closed. Lift those sections recursively using Lemma 3.1. At each
level they generate \(L_n^d\), because their cokernel reduces to zero
modulo the nilpotent ideal \(I\), and Nakayama applies. They define
compatible maps \[
\psi_n:T_n\longrightarrow\mathbf P^N_{A/I^n}.
\] Their reductions are \(\psi_1\), a closed immersion. The nilpotent
closed-immersion criterion {[}Stacks, Tag 0896{]} makes every \(\psi_n\)
a closed immersion. The squares are cartesian because the original
systems and the projective spaces both have the specified base changes.

Apply Theorem 1.1 in \(\mathbf P^N_A\). It gives a closed subscheme
\(T\), hence a projective scheme, with precisely the desired reductions.
The coherent formal module \((L_n)\) on \(T\) algebraizes by
Grothendieck existence to a coherent \(L\). The inverse line bundles
\((L_n^{-1})\) also algebraize to \(G\). Their formal evaluation
isomorphism algebraizes, with its inverse, to
\(L\otimes G\cong\mathcal O_T\). This makes \(L\) invertible. Locally
its tensor inverse forces both finite modules to have one generator
modulo the maximal ideal; writing them as quotients of the local ring
shows their defining ideals must both be zero.

The embedding constructed from the lifted sections identifies \(L_n^d\)
with \(\mathcal O_T(1)|_{T_n}\). Full faithfulness identifies \(L^d\)
with \(\mathcal O_T(1)\). This is ample; ampleness of a positive power
implies ampleness of \(L\). Thus the theorem recovers the given
polarization itself, not just its embedding power. \(\square\)

This is the full form of {[}Stacks, Tag 089A{]}. Its projective
conclusion follows here from an actual closed immersion into a fixed
finite projective space. Different compatible choices of sections
produce the same algebraized system up to a unique isomorphism inducing
the prescribed identifications: Theorem 2.2 algebraizes the compatible
isomorphisms and their inverses. Full faithfulness also identifies the
polarizations.

\subsection{5. Lifting line bundles and the role of
flatness}\label{lifting-line-bundles-and-the-role-of-flatness}

For a square-zero thickening \(T\hookrightarrow T'\) with ideal \(J\),
there is an exact sequence of abelian sheaves on their common space \[
0\to J\xrightarrow{a\mapsto1+a}\mathcal O_{T'}^*
\to\mathcal O_T^*\to1.
\tag{2}
\] The first map is additive because \(J^2=0\). A unit modulo \(J\)
lifts locally to a unit: lift it and an inverse, whose product is
\(1+a\), and multiply by \(1-a\). Hence the sequence is exact as a
sequence of sheaves. Since
\(\operatorname{Pic}(T)=H^1(T,\mathcal O_T^*)\), its long exact sequence
gives \[
H^1(T,J)\to\operatorname{Pic}(T')\to\operatorname{Pic}(T)
\xrightarrow{\partial}H^2(T,J).
\tag{3}
\] Thus \(\partial(L)\) is the exact obstruction to lifting \(L\). When
it vanishes, lift classes form a torsor under the quotient of
\(H^1(T,J)\) by the image of the preceding unit boundary. This proves
the square-zero lifting statement directly, including its correct
obstruction group.

\textbf{Theorem 5.1 (the flat \(H^2\) criterion).} Let
\((A,\mathfrak m,k)\) be complete Noetherian local. Suppose the
cartesian system \(T_n\to\operatorname{Spec}(A/\mathfrak m^n)\) is flat
at every level, \(T_1\) is projective over \(k\), and
\(H^2(T_1,\mathcal O_{T_1})=0\). Then it algebraizes to a projective
flat scheme over \(A\).

\textbf{Proof.} The ideal of \(T_n\) in \(T_{n+1}\) is square-zero,
since \(2n\geq n+1\). Flatness makes its ideal layer \[
J_n\cong\mathcal O_{T_1}\otimes_k
(\mathfrak m^n/\mathfrak m^{n+1}).
\] This follows by tensoring the defining ideal sequence of the base
with the flat structure sheaf; the vector space on the right is
finite-dimensional. Consequently \(H^2(T_n,J_n)=0\). Starting with an
ample \(L_1\), sequence (3) constructs \(L_{n+1}\) with a chosen
identification on \(T_n\), recursively. Theorem 4.1 algebraizes the
resulting polarized system to projective \(T\).

At a point of the closed fiber, its Noetherian local algebra is flat
over \(A\): all quotients modulo \(\mathfrak m^n\) are flat over
\(A/\mathfrak m^n\), and the earlier \emph{Faithful flatness and the
local criterion for flatness}, Proposition 4.5 (all infinitesimal
quotients), applies to this local Noetherian map with its finite module.
The flat locus is open by the earlier \emph{Flatness criteria}, Theorem
5.2, for a finite-type algebra and finite module over a Noetherian base.
Its closed complement has closed image in the local base by properness;
if nonempty, that image contains the closed point, contrary to the
flatness just proved. The complement is empty, so \(T\) is flat
everywhere. \(\square\)

Flatness cannot simply be removed from the conversion of
\(H^2(\mathcal O)\) into the obstruction vanishing. We give a concrete
first-order example. Work over a field of characteristic zero, take
\(P=\mathbf P^3_k\), and a quartic hypersurface \(Z\subset P\); put
\(J=\mathcal O_Z\), regarded as an \(\mathcal O_P\)-module.
Projective-space cohomology and the hypersurface sequences give \[
H^2(P,\mathcal O_P)=0,\quad H^2(P,J)=k,
\quad H^1(P,J(1))=H^2(P,J(1))=0.
\] For instance the middle equality is \(H^3(P,\mathcal O_P(-4))=k\).
Tensor the tangent Euler sequence with \(J\): \[
0\to J\to J(1)^4\to T_P\otimes J\to0.
\] Here \(T_P\) is the sheaf of derivations. On \(U_i=\{X_i\neq0\}\), a
tuple of homogeneous components induces the derivation of \(X_l/X_i\)
given by its \(l\)-th component minus \((X_l/X_i)\) times its \(i\)-th
component, in the frame \(X_i\). This map is onto, and its kernel is the
scalar coordinate vector, proving the displayed Euler sequence. Its
boundary gives an isomorphism \(H^1(P,T_P\otimes J)\cong H^2(P,J)\).
Choose a derivation-valued Čech cocycle \(\delta_{ij}\) representing a
nonzero class on the standard affine cover. Glue the split square-zero
rings \(\mathcal O_P\oplus J\) by \[
(f,a)\longmapsto(f,a+\delta_{ij}(f)).
\] The derivation rule makes these ring isomorphisms; the cocycle rule
gives gluing. They define a scheme \(P'\) with square-zero ideal \(J\).
The global section \((0,1)\) gives a map to
\(k[\epsilon]/(\epsilon^2)\), with reduction \(P\), since it generates
\(J\) as an \(\mathcal O_P\)-module.

The obstruction to lifting \(\mathcal O_P(1)\) is nonzero. Set
\(g_{ij}=X_j/X_i\), so \(g_{ij}g_{jk}=g_{ik}\), and lift these
transitions as \((g_{ij},0)\). Their triple product differs from one by
the additive cocycle \(z_{ijk}=\delta_{ij}(g_{jk})/g_{jk}\). Lift
\(\delta_{ij}\) to Euler components in frame \(X_i\) by taking its
\(i\)-th component zero and its other components
\(\delta_{ij}(X_l/X_i)\). The Euler connecting cocycle is then
\(\lambda_{ijk}=-\delta_{jk}(g_{ij})/g_{ij}\). For
\(a_{ij}=\delta_{ij}(g_{ij})/g_{ij}\), the cocycle identities give \[
z_{ijk}-\lambda_{ijk}=-(a_{jk}-a_{ik}+a_{ij}).
\] Thus the obstruction class equals the Euler boundary of \([\delta]\).
That boundary is an isomorphism, so the class is nonzero. This is a
nonflat first-order deformation with \(H^2(\mathcal O_P)=0\) and a line
bundle that does not lift. The flat hypotheses in Theorem 5.1 prevent
this discrepancy of ideal layers.

\subsection{6. Curves and exercises with
solutions}\label{curves-and-exercises-with-solutions}

A flat compatible formal deformation of a smooth projective curve over
\(k[[t]]\) algebraizes. Its closed fiber has dimension one, so
\(H^2(C,\mathcal O_C)=0\) by the cohomological dimension bound. Theorem
5.1 supplies a projective flat algebraization. It also algebraizes
compatible sheaves by Grothendieck existence and compatible maps to
separated finite-type targets by Theorem 2.2.

The curve need not be smooth for the existence argument. For example,
consider the compatible projective conics \[
T_n=\{xy=t z^2\}\subset\mathbf P^2_{k[t]/t^n},
\qquad L_n=\mathcal O_{T_n}(1).
\] Their homogeneous coordinate rings are flat over the parameter ring:
the equation is monic in the monomial \(xy\), and the monomials not
divisible by \(xy\) give a free basis. Localization and taking the
degree-zero summand preserve flatness on the projective charts. The
closed fiber is the union of two lines, still of dimension one, so its
second cohomology vanishes. The theorem algebraizes this system;
explicitly its algebraization is the same equation in
\(\mathbf P^2_{k[[t]]}\). The uniqueness theorem identifies any other
proper algebraization inducing the given formal identifications with
this one. The compatible hyperplane bundles algebraize as well, so the
equation and its polarization are recovered together.

\textbf{Exercise 6.1 (easy: uniqueness).} Prove uniqueness of an
algebraized closed subscheme, keeping its embedding in a fixed proper
\(X\).

\textbf{Solution.} The prescribed quotient systems give a formal
isomorphism of the two quotient sheaves respecting the maps from
\(\mathcal O_X\). Full faithfulness gives the unique algebraic
isomorphism, and makes compatibility with those maps an actual equality.
Their kernels coincide, so the embedded closed subschemes coincide. An
abstract isomorphism alone would not identify their embeddings.

\textbf{Exercise 6.2 (medium: projective subschemes).} Algebraize a
cartesian system of closed subschemes of \(\mathbf P^N_{k[t]/t^n}\).

\textbf{Solution.} Regard their structure sheaves as coherent formal
quotient modules on \(\mathbf P^N_{k[[t]]}\). Existence algebraizes the
quotient module; full faithfulness algebraizes the quotient map;
exactness and faithfulness kill its cokernel. Its coherent ideal kernel
cuts out the desired projective subscheme, and Theorem 1.1 proves
uniqueness. No flatness of the subschemes is required.

\textbf{Exercise 6.3 (medium: the obstruction).} Derive the obstruction
and ambiguity for lifting a line bundle across a square-zero thickening.

\textbf{Solution.} Use the unit sequence (2), whose kernel is additively
\(J\). The long exact sequence identifies the obstruction as
\(\partial(L)\in H^2(T,J)\). It vanishes exactly when \(L\) lifts. The
possible lift classes, if any, form a torsor under
\(\ker(\operatorname{Pic}(T')\to\operatorname{Pic}(T))\), which (3)
identifies with \(H^1(T,J)\) modulo the image of the unit boundary. This
is the precise ambiguity, rather than an unqualified torsor under all of
\(H^1(T,J)\).

\textbf{Exercise 6.4 (hard: a formal curve).} Algebraize a flat formal
deformation of a smooth projective curve over \(k[[t]]\), explaining how
its polarization is obtained.

\textbf{Solution.} Choose an ample line bundle on the closed curve. Each
ideal layer is \(\mathcal O_C\), since \((t^n)/(t^{n+1})\cong k\). Its
second cohomology vanishes in dimension one, so (3) lifts the line
bundle successively, with compatible identifications. Theorem 4.1
algebraizes the polarized system to a projective scheme. The flat
quotient criterion gives flatness on the closed fiber; openness and
properness over the local base give flatness everywhere. This is the
projective flat family required.

\textbf{Exercise 6.5 (hard: recovering the root).} In Theorem 4.1, why
is recovering \(L^d\) insufficient, and how is \(L\) itself recovered?

\textbf{Solution.} An embedding constructs the chosen positive power; it
does not by itself produce its specified root. Apply existence to
\((L_n)\) and \((L_n^{-1})\). Full faithfulness algebraizes the
evaluation isomorphism and its inverse, so their algebraizations are
tensor inverses and the first is invertible. Full faithfulness
identifies its \(d\)-th power with the embedding bundle. Ampleness of
that power then proves ampleness of the recovered \(L\).

\textbf{Exercise 6.6 (hard: finite algebras).} Explain why algebraizing
only the underlying module of a finite formal algebra is enough to
recover its algebra structure.

\textbf{Solution.} Completion commutes with finite tensor products. Thus
formal multiplication and unit are formal maps between completions of
coherent modules. Full faithfulness algebraizes them uniquely.
Associativity compares two maps from the triple tensor product,
commutativity compares two maps from the double tensor product, and the
unit axioms compare maps from the module itself. All agree after
completion, hence agree algebraically by faithfulness. Relative Spec
then recovers the finite scheme and all its reductions.

\subsection{References}\label{references}

\begin{itemize}
\tightlist
\item
  \textbf{{[}Stacks{]}} The Stacks project authors, \emph{The Stacks
  project}, in its AI Integrated Stacks Project edition:
  closed-subscheme algebraization
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-lemma-algebraize-formal-closed-subscheme}{Tag
  0899}, finite algebraization
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-lemma-algebraize-formal-scheme-finite-over-proper}{Tag
  09ZT}, morphisms
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-lemma-algebraize-morphism}{Tag
  0A42}, and polarized algebraization
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-theorem-algebraization}{Tag
  089A}.
\item
  Nilpotent checks
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-morphisms.html\#more-morphisms-lemma-thicken-property-morphisms-cartesian}{Tag
  09ZW},
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/morphisms.html\#morphisms-lemma-check-closed-infinitesimally}{Tag
  0896}; finite-fiber neighborhood
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-lemma-proper-finite-fibre-finite-in-neighbourhood}{Tag
  02OH}.
\item
  The Picard sequence is
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-morphisms.html\#more-morphisms-lemma-picard-group-first-order-thickening}{Tag
  0C6R}. Flatness uses
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html\#algebra-lemma-flat-module-powers}{Tag
  0523} and its open locus
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-morphisms.html\#more-morphisms-theorem-openness-flatness}{Tag
  0399}. The flat \(H^2\) criterion also appears in
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-lemma-algebraize-flat-formal-scheme-H2}{AI
  Integrated Stacks Project, coherent.tex,
  lemma-algebraize-flat-formal-scheme-H2}, with these same flatness
  hypotheses.
\item
  These open reference treatments retain GNU FDL 1.2. The proofs,
  exposition, examples and solutions here are independently written CC0,
  using the preceding course proofs and the exact routine open
  foundations named above.
\end{itemize}
\end{document}