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\begin{document}
\section{Grothendieck's existence
theorem}\label{grothendiecks-existence-theorem}

\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).}

Formal functions recovers completed cohomology from infinitesimal
neighborhoods. Grothendieck's existence theorem recovers the sheaves
themselves. Over a complete Noetherian base, a compatible coherent sheaf
on every thickening of a proper scheme comes from one coherent sheaf,
and compatible morphisms come from unique algebraic morphisms.

We use \href{the-theorem-on-formal-functions.md}{formal functions},
\href{serres-theorems-on-projective-schemes.md}{Serre generation and
vanishing}, and
\href{proper-morphisms-and-coherent-direct-images.md}{proper coherence}.
The existing earlier \href{../AG-CA/AG-CA-19.html}{\emph{Completion},
Theorems 3.1--3.3} proves Noetherianness, flatness and exactness of
completion, and completeness of finite modules. The proper-coherence
lesson gives Chow's construction in Lemma 4.0 and links the full
Noetherian variant to the earlier programme's \emph{Projective morphisms
and Chow's lemma}, Theorem 4.1. Internal derived Hom and its
identification with global Ext are proved in the earlier
\href{../derived-categories-and-sheaf-operations/internal-derived-hom-and-ext-sheaves.html}{\emph{Internal
derived Hom and Ext sheaves}, Theorem 3.1 and equations (3.2)--(3.3)}.
We prove the formal comparison and proper reduction below.

\subsection{1. Compatible coherent
systems}\label{compatible-coherent-systems}

For a Noetherian scheme \(X\) and a coherent ideal
\(J\subset\mathcal O_X\), a \textbf{coherent formal module} is a system
\(\mathfrak F=(F_n)_{n\geq1}\) of coherent sheaves, with \[
J^nF_n=0,\qquad F_{n+1}/J^nF_{n+1}\xrightarrow{\sim}F_n.
\tag{1}
\] Morphisms are compatible sheaf maps at every level. Equivalently,
\(F_n\) is a coherent sheaf on \(X_n=V(J^n)\). Iterating (1) gives
\(F_m/J^nF_m=F_n\) for \(m\geq n\). Write \(\operatorname{Coh}(X,J)\)
for this category.

\textbf{Lemma 1.1 (the affine description).} If
\(X=\operatorname{Spec}R\) and \(J=I\mathcal O_X\), this category is
equivalent to finite modules over \(\widehat R=\varprojlim R/I^n\). The
correspondence is \[
(M_n)\longmapsto M=\varprojlim M_n,
\qquad M\longmapsto(M/I^nM).
\]

\textbf{Proof.} The transition maps are onto, so a compatible sequence
lifting any chosen element of \(M_1\) can be constructed recursively.
Lift finitely many generators of \(M_1\) to \(M\). They define maps
\((R/I^n)^r\to M_n\), which are onto by Nakayama for the nilpotent ideal
\(I/I^n\).

Let \(K_n\) be their kernels. These kernels also have surjective
transitions. To lift \(v\in K_n\), lift it to \(w\in(R/I^{n+1})^r\). Its
image in \(M_{n+1}\) lies in \(I^nM_{n+1}\), since it vanishes in
\(M_n\). Surjectivity of the free-module map lets us subtract an element
of \(I^n(R/I^{n+1})^r\) with that same image. The corrected lift belongs
to \(K_{n+1}\).

Taking limits gives \[
0\to\varprojlim K_n\to\widehat R^{\,r}\to M\to0.
\] Surjectivity on the right follows by adjusting consecutive lifts
using the surjective kernel transitions. Thus \(M\) is finite. Since
\(\varprojlim K_n\to K_n\) is onto, reducing this presentation modulo
\(I^n\) gives \(M/I^nM=M_n\). Conversely finite modules over the
Noetherian complete ring \(\widehat R\) are complete, so are recovered
from their quotients. Compatible maps pass uniquely to and from limits,
proving the equivalence. \(\square\)

An affine open in a scheme over a complete ring need not have a complete
coordinate ring. For instance the \(t\)-adic completion of \(k[[t]][x]\)
is the ring \[
k[[t]]\langle x\rangle=\varprojlim_n(k[t]/t^n)[x]
\] of restricted power series: in \(\sum a_jx^j\), the coefficients
\(a_j\) tend to zero \(t\)-adically. This contains series of unbounded
degree in \(x\), unlike the polynomial ring.

\subsection{2. Exactness and the actual formal
kernel}\label{exactness-and-the-actual-formal-kernel}

\textbf{Theorem 2.1.} The category \(\operatorname{Coh}(X,J)\) is
abelian, exactness is local on \(X\), and the completion functor \[
\operatorname{Coh}(X)\longrightarrow\operatorname{Coh}(X,J),
\qquad F\longmapsto\widehat F=(F/J^nF)
\] is exact.

\textbf{Proof.} On an affine, Lemma 1.1 identifies the category with
finite modules over a Noetherian ring, an abelian category. We explain
why its kernels glue, since taking the kernels separately at each level
is generally wrong.

For a morphism \(\alpha:(F_n)\to(G_n)\), take the images \[
K'_{l,m}=\operatorname{im}(\ker\alpha_l\to F_m),\qquad l\geq m.
\] On an affine let \(u:M\to N\) be the corresponding completed-module
map, with kernel \(K\) and image \(H\). Artin--Rees for \(H\subset N\)
gives a constant \(c\) such that \[
u^{-1}(I^lN)\subset K+I^{l-c}M\quad(l\geq c).
\] Indeed \(H\cap I^lN\subset I^{l-c}H\), and elements of this last
module lift from \(I^{l-c}M\). Hence for \(l\geq m+c\), the image
\(K'_{l,m}\) is precisely \(K/(K\cap I^mM)\). It stabilizes. Artin--Rees
for \(K\subset M\) then shows that, for fixed \(n\) and sufficiently
large \(m\), \[
K'_m/I^nK'_m
=K/(K\cap I^mM+I^nK)=K/I^nK.
\tag{2}
\] All these images and quotients are constructions with coherent
sheaves and commute with restriction. A finite affine cover gives
uniform stabilization bounds, so they define coherent sheaves globally.
The resulting system is the formal kernel: a map of systems killed by
\(\alpha\) factors at every level through the stabilized images, then
through (2); the factorization is unique. Cokernels are the levelwise
cokernels, since quotient formation preserves (1). On affines these
constructions give the ordinary module kernel and cokernel, so image
equals coimage. They also show that restriction is exact and that
exactness can be checked on an open cover.

On an affine the completion functor is \(M\mapsto M\otimes_R\widehat R\)
on finite modules. Flatness of Noetherian completion makes it exact. The
local characterization just established proves exactness globally.
\(\square\)

For example multiplication by \(t\) on \(k[[t]]\) has zero kernel, but
multiplication by \(t\) on \(k[t]/t^n\) has kernel \(k\,t^{n-1}\). These
nonzero kernels map to zero at the preceding level. Their stabilized
formal kernel is zero. An exact sequence in the formal category
therefore need not give a left-exact sequence at each fixed quotient
level.

A map of formal modules is onto exactly when its first-level map is
onto. In the affine description, its cokernel is a finite completed
module \(C\); if \(C/IC=0\), Nakayama gives \(C=0\). Here
\(I\widehat R\) lies in the Jacobson radical: \(1-a\) is invertible for
\(a\in I\widehat R\), by its convergent geometric series. This also
proves the assertion locally on \(X\).

\subsection{3. Full faithfulness from formal
functions}\label{full-faithfulness-from-formal-functions}

Now let \(A\) be Noetherian and complete for an ideal \(I\), let \(X\)
be proper over \(A\), and set \(J=I\mathcal O_X\).

\textbf{Lemma 3.1 (completed Hom).} For coherent \(F,G\), with
\(H=\mathcal Hom(F,G)\), \[
\operatorname{Hom}_{\operatorname{Coh}(X,J)}(\widehat F,\widehat G)
\cong\varprojlim_n\Gamma(X,H/J^nH).
\tag{3}
\]

\textbf{Proof.} Affine locally, \(F\) has a finite presentation.
Applying \(\operatorname{Hom}(-,G)\) describes Hom as the kernel of a
map between two finite direct sums of \(G\). Tensoring this kernel with
the flat completed ring preserves the kernel and gives \[
\operatorname{Hom}_R(M,N)\otimes_R\widehat R
=\operatorname{Hom}_{\widehat R}(\widehat M,\widehat N).
\] Lemma 1.1 identifies the right side with compatible maps on the
quotients. These natural affine identifications identify the sheaves of
completed Hom sections with the sheaves of compatible formal maps. They
glue on overlaps. Taking global sections, which commutes with inverse
limits of sheaves, gives (3). Notice that no equality of the separate
finite-level Hom modules was asserted. \(\square\)

\textbf{Theorem 3.2 (full faithfulness).} Completion on coherent sheaves
on \(X\) is fully faithful.

\textbf{Proof.} The sheaf \(H\) is coherent, so its global section
module is finite over \(A\). Formal functions identifies the right side
of (3) with \(\widehat{\Gamma(X,H)}\). A finite module over the complete
Noetherian ring \(A\) is already complete. Thus (3) identifies formal
maps with \(\Gamma(X,H)=\operatorname{Hom}_X(F,G)\), and the
identification is the completion map on morphisms. \(\square\)

In particular, an algebraization, when it exists, is unique up to the
unique isomorphism inducing a specified formal identification. Formal
isomorphisms algebraize because both directions algebraize and
faithfulness makes their composites identities.

\textbf{Proposition 3.3 (extension comparison, also with proper
supports).} If \(F,G\) are coherent on a separated finite-type \(X/A\),
and have proper support, then completion induces isomorphisms \[
\operatorname{Ext}^r_X(F,G)
\xrightarrow{\sim}
\operatorname{Ext}^r_{\operatorname{Coh}(X,J)}(\widehat F,\widehat G)
\qquad(r=0,1).
\] The group in degree one means short exact sequences in the formal
category with their specified ends. Thus every such formal extension is
algebraizable.

\textbf{Proof.} On an affine chart, resolve the finite module of \(F\)
by finite free modules, successively resolving its finite syzygies.
Noetherianness permits any finite prefix. Applying Hom into \(G\)
computes module Ext as a kernel modulo an image; these finite modules
sheafify, and localization commutes with the computation. Hence the
sheaves \(E^q=\mathcal Ext^q_X(F,G)\) are coherent and supported on the
proper intersection of the two supports. Flatness of completion
preserves every kernel and image in the finite prefix, giving the formal
local identity \[
\mathcal Ext^q(\widehat F,\widehat G)=\widehat{E^q}.
\] The identifications are compatible with restriction to principal
opens (where restriction is followed by completion), so glue.

We spell out the cohomological step in this formal identity. On a formal
affine, coherent modules are finite completed modules by Lemma 1.1.
Their Čech complex for a finite principal cover is the inverse limit of
the corresponding module complexes modulo \(I^n\), each exact in
positive degrees by affine acyclicity. The complexes have surjective
transitions in every term and in the augmented section term. The
elementary inverse-limit argument therefore preserves this exactness:
lift a preimage at one level and correct its next lift by a cocycle at
the preceding level. Equivalently the difference map
\(\prod C_n\to\prod C_n\), \((c_n)\mapsto(c_n-u_{n+1}c_{n+1})\), is onto
termwise, and its kernel is \(\varprojlim C_n\). The basis criterion in
the Čech lesson proves acyclicity for coherent formal modules on these
affine opens. Use a finite affine cover of \(X\), whose intersections
are affine by separatedness. For a coherent \(E\), the formal section
complex is \(\varprojlim C_n(E/I^nE)\). Formal functions proves
Mittag--Leffler stabilization of all its cohomology systems. The same
product-difference sequence now gives \[
H^p(X,\widehat E)=\varprojlim H^p(X,E/I^nE)
=\widehat{H^p(X,E)}.
\] Here the possible cokernel of the product-difference map on
\(H^{p-1}\) is zero: stabilized images admit recursive lifts, which
proves this assertion directly for a Mittag--Leffler system. For proper
support, formal functions is applied to its proper scheme-theoretic
support. The finite \(A\)-modules \(H^p(X,E)\) are already complete.

For completeness the local-to-global Ext spectral sequence follows from
exactly the first-quadrant filtered-complex construction of the Čech
lesson. Resolve \(G\) by injectives as module sheaves on the ringed
space, take internal Hom from \(F\), and apply the cohomology-sheaf
filtration and a flasque resolution. Internal Hom into an injective is
flasque: a map on an open extends using the injection from extension by
zero of \(F\) into \(F\). Thus the total global Hom complex computes Ext
and the second page is \(H^p(X,E^q)\). The formal ringed space is
constructed by the completed affine rings and their completed
restrictions, as in Lemma 1.1; the same injective construction applies
to its module sheaves. Its coherent modules are exactly our systems.
Extensions of coherent modules are coherent locally over the Noetherian
completed rings, so its degree-one Ext agrees with Yoneda Ext in our
formal category. Its second page is \(H^p(X,\widehat{E^q})\). The
preceding cohomology comparison identifies these second pages. Each
total degree has finitely many \(p,q\ge0\), so the finite-filtration
comparison gives the asserted isomorphisms in degrees zero and one.

Finally degree-one derived Ext classifies extensions: embed \(G\) in an
injective \(Q\), pull back \(Q\to Q/G\) along maps from \(F\), and
identify two pullbacks when their difference lifts to \(Q\). Conversely
injectivity extends \(G\to Q\) over any extension's middle module and
recovers such a map. This identifies Yoneda classes with the same Hom
cokernel, and commutes with completion. An algebraic extension of
coherent modules is coherent and has support in the union of the proper
supports. \(\square\)

\subsection{4. One twist for every
thickening}\label{one-twist-for-every-thickening}

Suppose \(X\) is projective over \(A\), and choose a relatively ample
invertible sheaf \(L\). For a formal module \(\mathfrak F=(F_n)\), put
\(F_0=0\) and \[
D_n=\ker(F_{n+1}\to F_n),\qquad n\geq0.
\]

\textbf{Lemma 4.1 (uniform vanishing).} There exists \(d_0\) such that
\[
H^1(X,D_n\otimes L^d)=0\quad\text{for every }n\geq0,
\quad d\geq d_0.
\]

\textbf{Proof.} Let \(B=\bigoplus_{n\geq0}I^n/I^{n+1}\), a Noetherian
graded algebra generated in degree one over \(A/I\). The sheaf
\(D=\bigoplus D_n\) is a graded module over \(\mathcal O_X\otimes_A B\).
On an affine, Lemma 1.1 identifies it with the associated graded module
\(\bigoplus J^nM/J^{n+1}M\) of a finite completed module. It is
generated over the associated graded ring in degree zero. The map from
the pulled-back \(B\) to that associated graded ring is surjective,
because \(J\) is generated by the image of \(I\). Consequently \(D\) is
of finite type over \(\mathcal O_X\otimes_A B\). Surjectivity is
sufficient; no flatness is needed to identify associated graded rings.

On \(X_B=X\times_A\operatorname{Spec}B\), with affine projection \(p\),
the affine sheaf correspondence realizes \(D=p_*E\) for a coherent
\(E\). The morphism \(X_B\to\operatorname{Spec}B\) is projective and
\(p^*L\) is relatively ample. Serre vanishing gives one bound for the
positive cohomology of \(E\otimes p^*L^d\). Affine pushforward, its
projection formula, and cohomology commuting with direct sums of
quasi-coherent sheaves on this separated quasi-compact scheme identify
that cohomology with \(\bigoplus_n H^1(X,D_n\otimes L^d)\). Every
summand is zero for the same bound. \(\square\)

The bound is uniform in the infinitesimal index. Applying Serre
vanishing independently to each \(F_n\) would give bounds depending on
\(n\), insufficient for a single finite presentation.

\subsection{5. Existence on a projective
scheme}\label{existence-on-a-projective-scheme}

\textbf{Theorem 5.1 (projective existence).} For projective \(X\) over
\(A\), completion gives an equivalence \[
\operatorname{Coh}(X)\simeq\operatorname{Coh}(X,I\mathcal O_X).
\]

\textbf{Proof.} Full faithfulness is Theorem 3.2. Let \(\mathfrak F\) be
a formal module. Choose \(d\) large enough both for Lemma 4.1 and for
finitely many sections of \(F_1\otimes L^d\) to generate it. The exact
sequences with kernel \(D_n\otimes L^d\) make \[
\Gamma(X,F_{n+1}\otimes L^d)\to\Gamma(X,F_n\otimes L^d)
\] surjective. Lift the chosen finite collection recursively to
compatible sections at every level. They give a formal morphism \[
\widehat P\twoheadrightarrow\mathfrak F,
\qquad P=(L^{-d})^r.
\] It is onto because it is onto at the first level. Its kernel
\(\mathfrak K\) is a coherent formal module by Theorem 2.1. Apply the
same construction to \(\mathfrak K\) to obtain
\(\widehat Q\twoheadrightarrow\mathfrak K\), with \(Q=(L^{-e})^s\) for
some \(e,s\). Thus \[
\widehat Q\longrightarrow\widehat P\longrightarrow\mathfrak F\to0
\] is a presentation in the formal category. Full faithfulness
algebraizes its first map to \(Q\to P\). Take its coherent cokernel
\(F\). Exactness of completion identifies \(\widehat F\) with
\(\mathfrak F\), proving essential surjectivity. \(\square\)

The two presentations use different twists if necessary. There is no
assertion that their kernel is a vector bundle, or that every coherent
sheaf has a finite resolution by line bundles.

\subsection{6. Proper schemes and proper
supports}\label{proper-schemes-and-proper-supports}

\textbf{Theorem 6.1 (Grothendieck existence).} Let \(A\) be Noetherian
and \(I\)-adically complete, and let \(X\) be separated and of finite
type over \(A\). Completion gives an equivalence between coherent
sheaves on \(X\) with support proper over \(A\) and coherent formal
modules whose first-level support is proper over \(A\). In particular,
when \(X\) is proper, all coherent sheaves and all coherent formal
modules participate.

\textbf{Proof.} Full faithfulness for proper supports is Proposition 3.3
in degree zero. We prove existence by Noetherian induction on closed
subschemes of \(X\). If there were a counterexample, the ascending chain
condition on coherent ideals would give a maximal defining ideal of a
closed subscheme on which existence fails. Replace \(X\) by that
subscheme. We may therefore assume existence for every \(V(K)\) with
\(K\ne0\). This is ideal induction and requires no finite dimension
bound on the Noetherian base.

Use the schematic, possibly nonreduced Noetherian Chow theorem proved in
the earlier \emph{Projective morphisms and Chow's lemma}, Theorem 4.1.
Its hypotheses are satisfied: \(A\) is Noetherian and \(X/A\) is
separated and of finite type. Its affine cover is chosen with every
member containing all generic points; their intersection \(U\) contains
these points. Schematic closure preserves the structure over \(U\). The
resulting projective proper surjection \(p:Y\to X\) is an isomorphism
over \(U\), with \(Y\) quasi-projective over \(A\). This assertion uses
the full schematic theorem, and does not replace \(X\) by its reduction.
If \(X\) is proper, \(Y\) is projective over \(A\), by the graph
argument.

Pull the formal module \(\mathfrak F\) to \(Y\). For proper \(X\),
Theorem 5.1 algebraizes it to \(E\). In the proper-support case, immerse
\(Y\) as an open of its projective closure \(\overline Y\). The
first-level pulled-back support is proper, since \(p\) is proper and the
original first-level support is proper. At higher levels its underlying
closed set is unchanged and its scheme structure is a nilpotent
thickening; Lemma 5.4 of the formal-functions lesson makes these
supports proper as well. Each is consequently closed in \(\overline Y\).
Extension by zero is coherent: near a point of its closed support we are
inside \(Y\), and outside that support the sheaf is zero. These
extensions and their quotient transitions give a coherent formal module
on \(\overline Y\). Theorem 5.1 algebraizes it to \(\overline E\).

The boundary intersection of the algebraic support of \(\overline E\) is
closed in the projective scheme and has closed image in
\(\operatorname{Spec}A\). It misses \(V(I)\), by the specified
first-level support. The ideal \(I\) lies in the Jacobson radical: the
geometric series inverts \(1-a\) for every \(a\in I\). Every nonempty
closed subset of \(\operatorname{Spec}A\) therefore meets \(V(I)\). The
boundary intersection is empty. Restriction gives \(E\) on \(Y\) with
proper support. In both cases \(H=p_*E\) is coherent by proper coherence
and has proper support. For the support assertion, its reduced support
is contained in the proper image of the support of \(E\); its
scheme-theoretic support is a finite nilpotent thickening of a closed
subset of that image, hence proper by Lemma 5.4.

The adjunction maps at all finite levels, followed by inverse limit,
define \[
\alpha:\mathfrak F\longrightarrow\widehat H.
\] Formal functions in degree zero for \(p\), checked on affine opens of
\(X\), identifies the completed direct image with this inverse limit.
Thus \(\alpha\) is a morphism of coherent formal modules, an isomorphism
over \(U\). We must prove a fixed power annihilating both defects.

Fix an affine \(V=\operatorname{Spec}R\subset X\), and let
\(\widehat R\) be its \(I\)-adic completion. Lemma 1.1 represents
\(\mathfrak F|_V\) by a finite \(\widehat R\)-module \(M\). The scheme
\(Y_{\widehat R}\) is projective over the complete Noetherian ring
\(\widehat R\). The pullbacks of \(E\) and \(p^*\widetilde M\) there
have the same specified formal completion, by their construction and
quotient identities. Theorem 3.2 on this projective scheme makes them
isomorphic algebraically. Flat base change along \(R\to\widehat R\),
proved in the base-change lesson, identifies \(H\otimes_R\widehat R\)
with \(p_*p^*\widetilde M\). Under these identifications \(\alpha\) is
the actual adjunction \[
M\longrightarrow\Gamma(Y_{\widehat R},p^*\widetilde M).
\] It is an isomorphism outside \(V(K\widehat R)\), where \(K\) is the
coherent ideal of \(X\setminus U\). Its kernel and cokernel are finite
modules over the Noetherian ring \(\widehat R\). The support-power proof
in the proper-coherence lesson annihilates them by a power of
\(K\widehat R\). A finite affine cover gives a common maximum exponent
\(c\), so both formal defects are annihilated by \(K^c\).

Write these defects as \(\mathfrak C,\mathfrak D\). Their first-level
supports are proper: the formal kernel has support inside the
first-level support of \(\mathfrak F\), since the affine Nakayama test
makes the formal module zero wherever its first member is zero; the
cokernel has support inside that of \(\widehat H\). They are now formal
modules on \(V(K^c)\). The ideal \(K^c\) is nonzero, since \(K\) is the
unit ideal at every generic point contained in \(U\). Induction
algebraizes the defects to coherent proper-support \(C,D\) on \(X\),
annihilated by \(K^c\). Full faithfulness algebraizes the formal
quotient \(\widehat H\to\mathfrak D=\widehat D\) to \(H\to D\). This is
onto: its coherent proper-support cokernel has zero completion, and full
faithfulness detects zero. Put \(Q=\ker(H\to D)\). Exact completion
identifies \(\widehat Q\) with the image of \(\alpha\). We obtain the
formal sequence \[
0\to\widehat C\to\mathfrak F\to\widehat Q\to0.
\] Proposition 3.3 lifts its extension class to
\(0\to C\to F\to Q\to0\). The middle sheaf is coherent with proper
support; equality of extension classes gives the specified
\(\widehat F\cong\mathfrak F\). This contradicts the counterexample and
completes the induction for both proper schemes and proper supports.
\(\square\)

\subsection{7. Examples: what can fail on an affine
line}\label{examples-what-can-fail-on-an-affine-line}

On \(\mathbf P^1_{k[[t]]}\), the systems \(\mathcal O_{X_n}(d)\)
algebraize to \(\mathcal O_X(d)\). The extension system \[
0\to\mathcal O_{X_n}(-2)\to E_n\to\mathcal O_{X_n}\to0
\] with compatible class \(a_n\in k[t]/t^n\) algebraizes to the
extension with class \(a=\varprojlim a_n\in k[[t]]\), because
\(H^1(\mathcal O_X(-2))=k[[t]]\). Choosing \(a=t\) recovers our
cohomology-jump example.

For \(X=\mathbf A^1_{k[[t]]}\), completion is not fully faithful. The
restricted series \[
h(x)=\sum_{j\geq0}t^jx^j
\] defines a compatible multiplication map on the systems
\(\mathcal O_{X_n}\), but is not a polynomial in \(x\). It therefore
does not come from an endomorphism of \(\mathcal O_X\). Faithfulness
also fails: the nonzero coherent module \(k[[t]][x]/(tx-1)\) has zero
completion, since \(tx-1\) is a unit modulo every power of \(t\).

These are failures of morphism recovery and uniqueness, not examples of
nonalgebraizable objects. In fact this particular affine line has a
stronger existence property.

\textbf{Proposition 7.1.} Every coherent formal module on
\(\mathbf A^1_{k[[t]]}\) is algebraizable.

\textbf{Proof.} Let \(R=k[[t]][x]\), \(B=\widehat R\), and let the
system correspond to a finite \(B\)-module \(M\). The ring \(B\) is a
Noetherian domain, and \(B/tB=k[x]\). Thus \((t)\) is prime, and
\(B_{(t)}\) is a discrete valuation ring: its maximal ideal is generated
by the nonzero element \(t\). The structure theorem over this ring gives
\[
M_{(t)}\cong B_{(t)}^{\,r}\oplus\bigoplus_{i=1}^s B_{(t)}/(t^{a_i}).
\] Finite presentations let this isomorphism and its inverse be defined
after inverting finitely many elements of \(B\setminus(t)\). The
identities defining inverse maps also hold after finitely many such
inversions. Let \(h\in k[x]\setminus\{0\}\) be the product of their
nonzero reductions modulo \(t\). In the \(t\)-adic completion of
\(R[1/h]\), each denominator is a unit, since its reduction modulo \(t\)
is a unit. Hence the formal system on \(D(h)\) is isomorphic to the
completion of the displayed free-plus-\(t\)-power-torsion module.

The zero set of \(h\) in \(\mathbf P^1_k\) is finite and lies in the
affine line: the homogenization has nonzero value at infinity. Let
\(V\subset\mathbf P^1_{k[[t]]}\) be the complement of that homogenized
zero set. It contains infinity and meets the affine chart in \(D(h)\).
On \(V\), take the formal completion of the same free-plus-torsion
module. Glue it to the given system on the affine chart using the formal
isomorphism on \(D(h)\). This gives compatible coherent sheaves on all
projective-line thickenings. Theorem 5.1 algebraizes them on
\(\mathbf P^1_{k[[t]]}\). Restricting that sheaf to the affine chart
algebraizes the original system. \(\square\)

Thus a claim that an arbitrary formal multiplication series
automatically gives a nonalgebraizable coherent module on this affine
line would be false. The theorem's equivalence, including morphisms and
unique recovery, genuinely needs its support hypotheses; essential
surjectivity alone can hold in a special nonproper situation.

\subsection{8. Exercises with solutions}\label{exercises-with-solutions}

\textbf{Exercise 8.1 (easy: exactness).} Explain the sense in which
completion of a coherent short exact sequence is exact, and exhibit the
failure of left exactness at a fixed quotient level.

\textbf{Solution.} On an affine, finite-module completion is tensoring
with the flat Noetherian completed ring, so is exact; Theorem 2.1 glues
this statement in the formal category. For multiplication by \(t\) on
\(k[[t]]\), the original map is injective but its reduction modulo
\(t^n\) has kernel \(k t^{n-1}\). These kernels are transient and have
stabilized formal kernel zero. Thus finite-level injectivity is not the
meaning of exact completion.

\textbf{Exercise 8.2 (medium: projective-line maps).} Compute the
compatible morphisms from \((\mathcal O_{X_n}(a))\) to
\((\mathcal O_{X_n}(b))\) on \(\mathbf P^1_{k[[t]]}\).

\textbf{Solution.} They are the limit of sections of
\(\mathcal O_{X_n}(b-a)\). If \(b-a<0\), each group is zero. Otherwise
its monomial basis gives \((k[t]/t^n)^{b-a+1}\), whose limit is
\(k[[t]]^{b-a+1}\). This is exactly the algebraic Hom group from
projective-space cohomology, with its coefficientwise restriction map.
It proves full faithfulness for these line bundles directly.

\textbf{Exercise 8.3 (medium: nonproper failure).} Give both a
compatible endomorphism on the affine-line formal structure sheaf that
does not algebraize as an endomorphism of \(\mathcal O_X\), and a
nonzero coherent sheaf with zero completion.

\textbf{Solution.} Multiplication by \(\sum_{j\geq0}t^jx^j\) is defined
modulo every \(t^n\), but a polynomial cannot have its unbounded nonzero
\(x\)-coefficients, so it does not algebraize as the specified
endomorphism. The sheaf associated to \(R/(tx-1)\) is nonzero: its ring
is \(k((t))\), with \(x=t^{-1}\). Its reduction modulo \(t^n\) is zero
because \(t\) is invertible there. Proposition 7.1 explains why neither
example should be misdescribed as a nonalgebraizable formal object.

\textbf{Exercise 8.4 (medium: algebraizing a presentation).} Recover
projective existence from uniform vanishing, full faithfulness and
exactness.

\textbf{Solution.} Use a uniform twist to lift a finite generating
collection for \(F_1\), giving
\(\widehat P\twoheadrightarrow\mathfrak F\). Its formal kernel is
coherent. Do the same for that kernel to obtain
\(\widehat Q\to\widehat P\to\mathfrak F\to0\). Full faithfulness
algebraizes \(Q\to P\), and exactness identifies the completion of its
coherent cokernel with \(\mathfrak F\). The construction never assumes
the finite-level kernels themselves form the formal kernel.

\textbf{Exercise 8.5 (hard: reconstruct the proper reduction).} Explain
why a projective modification and a bounded formal comparison suffice to
algebraize a coherent formal module, and prove the fixed bound on its
two defects.

\textbf{Solution.} Pull the formal module to the quasi-projective Chow
modification and algebraize it on its projective closure. Proper support
and the Jacobson-radical argument remove the boundary. Let \(H\) be its
proper direct image. Formal functions gives the adjunction comparison
\(\mathfrak F\to\widehat H\). On an affine of the target, complete the
coordinate ring; projective full faithfulness identifies the algebraized
pullback with the actual pullback of the finite completed module. Flat
base change turns the comparison into an actual adjunction of coherent
modules, an isomorphism off the exceptional closed set. The
support-power criterion supplies a local power killing kernel and
cokernel; a finite cover gives one power. Induction on its exceptional
thickening algebraizes both defects. Full faithfulness algebraizes the
quotient \(H\to D\), and exactness identifies its kernel's completion
with the comparison image. Proposition 3.3 algebraizes the remaining
extension by its kernel defect. This is an explicit reconstruction and
uses no unproved descent of sheaves along a proper surjection.

\textbf{Exercise 8.6 (hard: a special affine existence theorem).}
Explain the two features of the affine line that make Proposition 7.1
work.

\textbf{Solution.} At the generic point of its closed fiber, the
completed ring localized at \((t)\) is a discrete valuation ring. A
finite module there has a free-plus-\(t\)-power-torsion normal form,
which spreads to \(D(h)\) after completing, using finitely many
denominators with nonzero reductions. The zero set of the
single-variable polynomial \(h\) has projective closure disjoint from
infinity. Thus the standard module can be put on an open neighborhood of
the entire boundary, and glued to the original system. Projective
existence finishes. In higher dimension a hypersurface's closure can
meet the boundary, so this argument does not establish unrestricted
affine existence in general.

\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: affine formal
  modules
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-lemma-inverse-systems-affine}{Tag
  087W}, their abelian category
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-lemma-inverse-systems-abelian}{Tag
  087X}, exact completion
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-lemma-exact}{Tag
  0881}, completed Hom
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-lemma-completion-internal-hom}{Tag
  0882}, full faithfulness
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-lemma-fully-faithful}{Tag
  0883}.
\item
  Uniform vanishing
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-lemma-vanishing-projective}{Tag
  0884}, and projective existence
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-lemma-existence-projective}{Tag
  0885}.
\item
  Parallel open treatments of proper existence are the bounded
  comparison
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-lemma-inverse-systems-push-pull}{Tag
  088B}, change of completion
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-lemma-existence-tricky}{Tag
  088A}, and bounded formal gluing
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-lemma-existence-easy}{Tag
  0889}. Those references assemble into proper existence
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-proposition-existence-proper}{Tag
  088C} and proper-support existence
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-theorem-grothendieck-existence}{Tag
  088E}.
\item
  These linked reference proofs retain GNU FDL 1.2. Sections 1--6, the
  affine-line argument, examples and solutions are independently written
  CC0. Proposition 3.3 proves extension comparison and Section 6 proves
  the bounded comparison and proper-support reconstruction in the full
  stated generality.
\end{itemize}
\end{document}