# Artin’s theorems in supergeometry

Nadia Ott111 _Email address_<EMAIL_ADDRESS>
University of Pennsylvania, Philadelphia, PA

###### Abstract

We generalize Artin’s three main algebraicity theorems to the setting of
supergeometry: approximation [Art69a], algebraization of formal deformations
[Art69b], and algebraization of stacks [Art74].

###### Contents

  1. 1 Introduction
  2. 2 Notations and conventions
  3. 3 Preliminaries
  4. 4 Schlessinger’s conditions
  5. 5 Artin approximation
  6. 6 Artin algebraization
  7. 7 Flenner’s criterion for openness of formal versality
  8. 8 Formal smoothness from formal versality
  9. A Proof of Schlessinger’s theorem
  10. B Proof of Flenner’s criterion

##  1 Introduction

Artin’s theorems on approximation [Art69a], algebraization of formal
deformations [Art69b], and stacks [Art74] give general criteria for functors
to be, in various senses, described by algebraic objects. It has long been
expected that analogous results hold in supergeometry; for instance, this was
indicated in Deligne’s sketch construction of the moduli space of stable super
Riemann surfaces [Del87]. Some works have made progress in this direction
[Vai90, FS92, MZ19], and the existence of the moduli space of stable super
Riemann surfaces has since been established by other means [MZ19, FKP20].
However, proofs of the full suite of Artin theorems have remained absent from
the supergeometry literature.222One reason for this absence may be the sense
that establishing the Artin theorems in supergeometry would require the
tedious repetition of various difficult arguments in commutative algebra,
deformation theory, and algebraic geometry, e.g. for the original approach
[Sch68, Rim72, Nér64, Art69a, Art69b, Art74] (with some of these
substituteable by later approaches, see e.g. [Fle81, Pop86, CdJ02, Hal17,
HR19]). In any case one can find in the literature various assertions that
certain results hold in supergeometry “by the same argument as” something in
these works. Here we fill this gap by proving the Artin theorems.

We restrict ourselves to working over $\operatorname{Spec}\mathbb{C}$ due to
the needs of the subject and the comfort of the author. We provide complete
proofs, adhering to the principle that we are happy to cite statements in the
(bosonic) algebraic geometry literature, but not proofs: if an argument must
be repeated with the word ‘super’ added, we repeat the argument.333Usually
with some details added and of course with reference and credit given to the
original source. This is done so that we can verify that the various
foundational results needed to run these arguments are in fact known in
supergeometry (and so the supergeometer need not be referred to various other
works and verify at every stage that they still hold with odd coordinates).
Fortunately, not all arguments are such repetitions: at several key points we
are able to reduce to the (known) bosonic case by an argument which is
significantly simpler than the original bosonic argument. This is possible
because superschemes are nilpotent thickenings of bosonic schemes, so the
difference between them does not involve the difficult ‘convergence’ questions
which are often at the heart of the Artin theorems.

We now describe the results proven in this article. Recall that a superscheme
$X$ gives rise to a functor from affine superschemes to sets by
$X(S):=\operatorname{Hom}(S,X)$; such a functor is called representable and
necessarily satisfies étale descent. More generally, a superstack is an
arbitrary 2-functor from the category of affine superschemes to the category
of groupoids, satisfying étale descent. (In the body of the text we will as
usual pass to the 1-categorical formulation in terms of categories fibered in
groupoids; see Def. 3.13.)

We say a superstack is limit preserving if, for a directed systems of
algebras,
$\varinjlim\mathcal{X}(\operatorname{Spec}A_{i})\xrightarrow{\sim}\mathcal{X}(\operatorname{Spec}\varinjlim
A_{i})$. As all rings are direct limits of its finitely generated sub-rings,
the ‘limit preserving’ condition ensures superstacks are determined by their
values over a finite type superschemes.

We now state the Artin theorems. The first – Artin approximation – asserts
that object associated to the completion of a local ring of a finite type
superscheme are approximated to any fixed finite order by some object over
some finite type étale cover, depending on the order.

######  Theorem 1.1 (Artin approximation).

Let $S$ be finite type affine superscheme and let $\mathcal{X}$ be a limit-
preserving superstack. Let $s$ be a closed point in $S$ and let
$\widehat{\xi}\in\mathcal{X}(\widehat{\mathcal{O}}_{S,s})$. Then for any
integer $n\geq 1$, there exists an étale morphism
$(S^{\prime},s^{\prime})\to(S,s):s^{\prime}\mapsto s$ and an object
$\xi\in\mathcal{X}(S^{\prime})$ such that
$\widehat{\xi}|_{\operatorname{Spec}(\mathcal{O}_{S,s}/\mathfrak{m}_{s}^{n})}$
and
$\xi|_{\operatorname{Spec}(\mathcal{O}_{S^{\prime},s^{\prime}}/\mathfrak{m}_{s^{\prime}}^{n})}$
are isomorphic as objects in $\mathcal{X}$ over
$\operatorname{Spec}(\mathcal{O}_{S,s}/\mathfrak{m}_{s}^{n})\cong\operatorname{Spec}(\mathcal{O}_{S^{\prime},s^{\prime}}/\mathfrak{m}_{s^{\prime}}^{n})$.

The second – Artin algebraization – shows that if we require the initial data
$\widehat{\xi}$ to be _formally versal_ , then a far stronger conclusion
holds: One can find a single cover and a single algebraic object which
approximates the original object _up to any order._ We recall that an object
$\widehat{\xi}$ is called formally versal if it satisfies the following
infinitesimal lifting property: For any surjection of Artin superalgebras,
$B^{\prime}\to B$, and map $R\to B$, there exists an arrow completing the
diagram

${\operatorname{Spec}B}$${\operatorname{Spec}R}$${\operatorname{Spec}B^{\prime}}$${\mathcal{X}}$$\scriptstyle{\widehat{\xi}}$
(1)

Note here that for any superscheme $\operatorname{Spec}A$ the map
$\operatorname{Spec}A\to\mathcal{X}$ is by definition a map of functors
$\hom(\,\cdot\,,\operatorname{Spec}A)\to\mathcal{X}$, which by 2-Yoneda is
identified with an object in $\mathcal{X}(\operatorname{Spec}A)$.

######  Theorem 1.2 (Artin algebraization).

Let $\mathcal{X}$ be a limit-preserving superstack and let
$(R,\mathfrak{m}_{R})$ be a local, complete, Noetherian superalgebra. Let
$\widehat{\xi}\in\mathcal{X}(\operatorname{Spec}R)$ be formally versal. Then
there exists

  1. 1.

an affine finite type superscheme $\operatorname{Spec}A$ and a closed point
$u\in\operatorname{Spec}A$,

  2. 2.

an object $\xi_{A}\in\mathcal{X}(\operatorname{Spec}A)$,

  3. 3.

an isomorphism
$\alpha:R\overset{\sim}{\longrightarrow}\widehat{A}_{\mathfrak{m}_{u}}$ of
superalgebras, and

  4. 4.

a compatible family of isomorphisms
$\widehat{\xi}|_{\operatorname{Spec}(R/\mathfrak{m}_{R}^{n})}\cong\xi_{A}|_{\operatorname{Spec}(A_{\mathfrak{m}_{u}}/\mathfrak{m}_{u}^{n})}$
in $\mathcal{X}$ over $R/\mathfrak{m}_{R}^{n}\cong
A_{\mathfrak{m}_{u}}/\mathfrak{m}_{u}^{n}$ for all $n\geq 1$.

Formal versality is about what happens over Artin rings and so we can study it
using deformation theory. When using results from deformation theory we pass
to the functor $\overline{\mathcal{X}}_{x}$ over the category $C$ of local,
Artin superalgebras parameterizing isomorphism classes of _extensions_ of a
fixed object $x\in\mathcal{X}(\operatorname{Spec}k)$. For a given morphism
$\operatorname{Spec}k\to S$, we say that $y\in\mathcal{X}(S)$ is an extension
of $x$ over $S$ if there is an arrow $x\to y$ in $\mathcal{X}$ over
$\operatorname{Spec}k\to S$. In general, it not possible to determine formal
versality outright; and so we pass to the functor
$\widehat{\overline{\mathcal{X}}}_{x}$ sending a complete, local, Noetherian
superalgebra $R$ to the set

$\widehat{\overline{\mathcal{X}}}_{x}(\operatorname{Spf}R):=\varprojlim\overline{\mathcal{X}}_{x}(\operatorname{Spec}R/\mathfrak{m}_{R}^{n}).$

of _formal objects_ over the formal superscheme $\operatorname{Spf}R$.

We will now state Schlessinger’s theorem [Sch68, Theorem 2.11] (which predated
and motivated Artin’s work) on the existence of formally versal, formal
moduli.

.

######  Theorem 1.3 (Schlessinger’s theorem).

Let $A^{\prime}\to A$ and $A^{\prime\prime}\to A$ be morphisms in $C$ and
consider the map

$\overline{\mathcal{X}}_{x}(\operatorname{Spec}A^{\prime}\bigsqcup_{\operatorname{Spec}A}\operatorname{Spec}A^{\prime\prime})\to\overline{\mathcal{X}}_{x}(\operatorname{Spec}A^{\prime})\times_{\overline{\mathcal{X}}_{x}(\operatorname{Spec}A)}\overline{\mathcal{X}}_{x}(\operatorname{Spec}A^{\prime\prime})$
(2)

Then there exists a complete, local, Noetherian superalgebra $R$ and a
formally versal, formal object
$\widehat{\xi}\in\widehat{\overline{\mathcal{X}}}_{x}(\operatorname{Spf}R)$ if
and only if the following conditions hold.

  * _(S1)a:_

(2) is a surjection whenever $A^{\prime\prime}\to A$ is a small extension.

  * _(S1)b:_

(2) is a bijection whenever $A=k,A^{\prime\prime}=k[\epsilon,\eta]$ .

  * _(S2):_

$\overline{\mathcal{X}}_{x}$ has a finite-dimensional tangent space.

The formal object $\widehat{\xi}$ is a sequence
$(x=\widehat{\xi}_{0},\widehat{\xi}_{1},\cdots),\
\widehat{\xi}_{n}\in\overline{\mathcal{X}}_{x}(\operatorname{Spec}R/\mathfrak{m}_{R}^{n})$
of extensions of $x$. It is useful to view a formal object as a ‘power
series’; the first order term of which is an object in the tangent space to
$\overline{\mathcal{X}}_{x}$. The tangent space to
$\overline{\mathcal{X}}_{x}$ is
$\overline{\mathcal{X}}_{x}(\mathrm{Spec}(k[\epsilon,\eta]))$, where
$k[\epsilon,\eta]$ denotes the _super dual numbers_. Any finite order
extension of $x$ is built from iterating the first order extension and we can
use this fact to approximate the extensions of $x$ up to finite order. A
formal object is what we get when we take this process to the limit.

This leaves open the important question of convergence. We say that the formal
object is effective if it is contained in the image of the natural map

$\mathcal{X}(\operatorname{Spec}R)\to\varprojlim\mathcal{X}(\operatorname{Spec}R/\mathfrak{m}_{R}^{n}).$
(3)

We call the superstack $\mathcal{X}$ effective if (3) is an equivalence of
categories.

######  Remark 1.4.

Effectivity will be a hypothesis in Artin’s third theorem on algebraicity of
stacks. Often, effectivity can be checked using the Grothendieck existence
theorem; a super version of which is proved in _[MZ19]_.

While the preceding results have been of a local nature, the third Artin
theorem is a global result. To state it, let us recall that an algebraic space
is a quotient of a scheme by an étale equivalence relation, and that a map of
stacks $f:\mathcal{X}\to\mathcal{Y}$ is said to be representable by a scheme
(space) if for any $B\to\mathcal{Y}$, one has that
$\mathcal{X}\times_{\mathcal{Y}}B$ is a scheme (space). Properties of maps of
schemes which are stable under base change, such as smoothness, give rise to
properties of representable morphisms by demanding all base changes satisfy
the original property. Representability of the diagonal
$\Delta:\mathcal{X}\to\mathcal{X}\times\mathcal{X}$ implies that any map from
a scheme is representable, and so in particular we may always ask whether a
map from a scheme is smooth, étale, etc.

As in [Art74], we say:

######  Definition 1.5.

A superstack $\mathcal{X}$ is an algebraic superstack if the following
conditions hold:

  * _(1)_

The diagonal morphism $\Delta:\mathcal{X}\to\mathcal{X}\times\mathcal{X}$ is
representable by algebraic superspaces.

  * _(2)_

There exists a smooth, surjective morphism $\pi:U\to\mathcal{X}$ with $U$ a
superscheme.

We say that the algebraic superstack $\mathcal{X}$ is locally of finite type
if there exists a smooth surjection $\pi:U\to\mathcal{X}$ with $U$ a
superscheme locally of finite type.

From Schlessinger’s theorem and Artin algebraization, we deduce the (not yet
useful) algebraicity criterion:

######  Corollary 1.6.

$\mathcal{X}$ is an algebraic superstack locally of finite type if and only if

  1. 1.

The Schlessinger conditions _(S1)_ and _(S2)_ are satisfied.

  2. 2.

Effectivity: If $A$ is a complete, local, Noetherian superalgebra, then the
natural map

$\mathcal{X}(\operatorname{Spec}A)\to\varprojlim\mathcal{X}(\operatorname{Spec}A/\mathfrak{m}_{A}^{n})$

is an equivalence of categories.

  3. 3.

$\mathcal{X}$ is limit preserving.

  4. 4.

The diagonal map $\Delta:\mathcal{X}\to\mathcal{X}\times\mathcal{X}$ is
representable.

  5. 5.

The maps $\xi_{R}:(\operatorname{Spec}R,u)\to\mathcal{X}$ obtained by applying
Artin algebraization to each of the formally versal, formal objects produced
in Schlessinger’s theorem are smooth in some neighborhood of $u$.

###### Proof.

Note first that by effectivity, the Schlessinger formally versal, formal
objects are in fact formally versal objects. Thus we are permitted to apply
Artin approximation and get a formally versal object
$\xi_{R}\in\mathcal{X}(\operatorname{Spec}R)$. Now ‘if’ is obvious: Since we
assumed the diagonal is representable, the $\xi_{R}$ give us a chart at each
point of $\mathcal{X}$, and we have assumed that these are all smooth.

We omit the proof of ‘only if’ as we do not use it, but let us just remark
about why the maps $\xi_{A}$ must be smooth. As we assume $\mathcal{X}$ is
algebraic of locally finite type, we may study $\xi_{A}$ after base change to
some map between finite type schemes. By versality, this map satisfies the
lifting condition with respect to maps from spectra of Artin rings (at $u$).
For maps of schemes of finite type, this is equivalent to smoothness at $u$. ∎

To turn this into a useful criterion, we must give criteria for showing that
the formally versal object $\xi_{R}$ is in fact smooth. A refinement of this
idea is to give criteria which ensure that (1) formal versality is an open
condition, and that (2) formal versality and formal smoothness are equivalent
notions.

We recall that the object $\xi_{R}$ is formally versal at a closed point
$t\in\operatorname{Spec}R$ if it satisfies the following infinitesimal lifting
property: For every surjection of Artin superalgebras, $B^{\prime}\to B$,
there exists an arrow completing the diagram

${\operatorname{Spec}k(t)}$${\operatorname{Spec}B}$${\operatorname{Spec}R}$${\operatorname{Spec}B^{\prime}}$${\mathcal{X}}$$\scriptstyle{\xi_{R}}$
(4)

where $k(t)$ is the residue class field of $t$. Formal versality is an open
condition if $\xi_{R}$ formally versal at $t\in\operatorname{Spec}R$ implies
that there exists a Zariski open subset $t\in U\subset\operatorname{Spec}R$
such that $\xi_{R}$ is formally versal at every closed point $u\in U$.

Following Flenner [Fle81] we deduce criteria for openness of formal versality
by considering a certain sheaf $\mathcal{E}x(\xi_{R},\mathcal{M})$ on the
superscheme $\operatorname{Spec}R$ whose vanishing locus is the set of points
at which $\xi_{R}$ is formally versal. Observe that if
$\mathcal{E}x(\xi_{R},\mathcal{M})$ were a coherent module, then openness of
formal versality would follow automatically from the fact that coherent
modules have closed support; however, in general,
$\mathcal{E}x(\xi_{R},\mathcal{M})$ is just a sheaf of pointed sets. Flenner
deals with this problem by identifying certain coherent modules on
$\operatorname{Spec}R$, naturally associated to
$\mathcal{E}x(\xi_{R},\mathcal{M})$, and deducing a set of criteria which
ensure that the union of the support of these modules is equal to the support
of $\mathcal{E}x(\xi_{R},\mathcal{M})$. Let us now describe these modules.

Let $T=\operatorname{Spec}R$, and let $\mathcal{M}$ be a coherent
$\mathcal{O}_{T_{\operatorname{red}}}$-module. Let
$\mathcal{E}x(T,\mathcal{M})$ be the sheaf on $T$ classifying extensions of
$T$ by $\mathcal{M}$ and let $\mathcal{D}(\mathcal{M})$ be the sheaf on $T$
classifying extensions of $\xi_{R}$ over trivial extensions of $T$ by
$\mathcal{M}$. The sheaf $\mathcal{E}x(T,\mathcal{M})$ is well-known to be
coherent, and $\mathcal{D}(\mathcal{M})$ is coherent under the Schlessinger
conditions. There is a natural map
$\pi:\mathcal{E}x(\xi_{R},\mathcal{M})\to\mathcal{E}x(T,\mathcal{M})$ with
quotient $\mathcal{E}x(T,\mathcal{M})/\pi(\mathcal{E}x(\xi_{R},\mathcal{M}))$
classifying obstructions to extending $\xi_{R}$. We define
$\mathcal{O}b(\mathcal{M})$ to be the quotient
$\mathcal{E}x(T,\mathcal{M})/\pi(\mathcal{E}x(\xi_{R},\mathcal{M}))$.

We will now state Flenner’s criterion for openness of formal versality
(formulated in the super case).

######  Theorem 1.7 (Openness of formal versality).

Suppose $\mathcal{X}$ is a superstack satisfying the following conditions:

  1. 1.

Schlessinger’s conditions _(S1)_ and _(S2)_

  2. 2.

_Coherence of $\mathcal{O}b$_: For every affine superscheme $T$, object
$v\in\mathcal{X}(T)$ and coherent
$\mathcal{O}_{T_{\operatorname{red}}}$-module $\mathcal{M}$,
$\mathcal{O}b_{v}(\mathcal{M})$ is a coherent $\mathcal{O}_{T}$-module.

  3. 3.

_Constructibility_ : Let $v,T$ be as above and suppose that
$T^{\prime}\subseteq T$ is an irreducible reduced subspace of $T$. Then there
is a Zariski open dense subset $V^{\prime}\subseteq T^{\prime}$ such that for
each $t\in V^{\prime}$, the canonical morphisms

     * _(a)_

$\mathcal{D}_{v}(\mathcal{O}_{T^{\prime}})\otimes
k(t)\to\mathcal{D}_{v}(\mathcal{O}_{T^{\prime}}\otimes k(t))_{t}$ and
$\mathcal{D}_{v}(\Pi\mathcal{O}_{T^{\prime}})\otimes
k(t)\to\mathcal{D}_{v}(\Pi\mathcal{O}_{T^{\prime}}\otimes k(t))_{t}$ are
bijective, and

     * _(b)_

$\mathcal{O}b_{v}(\mathcal{O}_{T^{\prime}})\otimes
k(t)\to\mathcal{O}b_{v}(\mathcal{O}_{T^{\prime}}\otimes k(t))_{t}$ and
$\mathcal{O}b_{v}(\Pi\mathcal{O}_{T^{\prime}})\otimes
k(t)\to\mathcal{O}b_{v}(\Pi\mathcal{O}_{T^{\prime}}\otimes k(t))_{t}$ are
injective.

Then for every superscheme $T$ and every $v\in\mathcal{X}(T)$, the subset of
points $t$ in $T$ which are formally versal for $\mathcal{X}$ form a Zariski-
open subset of $T$.

It is also possible to deduce formal smoothness from formal versality given
some additional hypotheses on $\mathcal{X}$. We recall that the object
$\xi_{R}$ is _formally smooth_ if for any infinitesimal extension
$B^{\prime}\to B$ of Noetherian superalgebras and map
$\operatorname{Spec}B\to\operatorname{Spec}R$ there exists an arrow completing
the diagram

${\operatorname{Spec}B}$${\operatorname{Spec}R}$${\operatorname{Spec}B^{\prime}}$${\mathcal{X}}$$\scriptstyle{\xi_{R}}$
(5)

######  Theorem 1.8.

Let $\mathcal{X}$ be a superstack satisfying the conditions in Theorem 1.7.
Let $v\in\mathcal{X}(T)$ be algebraic and suppose $\mathcal{X}$ satisfies the
following additional conditions:

  1. 1.

The sheaf $\mathcal{D}$ is compatible with completion: If $T_{0}$ is a reduced
finite type scheme, $a_{0}\in\mathcal{X}(T_{0})$, $t\in T_{0}$ and
$\mathcal{M}$ is a coherent $\mathcal{O}_{T_{0}}$-module, then

$\mathcal{D}_{a_{0}}(\mathcal{M})\otimes_{\mathcal{O}_{T_{0}}}\widehat{\mathcal{O}}_{T_{0},t}\overset{\sim}{\longrightarrow}\varprojlim\mathcal{D}_{a_{0}}(\mathcal{M}/\mathfrak{m}_{t}^{n}\mathcal{M})_{t}$

  2. 2.

The sheaf $\mathcal{D}$ is compatible with étale localization: If $e:S_{0}\to
T_{0}$ is an étale morphism of reduced schemes, where $e^{*}(a_{0})=b_{0}$,
and $\mathcal{M}$ a coherent $\mathcal{O}_{T_{0}}$-module, then

$\mathcal{D}_{b_{0}}(\mathcal{M}\otimes_{\mathcal{O}_{T_{0}}}\mathcal{O}_{S_{0}})\cong\mathcal{D}_{a_{0}}(\mathcal{M})\otimes_{\mathcal{O}_{T_{0}}}\mathcal{O}_{S_{0}}$

Then $v$ is formally smooth if and only it is formally versal at every closed
point $t\in T$.

Given the above results on versality and smoothness we can turn the non-useful
Corollary 1.6 into the more checkable criterion which is the third Artin
theorem. The conditions (formulated in the super case) are as follows.

######  Theorem 1.9.

Let $\mathcal{X}$ be a limit preserving superstack and assume that conditions
_(S1)_ and _(S2)_ hold. Then $\mathcal{X}$ is a algebraic superstack locally
of finite type if and only if

  1. _(A1)_.

The diagonal morphism $\Delta:\mathcal{X}\to\mathcal{X}\times\mathcal{X}$ is
represented by an algebraic superspace locally of finite type.

  2. _(A2)_.

_Effectivity_ : If $R$ is a complete, local, Noetherian superalgebra, then the
natural map

$\mathcal{X}(\operatorname{Spec}R)\to\varprojlim\mathcal{X}(\operatorname{Spec}R/\mathfrak{m}_{R}^{n})$

is an equivalence of categories.

  3. _(A3)_.

_Coherence of $\mathcal{O}b$_: For every superscheme $T$, object
$v\in\mathcal{X}(T)$ and coherent
$\mathcal{O}_{T_{\operatorname{red}}}$-module $\mathcal{M}$,
$\mathcal{O}b_{v}(\mathcal{M})$ is a coherent $\mathcal{O}_{T}$-module.

  4. _(A4)_.

_Constructibility_ : Let $v,T$ be as above and suppose that
$T^{\prime}\subseteq T$ is an irreducible reduced subspace of $T$. Then there
is a Zariski open dense subset $V^{\prime}\subseteq T^{\prime}$ such that for
each closed point $t\in V^{\prime}$, the canonical morphisms

     * _(a)_

$\mathcal{D}_{v}(\mathcal{O}_{T^{\prime}})\otimes
k(t)\to\mathcal{D}_{v}(\mathcal{O}_{T^{\prime}}\otimes k(t))_{t}$ and
$\mathcal{D}_{v}(\Pi\mathcal{O}_{T^{\prime}})\otimes
k(t)\to\mathcal{D}_{v}(\Pi\mathcal{O}_{T^{\prime}}\otimes k(t))_{t}$ are
bijective, and

     * _(b)_

$\mathcal{O}b_{v}(\mathcal{O}_{T^{\prime}})\otimes
k(t)\to\mathcal{O}b_{v}(\mathcal{O}_{T^{\prime}}\otimes k(t))_{t}$ and
$\mathcal{O}b_{v}(\Pi\mathcal{O}_{T^{\prime}})\otimes
k(t)\to\mathcal{O}b_{v}(\Pi\mathcal{O}_{T^{\prime}}\otimes k(t))_{t}$ are
injective.

  5. _(A5)_.

The sheaf $\mathcal{D}$ is compatible with completion: If $T_{0}$ is a reduced
finite type scheme, $a_{0}\in\mathcal{X}(T_{0})$, $t\in T_{0}$ and
$\mathcal{M}$ is a coherent $\mathcal{O}_{T_{0}}$-module, then

$\mathcal{D}_{a_{0}}(\mathcal{M})\otimes_{\mathcal{O}_{T_{0}}}\widehat{\mathcal{O}}_{T_{0},t}\overset{\sim}{\longrightarrow}\varprojlim\mathcal{D}_{a_{0}}(\mathcal{M}/\mathfrak{m}_{t}^{n}\mathcal{M})_{t}$

  6. _(A6)_.

The sheaf $\mathcal{D}$ is compatible with étale localization: If $e:S_{0}\to
T_{0}$ is an étale morphism of reduced schemes, with $e^{*}(a_{0})=b_{0}$, and
$\mathcal{M}$ is a coherent $\mathcal{O}_{T_{0}}$-module, then

$\mathcal{D}_{b_{0}}(\mathcal{M}\otimes_{\mathcal{O}_{T_{0}}}\mathcal{O}_{S_{0}})\cong\mathcal{D}_{a_{0}}(\mathcal{M})\otimes_{\mathcal{O}_{T_{0}}}\mathcal{O}_{S_{0}}$

###### Proof.

We prove this result using the statements of the above theorems, whose proofs
we provide in later sections.

Since (S1) and (S2) hold there exists a formally versal, formal object
$\widehat{\xi}\in\widehat{\overline{\mathcal{X}}}_{x}(\operatorname{Spf}R)$
for each $x\in\mathcal{X}(\operatorname{Spec}k)$ by Schlessinger’s Theorem
(Theorem 1.3). By (A2), the formal object $\widehat{\xi}$ can be approximated
with a formally versal object $\xi\in\mathcal{X}(\operatorname{Spec}R)$. We
can approximate $\xi$ with a formally versal, algebraic object,
$\xi_{R}\in\mathcal{X}(\operatorname{Spec}R)$ by Artin algebraization (Theorem
1.2). The set of closed points in $\operatorname{Spec}R$ at which $\xi_{R}$ is
formally versal is Zariski open by (A3) and (A4) and Theorem 1.7. Denote this
locus by $U\subset\operatorname{Spec}R$—possibly shrinking $U$ to a
superscheme of finite type—and let $\xi_{R}$ continue to denote the pullback
of $\xi_{R}$ to an object over $U$. $\xi_{R}$ is formally smooth by (A5) and
(A6) and Theorem 1.8. The associated map $U\to\mathcal{X}$ is representable by
(A1). This map is smooth because it is formally smooth and $U$ is of finite
type. Repeating this process for each $x\in\mathcal{X}(\operatorname{Spec}k)$,
let $\mathcal{U}=\bigsqcup U_{i}$ and $u=\sqcup\xi_{R}$, where the index $i$
is over objects $x\in\mathcal{X}(\operatorname{Spec}k)$. Then
$\mathcal{U}\to\mathcal{X}$ is a smooth, and representable cover of
$\mathcal{X}$. Furthermore, $\mathcal{U}$ is locally of finite type since each
$U_{i}$ is of finite type. Thus, $\mathcal{X}$ is an algebraic superstack
locally of finite type. ∎

Strategy and outline. Artin approximation (Theorem 1.1 above) is proven as
Theorem 5.3 below; we follow [CdJ02] using the observation of [MZ19] that the
(deep and difficult) Néron-Popescu theorem [Nér64, Pop86] can be extended to
super commutative algebra by a simple argument.

Artin algebraization (Theorem 1.2 above) is proven as Theorem 6.2 below. We
adapt some ideas from [CdJ02], but are able to avoid many difficult
convergence arguments by reducing to the bosonic case.

The proof of Schlessinger’s theorem (Theorem 1.3 above) is identical to the
bosonic case; we provide it in Appendix A for completeness.

Openness of versality (Theorem 1.7 above) is proven as Theorem 7.7 below. We
follow [Fle81], except that the proof of the key lemma about half-exact
functors (Lemma B.6, the counterpart of [Fle81, Theorem 6.1]) is done by
reduction to the bosonic case.444 It is asserted in [FS92] that Flenner’s
results on openness of versality generalize to the superanalytic setting, but
few proofs are explicitly written; missing in particular is any argument that
the nontrivial [Fle81, Lemma 4.1] holds in supercommutative algebra.

We show in Theorem 8.2 that (under appropriate hypotheses), formal versality
in fact implies versality; here we begin by following Artin’s original proof
[Art74, Theorem 3.3], but then avoid a convergence question by reducing to the
bosonic case.

Finally, the relationship between formal versality and formal smoothness
(Theorem 1.8 above) is proven as Theorem 8.5 below, following Artin’s original
proof [Art74, Proposition 4.3].

Acknowledgements. I would like to thank Jack Hall for many helpful
explanations. I would also like to thank Vivek Shende, Ron Donagi, and
Alexander Voronov for helpful discussions.

##  2 Notations and conventions

The letter $k$ will denote an algebraically closed field of characteristic
zero. We consider the following categories.

> $\operatorname{sAffSch}$= category of Noetherian affine superschemes over
> $\operatorname{Spec}k$ whose residue fields at closed points are all equal
> to $k$.

> $C$=category of local, Artin superalgebras over $k$ with residue field at
> the closed point equal to $k$.

> $C^{*}$=category of local, Noetherian, Henselian superalgebras over $k$ with
> residue field at the closed point equal to $k$.

All super vector spaces are assumed to be over $k$. Morphisms of super vector
spaces are grading-preserving. All superalgebras are over $k$, and are assumed
to be Noetherian and supercommutative. Morphisms of local rings are always
local.

We denote the $\mathbb{Z}_{2}$-grading on a superalgebra $A$ by $A=A^{+}\oplus
A^{-}$ and write $\mathcal{J}_{A}$ for the ideal generated by $A^{-}$ in $A$.
For a homogeneous element $a\in A$, $|a|=i\ \operatorname{mod}2$ will denote
the parity of $a$. We say that $a$ is even (resp. odd) if $|a|=0$ (resp.
$|a|=1$). We write $A_{\operatorname{bos}}$ for the ordinary $k$-algebra
$A/\mathcal{J}_{A}$ and $A_{\operatorname{red}}$ for the quotient of $A$ by
the ideal of _all_ nilpotents in $A$. Given a superscheme $S$, we denote the
$\mathbb{Z}_{2}$-graded structure sheaf of $S$ by
$\mathcal{O}_{S}=\mathcal{O}_{S}^{+}\oplus\mathcal{O}_{S}^{-}$ and write
$\mathcal{J}_{S}=\langle\mathcal{O}_{S}^{-}\rangle$ for the sheaf of odd
nilpotents in $S$.

##  3 Preliminaries

###  3.1 Super algebraic geometry

A _super vector space_ over $k$ is a vector space $V$ with a
$\mathbb{Z}_{2}$-grading, $V=V^{+}\oplus V^{-}$ where $V^{+}$ and $V^{-}$ are
$k$-vector spaces in the ordinary sense. The super dimension of $V$, denoted
by $\operatorname{sdim}_{k}(V)$, is a pair of non-negative integers $r|s$
where $r=\operatorname{dim}_{k}V^{+}$ and $s=\operatorname{dim}_{k}V^{-}$. We
say that $V$ is finite dimensional if $r<\infty$ and $s<\infty$.

A _superalgebra_ over $k$ is a super vector space which is an associative
algebra with unit $1$ and such that the multiplication $A\otimes A\to A$ is a
morphism of super vector spaces. A superalgebra is supercommutative if for all
$a,b\in A$ we have that

$ab=(-1)^{|a||b|}ba.$

A superalgebra is Noetherian if it satisfies the ascending chain condition on
ideals.

Henceforth, the word superalgebra will mean a Noetherian, supercommutative
$k$-superalgebra.

A super module $M$ over a superalgebra $A$ is a super vector space on which
$A$ acts from the left, where the action $a\otimes m\mapsto a\cdot m,\ \ (a\in
A,m\in M)$ is a morphism of super vector spaces, that is, $|a\cdot
m|=|a|+|m|$. Henceforth, we write $A$-module to mean $A$ super module.

Every superalgebra $A$ has an ideal $\mathcal{J}$ of odd nilpotents generated
by $A^{-}$. $\mathcal{J}$ has a canonical filtration:

$\mathscr{J}:\mathcal{J}\supset\mathcal{J}^{2}\supset\mathcal{J}^{3}\supset\cdots\supset\mathcal{J}^{r},$

where $r$ is the smallest integer for which $\mathcal{J}^{r+1}=0$. The
_$\mathcal{J}$ -associated graded of $A$_ is the
$\mathbb{Z}\times\mathbb{Z}_{2}$-graded $A/\mathcal{J}$-module

$\operatorname{gr}_{\mathcal{J}}(A):=A/\mathcal{J}\oplus\mathcal{J}/\mathcal{J}^{2}\oplus\cdots\oplus\mathcal{J}^{r}$

.

A _superspace_ is a _locally super ringed space_ $(X,\mathcal{O}_{X})$
consisting of a topological space $X$ and a structure sheaf

$\mathcal{O}_{X}=\mathcal{O}_{X}^{+}\oplus\mathcal{O}_{X}^{-}$

of superalgebra such that the stalks are local superrings.

A _superscheme_ $(X,\mathcal{O}_{X})$ is a superspace $(X,\mathcal{O}_{X})$
such that $(X,\mathcal{O}_{X}^{+})$ is an ordinary scheme and
$\mathcal{O}_{X}^{-}$ is a quasi-coherent sheaf of
$\mathcal{O}_{X}^{+}$-modules. A _morphism of superschemes_
$f:(X,\mathcal{O}_{X})\to(Y,\mathcal{O}_{Y})$ is a continuous map $f:X\to Y$
of topological spaces along with a morphism $f^{\\#}:\mathcal{O}_{Y}\to
f_{*}\mathcal{O}_{X}$ of sheaves of superalgebras inducing a local morphism on
stalks.

A superscheme $X$ is affine if it is the $\operatorname{Spec}$ of a
superalgebra $A$. An affine superscheme $\operatorname{Spec}A$ is of finite
type, Noetherian, etc. if the superalgebra $A$ is is finite type, Noetherian,
etc.

######  Definition 3.1 (Flat morphism).

A morphism $f:X\to Y$ of superschemes is _flat_ if $f^{-1}\mathcal{O}_{Y}$ is
a flat $\mathcal{O}_{X}$-module.

######  Definition 3.2 (Smooth morphism).

A morphism $f:X\to Y$ of superschemes is _smooth_ at $x\in X$ if the following
hold:

  * (a)

$f$ is of finite type at $x$.

  * (b)

$f$ is flat at $x$.

  * (c)

If $y=f(x)$, then $X_{y}=X\times_{Y}\operatorname{Spec}k(y)$ is regular at
$x$.

We say that $f$ is _smooth of relative dimension $(m|n)$_ if $f$ is smooth and
for each $y=f(x)$, $\operatorname{sdim}_{k(y)}X_{y}=(m|n)$.

######  Definition 3.3 (Étale morphism).

A smooth morphism $f:X\to Y$ of superschemes is $\acute{e}$tale if for every
$y=f(x)$, $\operatorname{sdim}_{k(y)}X_{y}=(0|0)$. We say that $f:X\to Y$ is
an $\acute{e}$tale covering of $Y$ if $f$ is also surjective.

######  Lemma 3.4.

Let $f:X\to Y$ be a smooth morphism of superschemes of relative dimension
$(m|n)$ and let $y\in Y$ be a closed point of $Y$, let $x\in X_{y}$ be a point
in the fiber over $y$. Then there exists a Zariski open subset $x\in
U\subseteq X$ and an étale morphism of $Y$-superschemes
$U\to\mathbb{A}_{Y}^{m|n}$ such that the following diagram commutes

${U}$${\mathbb{A}_{Y}^{m|n}}$${Y}$$\scriptstyle{f|_{U}}$$\scriptstyle{\operatorname{pr}}$

###  3.2 Extensions and deformations

An _infinitesimal extension_ $(R^{\prime},\phi)$ of a superalgebra $R$ is a
surjection $\phi:R^{\prime}\to R$ of superalgebras such that
$\operatorname{ker}(R^{\prime}\to R)=I$ is nilpotent. An infinitesimal
extension of a superscheme $T$ is a closed immersion $i:T\hookrightarrow
T^{\prime}$ of superschemes defined by a sheaf of ideals
$\mathcal{I}\subset\mathcal{O}_{T^{\prime}}$ such that $\mathcal{I}^{n}=0$ for
$n>1$.

An morphism of infinitesimal extensions
$(R^{\prime},\phi)\to(R^{\prime\prime},\phi^{\prime})$ of $R$ is a commutative
diagram

${0}$${I}$${R^{\prime}}$${R}$${0}$${0}$${I}$${R^{\prime\prime}}$${R}$${0.}$$\scriptstyle{\phi}$$\scriptstyle{\alpha}$$\scriptstyle{\phi^{\prime\prime}}$
(6)

The map $\alpha$ is an isomorphism by the Snake lemma. It follows that the
extensions $(R^{\prime},\phi)$ of $R$ by $I$ form a groupoid.

As in the classical case, any infinitesimal extension $(R^{\prime},\phi)$ of
$R$ can be factored into a finite composite of square-zero extensions, the
ideal $I$ of which is naturally a finitely generated $R$-module.

Let $I$ be a finitely-generated $R$-module and let $\operatorname{Ex}(R,I)$
denote the set of isomorphism classes of square-zero extensions of $R$ by $I$.
This gives a functor
$\operatorname{Ex}(R,-):\operatorname{Coh}(R)\to\operatorname{Set}$ sending a
finitely generated $R$-module $I$ to the set $\operatorname{Ex}(R,I)$. There
is an $R$-module structure on $\operatorname{Ex}(R,I)$ which is described
exactly as in the classical case, (see e.g, [Ser07]): In particular,
$\operatorname{Ex}(R,-):\operatorname{Coh}(R)\to\operatorname{Coh}(R)$.

A square-zero extension $(R^{\prime},\phi)$ of $R$ is called _trivial_ if it
has a section $s:R\to R^{\prime}$ such that $\phi s=1_{R}$. Given an
$R$-module $I$ we can always construct a trivial extension of $R$ by $I$:
Define on $R\oplus I$ the multiplication

$(r,i)\cdot(r^{\prime},i^{\prime})=(rr^{\prime},r^{\prime}i+i^{\prime}r),\ \
r,r^{\prime}\in R,\ i,i^{\prime}\in I.$ (7)

Observe that $I^{2}=0$. Henceforth, we set $R[I]:=(R\oplus I,\cdot)$. We
identify $R[I]$ with the zero element in $\operatorname{Ex}(R,I)$.

######  Lemma 3.5.

Let $(R^{\prime},\phi)$ be a square-zero extension of $R$ and let $f:B\to R$
be a morphism of superalgebras. Let $f_{1},f_{2}:B\to R^{\prime}$ be two
superalgebra morphisms such that the following diagram commutes.

${B}$${0}$${I}$${R^{\prime}}$${R}$${0}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{2}}$$\scriptstyle{f}$$\scriptstyle{\phi}$
(8)

Then the induced map $f_{1}-f_{2}:B\to I$ is a grading-preserving $k$-linear
derivation.

The maps $f_{1}$ and $f_{2}$ in Lemma 3.5 are called _lifts_ of $f$. In
particular, Lemma 3.5 shows that the set of lifts of $f$ form a
$\operatorname{Der}_{k}(B,I)$-torsor. In the case that $B=R$ and $f=1_{R}$,
giving a lift of $1_{R}$ is equivalent to giving a section $s:R\to R^{\prime}$
such that $\phi s=1_{r}$.

####  3.2.1 Artin superalgebras

Let $C$ denote the category of local, Artin superalgebras over $k$ with
residue field at the unique closed point equal to $k$.

######  Example 3.6.

An important example of a local, Artin superalgebra are the super dual numbers
$k[\epsilon,\eta]/(\epsilon^{2},\epsilon\eta),\ |\epsilon|=0,\ |\eta|=1$.

As in the ordinary case, any surjection $A^{\prime}\to A,\
\operatorname{ker}(A^{\prime}\to A)=I$ in $C$ is nilpotent. We say that a
square-zero extension $A^{\prime}\to A,\ \operatorname{ker}(A^{\prime}\to
A)=I$ with $I=(t_{1},\cdots,t_{r},\theta_{1},\cdots,\theta_{s}),\ |t_{i}|=0,\
|\theta_{i}|=1$ is $(r|s)$-small if $\mathfrak{m}_{A^{\prime}}\cdot I=0$ so
that $I$ is naturally an $(r|s)$-dimensional super vector space over
$k=A^{\prime}/\mathfrak{m}_{A^{\prime}}$.

######  Example 3.7.

The surjection $k[\eta]\to k,\ |\eta|=1,\ \operatorname{ker}(k[\eta]\to
k)=(\eta)$ is a $(0|1)$-small extension of $k$ by $(\eta)\cong\Pi k$. The
surjection $k[\epsilon,\eta]\to k,\ \operatorname{ker}(k[\epsilon,\eta]\to
k)=(\epsilon,\eta)$ is a $(1|1)$-small extension of $k$ by
$(\epsilon,\eta)\cong k\times\Pi k$.

######  Lemma 3.8.

Let $f:A^{\prime}\to A,\ \operatorname{ker}(A^{\prime}\to A)=I$ be a square-
zero extension in $C$. Then $f$ can be factored as a composite of $(1|0)$\-
and $(0|1)$-small extensions.

###### Proof.

The maximal ideal $\mathfrak{m}_{A^{\prime}}$ is nilpotent since $A^{\prime}$
is Artin and so there exists some integer $m\geq 0$ such that
$\mathfrak{m}_{A^{\prime}}^{m}=0$. From this we get a factorization,

$A^{\prime}=A^{\prime}/I\mathfrak{m}_{A^{\prime}}^{m-1}\to
A^{\prime}/I\mathfrak{m}_{A^{\prime}}^{m-2}\to\cdots\to A^{\prime}/I=A.$

of surjective maps whose kernels
$\mathfrak{m}_{A^{\prime}}^{i}I/\mathfrak{m}_{A^{\prime}}^{i+1}I$ are all
annihilated by $\mathfrak{m}_{A^{\prime}}$. That is, $f$ can be factored as a
sequence of small $(r_{i}|s_{i})$-extensions, for $i=1,\cdots,m-1$, where
$(r_{i}|s_{i})=\operatorname{sdim}_{k}(\mathfrak{m}_{A^{\prime}}^{i}I/\mathfrak{m}_{A^{\prime}}^{i+1}I)$.
Thus it suffices to prove the lemma when $f$ is a $(r|s)$-small extension.
Take a basis $(t_{1},\dots,t_{r},\theta_{1},\dots,\theta_{s})$ for
$\operatorname{ker}(f)=I$ as a super vector space over $k$. From this choice
we get a factorization,

$A^{\prime}\to A^{\prime}/(t_{1})\to A^{\prime}/(t_{1},t_{2})\to\cdots\to
A^{\prime}/(t_{1},\dots,t_{r})\to
A^{\prime}/(t_{1},\dots,t_{r},\theta_{1})\to\cdots\to
A^{\prime}/(t_{1},\dots,\theta_{s})=A$

into $s$ $(0|1)$-small extensions and $r$ $(1|0)$-small extensions.

∎

######  Definition 3.9.

An infinitesimal deformation of a superscheme $Y$ is a cartesian diagram of
morphisms of superschemes

${Y}$${\mathcal{Y}}$${\operatorname{Spec}k}$${S}$$\scriptstyle{\pi}$$\scriptstyle{s}$
(9)

with $S$ an Artin superscheme, and $\pi$ flat and surjective.

###  3.3 Categories fibered in groupoids

A _category fibered in groupoids_ over the category $\operatorname{sAffSch}$
is a functor $p:\mathcal{X}\to\operatorname{sAffSch}$ with the following
properties:

  1. 1.

For every morphism $f:S\to T$ in $\operatorname{sAffSch}$ and for every object
$t$ in $\mathcal{X}$ over $T$, there exists an object $s\in\mathcal{X}(S)$ and
a cartesian arrow $\overline{f}:s\to t$.

  2. 2.

Compositions of cartesian arrows are cartesian.

  3. 3.

$\overline{f}:s\to t$ is an isomorphism if $p(\overline{f})$ is.

The fiber $\mathcal{X}(T)$ is the groupoid whose objects are the objects in
$\mathcal{X}$ over $T$ and whose morphisms are arrows in $\mathcal{X}$ which
go to $\operatorname{id}_{T}$ under $p$. Henceforth, we will refer to
$\mathcal{X}$ as simply a groupoid.

######  Example 3.10.

The functor $\operatorname{Def}_{Y}:C\to\operatorname{Groupoid}$ sending an
Artin superscheme $S$ to the groupoid of infinitesimal deformations of $Y$
over $S$ is a category fibered in groupoids. To check this, let $f:S\to T$ be
a morphism in $C$ and let $\mathcal{Y}$ be a deformation of $Y$ over $T$. Then
the top horizontal arrow in the fiber product diagram

${\mathcal{Y}\times_{T}S}$${\mathcal{Y}}$${S}$${T}$$\scriptstyle{f}$

is a cartesian arrow $\mathcal{Y}\times_{T}S\to\mathcal{Y}$ over $f:S\to T$.
This proves condition 1. Conditions 2 and 3 are now obvious from properties of
the fiber product.

For a fixed object $v\in\mathcal{X}(T)$ and a morphism of superschemes $f:T\to
S$, we write $\mathcal{X}_{v}(S)$ for the groupoid of maps $v\to s$ in
$\mathcal{X}$ lying over $f$. An isomorphism $(v\to s_{1})\to(v\to s_{2})$ in
$\mathcal{X}_{v}$ is a commutative diagram

${s_{1}}$${v}$${s_{2}}$$\scriptstyle{\alpha}$

A bar will denote the set of isomorphism classes in $\mathcal{X}$. For
example, $\overline{\mathcal{X}}_{v}(S)$ is the set of isomorphism classes in
the groupoid $\mathcal{X}_{v}(S)$.

######  Example 3.11.

$\overline{\operatorname{Def}}_{Y}:C\to\operatorname{Set}$ is a functor
sending an Artin superscheme $S$ to the _set_ of isomorphism classes of
deformations of $Y$ over $S$.

######  Definition 3.12.

We say that a groupoid $\mathcal{X}$ _limit-preserving_ if for every directed
system $\\{A_{i}\\}_{i}$ of superalgebras the natural map

$\varinjlim\mathcal{X}(\operatorname{Spec}A_{i})\to\mathcal{X}(\operatorname{Spec}\varinjlim
A_{i})$

is an equivalence of categories. 555It is customary to call this property
limit-preserving instead of the more accurate colimit-preserving.

The following definitions of a superstack and an algebraic superstack are
straightforward generalizations of their ordinary counterparts.

######  Definition 3.13 (Superstack).

A groupoid $\mathcal{X}$ is a _superstack_ if it satisfies the following
conditions:

  1. 1.

For every pair $x,y\in\mathcal{X}(T)$, the isomorphism functor

$\operatorname{Isom}(x,y):\operatorname{sAffSch}/T\to\operatorname{Set}:(f:S\to
T)\longmapsto\\{\phi:f^{*}x\overset{\sim}{\longrightarrow}f^{*}y\ |\
p(\phi)=\operatorname{id}_{T}\\}$

is a sheaf in the étale topology on $T$.

  2. 2.

For any étale covering $\\{A\overset{\phi_{i}}{\longrightarrow}B_{i}\\}$,
every descent datum for $\mathcal{X}$ relative to $\phi_{i}$ is effective.

##  4 Schlessinger’s conditions

In this section we provide super versions of Artin’s generalization [Art74] of
Schlessinger’s conditions [Sch68].

To generalize Schlessinger’s conditions, we consider arbitrary maps of
infinitesimal extensions of a given reduced algebra $R_{0}$ and, in
particular, diagrams

$R^{\prime}\to R\to R_{0},\ \ \operatorname{ker}(R^{\prime}\to R)=M$ (10)

of infinitesimal extensions of $R_{0}$, where $R^{\prime}\to R$ is surjective
and $M$ is naturally a finite $R_{0}$-module.

######  Remark 4.1.

Schlessinger’s original conditions 1.3 are for extensions of $k$. Artin’s
conditions, as we will see, are formulated for extensions of an arbitrary
reduced ring $R_{0}$. The condition that $M$ be a finite $R_{0}$-module is
coming from the fact that any surjection $f:R\to R_{0}$ can be factored into a
finite composite of extensions by finite $R_{0}$-modules. This is a
generalization of the fact that any surjection of local, Artin superalgebras
can be factored into a finite composite of extensions by finite dimensional
super vector spaces (Lemma 3.8).

In terms of superschemes, we consider arbitrary maps of infinitesimal
extensions of a given reduced scheme $T_{0}$, and, in particular, diagrams

$T_{0}\hookrightarrow T\hookrightarrow T^{\prime},\
\operatorname{ker}(i^{-1}\mathcal{O}_{T^{\prime}}\to\mathcal{O}_{T})=\mathcal{M}$
(11)

of infinitesimal extensions of $T_{0}$, where $T\hookrightarrow T^{\prime}$ is
a closed immersion and $\mathcal{M}$ is a coherent
$\mathcal{O}_{T_{0}}$-module.

The following conditions are super versions of Artin’s conditions (S1) and
(S2) in [Art74].

Condition. (S1). (a) Let $\mathcal{X}$ be a superstack over
$\operatorname{sAffSch}$. Let

${T_{0}}$${T}$${Y}$${T^{\prime}}$ (12)

be a diagram in $\operatorname{sAffSch}$, where $T_{0}\hookrightarrow T\to
T^{\prime}$ is as in (11). Assume that the composition $T_{0}\hookrightarrow
T\to Y$ is a closed immersion. Let $t\in\mathcal{X}(T)$. Then the natural map

$\overline{\mathcal{X}}_{t}(T^{\prime}\sqcup_{T}Y)\to\overline{\mathcal{X}}_{t}(T^{\prime})\times\overline{\mathcal{X}}_{t}(Y),$
(13)

is surjective.

Condition. (S1). (b) Let $i:T_{0}\hookrightarrow Y$ be a closed immersion,
$T_{0}$ a reduced scheme, and let $\mathcal{M}$ be a coherent
$\mathcal{O}_{T_{0}}$-module. Let $y\in\mathcal{X}(Y)$ such that $t_{0}\to y$
is an arrow in $\mathcal{X}$ over $i$. Then the natural map

$\overline{\mathcal{X}}_{y}(Y[\mathcal{M}])\to\overline{\mathcal{X}}_{t_{0}}(T_{0}[\mathcal{M}]))$
(14)

is bijective.

Set

$\overline{\mathcal{X}}_{t_{0}}(T_{0}[\mathcal{M}])=D_{t_{0}}(\mathcal{M})$
(15)

We check carefully in Lemma 4.2 that (S1)b gives $D_{t_{0}}(\mathcal{M})$ a
natural $\Gamma(\mathcal{O}_{T_{0}})$-module structure, and that when (S1)a
holds the additive group $D_{t_{0}}(\mathcal{M})$ acts transitively on the set
$\overline{\mathcal{X}}_{t}(T^{\prime})$.

Condition. (S2) $D_{t_{0}}(\mathcal{M})$ is a coherent
$\Gamma(\mathcal{O}_{T_{0}})$-module.

Conditions (S1) and (S2) have the obvious reformulations in terms of
superalgebras, _i.e._ , replace $\mathcal{M}$ with its associated $R_{0}$
module $M$, $T_{0}\hookrightarrow T$ with the associated surjection $R\to
R_{0}$, etc.

######  Lemma 4.2.

_(S1)b_ gives $D_{t_{0}}(\mathcal{M})$ a natural
$\Gamma(\mathcal{O}_{T_{0}})$-module structure, and when _(S1)a_ holds the
additive group $D_{t_{0}}(\mathcal{M})$ acts transitively on the set
$\overline{\mathcal{X}}_{t}(T^{\prime})$.

###### Proof.

Let $\phi:B\to R_{0}$, $M$ and $t_{0}$ be as in the hypotheses of (S1)b and
set $F_{t_{0}}(-)=\overline{\mathcal{X}}_{t_{0}}^{op}(-)$. Our first goal is
to prove that the natural map

$F_{t_{0}}(B\times_{R_{0}}R_{0}[M])\to F_{t_{0}}(B)\times
F_{t_{0}}(R_{0}[M]).$ (16)

is bijective by (S1)b.

First observe that the map $B\times_{R_{0}}R_{0}[M]\to
B[M]:(b,\phi(b)+rm)\mapsto(b,rm),\ (b\in B,r\in R_{0},m\in M)$ is an
isomorphism of superalgebras.

For each $b\in F_{t_{0}}(B)$, the natural map

$F_{b}(B\times_{R_{0}}R_{0}[M])\to F_{b}(B)\times F_{t_{0}}(R_{0}[M])$ (17)

is a bijection since $F_{b}(B)=b$ and since
$F_{b}(B\times_{R_{0}}R_{0}[M])=F_{b}(B[M])$ is equal to $F_{t_{0}}(R_{0}[M])$
by (S1)b. This proves that (16) is a bijection.

We will now define a $R_{0}$-module structure on
$D_{t_{0}}(M)=F_{t_{0}}(R_{0}[M]\times_{R_{0}}R_{0}[M])$ using the bijection
(16) with $B=R_{0}[M]$.

For the additive structure: For each $t\in D_{t_{0}}(M)$, there is an arrow
$s^{*}(t)\to t$ in $D_{t_{0}}(M)$ over

$s:R_{0}[M]\times_{R_{0}}R_{0}[M]\to
R_{0}[M]:((r,m),(r,m^{\prime}))\mapsto(r,m+m^{\prime}),\ \ r,r^{\prime}\in R,\
m,m\in M.$

In particular, we have a map $+:D_{t_{0}}(M\times M)\to D_{t_{0}}(M):t\to
s^{*}(t)$. Since $D_{t_{0}}(M\times M)=D_{t_{0}}(M)\times D_{t_{0}}(M)$ by
(S1)b, the map $+$ defines an additive structure on $D_{t_{0}}(M)$.
Multiplication by $c\in R_{0}$ is induced by the morphism

$R_{0}[M]\to R_{0}[M]:(a,m)\mapsto(a,cm).$

The additive identity (or zero object) in $D_{t_{0}}(M)$ is just the trivial
extension of $t_{0}$ over $R_{0}[M]$.

The proof that $D_{t_{0}}(M)$ acts transitively on
$\overline{\mathcal{X}}_{t}(T^{\prime})$ can be proved exactly as in [Sch68,
Remark 2.17].

∎

Schlessinger’s theorem [Sch68, Theorem 2.11] asserts that (S1) and (S2) are
enough to ensure the existence of formally versal, formal objects for
$\mathcal{X}$.

######  Theorem 4.3.

Suppose _(S1)_ and _(S2)_ hold for $\overline{\mathcal{X}}_{x}$. Then there
exists a complete, local, Noetherian superalgebra $R$ and a formally versal,
formal object
$\widehat{\xi}\in\widehat{\overline{\mathcal{X}}}_{x}(\operatorname{Spf}R)$.

We check carefully in Appendix A that this result can be established for
superalgebras by the same argument as in the ordinary case, with care to take
account also for odd directions.

##  5 Artin approximation

Following [CdJ02], we will deduce Artin approximation from the Popescu
desingularization theorem. The latter is a very difficult result in
commutative algebra. Fortunately, as observed by [MZ19], the supercommutative
version of Popescu is easily reduced to the original theorem. We begin by
recalling this result:

######  Lemma 5.1 ([MZ19, Lemma 7.9]).

Let $X$ and $Y$ be affine superschemes and let $f:X\to Y$ be a flat morphism
such that $f^{-1}(\mathcal{J}_{Y})$ generates $\mathcal{J}_{X}$. Then the
commutative diagram

${X}$${Y}$${X_{\operatorname{ev}}}$${Y_{\operatorname{ev}}}$$\scriptstyle{f}$$\scriptstyle{f_{\operatorname{ev}}}$

is cartesian, and $f_{\operatorname{ev}}$ is flat.

######  Theorem 5.2 ([MZ19], Lemma 7.30).

Let $(A,\mathfrak{m})$ be a local Noetherian superring and suppose that the
natural morphism $A^{+}\to\widehat{A}^{+}$ is geometrically regular.

Then,

$\widehat{A}=\varinjlim_{\lambda}C_{\lambda}$

where $C_{\lambda}$ are smooth $A$-superalgebras.

###### Proof.

Here $\widehat{A}^{+}$ denotes the completion of $A^{+}$ with respect to the
even component $\mathfrak{m}^{+}$ of the maximal ideal $\mathfrak{m}$. Since
$A^{+}\to\widehat{A}^{+}$ is geometrically regular, we can use the ordinary
Neron-Popescu theorem to conclude that

$\widehat{A}^{+}=\varinjlim_{\lambda}C_{\lambda}$

with $C_{\lambda}$ smooth $A^{+}$ superalgebras. Let $f$ denote the natural
map $A\to\widehat{A}$. $f$ is well-known to be flat. Furthermore, it is easy
to see that $f^{-1}(\mathcal{J}_{\widehat{A}})$ generates $\mathcal{J}_{A}$.
Therefore, we can apply [MZ19, Lemma 7.9] to conclude that the diagram

${A^{+}}$${\widehat{A}^{+}=\varinjlim_{\lambda}C_{\lambda}}$${A}$${\widehat{A}}$$\scriptstyle{f_{0}}$$\scriptstyle{f}$

is cocartesian. Thus,
$\widehat{A}=\varinjlim_{\lambda}(C_{\lambda})\otimes_{A^{+}}A=\varinjlim_{\lambda}(C_{\lambda}\otimes_{A^{+}}A)$.
The result now follows since $C_{\lambda}\otimes_{A^{+}}A$ is smooth over
$A^{+}$ and so $C_{\lambda}\otimes_{A^{+}}A$ is smooth over $A$.

∎

######  Theorem 5.3 (Artin approximation).

Let $S$ be a finite type affine superscheme and let $\mathcal{X}$ be a limit-
preserving superstack. Let $s$ be a closed point in $S$ and let
$\widehat{\xi}\in\mathcal{X}(\operatorname{Spec}\widehat{\mathcal{O}}_{S,s})$.
Then for any integer $n\geq 1$, there exists an étale morphism
$(S^{\prime},s^{\prime})\to(S,s):s^{\prime}\mapsto s$ and an object
$\xi\in\mathcal{X}(S^{\prime})$ such that
$\widehat{\xi}|_{\operatorname{Spec}(\mathcal{O}_{S,s}/\mathfrak{m}_{s}^{n})}$
and
$\xi|_{\operatorname{Spec}(\mathcal{O}_{S^{\prime},s^{\prime}}/\mathfrak{m}_{s^{\prime}}^{n})}$
are isomorphic as objects in $\mathcal{X}$ over
$\operatorname{Spec}(\mathcal{O}_{S,s}/\mathfrak{m}_{s}^{n})\cong\operatorname{Spec}(\mathcal{O}_{S^{\prime},s^{\prime}}/\mathfrak{m}_{s^{\prime}}^{n})$.

###### Proof.

We follow [CdJ02], drawing also from the exposition [Alp15]. Since $S$ if of
finite type, the natural map
$\mathcal{O}_{S,s}^{+}\to\widehat{\mathcal{O}}_{S,s}^{+}$ is geometrically
regular, [Gro65, 7.4.4]. Thus,

$\widehat{\mathcal{O}}_{S,s}=\varinjlim B_{\lambda}$

where $B_{\lambda}$ are smooth $\mathcal{O}_{S,s}$-superalgebras, by Theorem
5.2. Since $\mathcal{X}$ is limit-preserving, there exist some large enough
$\lambda$, and an object
$\xi_{\lambda}\in\mathcal{X}(\operatorname{Spec}B_{\lambda})$ such that
$\widehat{\xi}\to\xi_{\lambda}$ is an arrow over the natural morphism
$\phi:\operatorname{Spec}\widehat{\mathcal{O}}_{S,s}\to\operatorname{Spec}B_{\lambda}$
of superschemes over $\operatorname{Spec}\mathcal{O}_{S,s}$.

Using this fact, we will now prove that for each $n\geq 1$, there exists a
pointed superscheme $(S^{\prime},s^{\prime})$ and an object
$\xi\in\mathcal{X}(S^{\prime})$ with the following properties:

  1. 1.

there is a closed immersion
$i:S^{\prime}\hookrightarrow\operatorname{Spec}B_{\lambda}:s^{\prime}\to s$,

  2. 2.

the composition
$S^{\prime}\hookrightarrow\operatorname{Spec}B_{\lambda}\to\operatorname{Spec}\mathcal{O}_{S,s}$
is étale, and

  3. 3.

$\xi=i^{*}(\xi_{\lambda})|_{\operatorname{Spec}(\mathcal{O}_{S^{\prime},s^{\prime}}/\mathfrak{m}_{s^{\prime}}^{n})}$
and
$\widehat{\xi}|_{\operatorname{Spec}(\widehat{\mathcal{O}}_{S,s}/\mathfrak{m}_{s}^{n})}$
are isomorphic as objects in $\mathcal{X}$ over
$\operatorname{Spec}(\mathcal{O}_{S^{\prime},s^{\prime}}/\mathfrak{m}_{s^{\prime}}^{n})\cong\operatorname{Spec}(\widehat{\mathcal{O}}_{S,s}/\mathfrak{m}_{s}^{n})\cong\operatorname{Spec}(\mathcal{O}_{S,s}/\mathfrak{m}_{s}^{n})$.

The remainder of the proof concerns the construction of $S^{\prime}$.

For (1): Since $B_{\lambda}$ is a smooth $\mathcal{O}_{S,s}$-superalgebra,
$\Omega_{B_{\lambda}/\mathcal{O}_{S,s}}^{1}$ is a locally free
$B_{\lambda}$-module. Let $s\in U\subset\operatorname{Spec}B_{\lambda}$ be an
affine Zariski open neighborhood $s$ on which
$\Omega_{B_{\lambda}/\mathcal{O}_{S,s}}^{1}$ trivializes. Choose a basis
$du_{1},\dots,du_{n},d\mu_{1},\dots,d\mu_{m},\ |du_{i}|=0,|d\mu_{i}|=1$ for
$\Omega_{B_{\lambda}/\mathcal{O}_{S,s}}^{1}|_{U}$. The map

$\mathcal{O}_{S,s}[x_{1},\dots,x_{n},\eta_{1},\dots,\eta_{m}]\to
B_{\lambda}:x_{i}\mapsto u_{i},\eta_{i}\mapsto\mu_{i}$ (18)

is an étale morphism of smooth $\mathcal{O}_{S,s}$-superalgebras.

Let $E=\mathcal{O}_{S,s}[x_{1},\dots,x_{n},\eta_{1},\dots,\eta_{m}]$ and let
$g$ denote the map $\mathcal{O}_{S,s}\hookrightarrow E$ making $E$ and
$\mathcal{O}_{S,s}$-superalgebra. We will now prove that $g$ has a section
$f^{\prime}:E\to\widehat{\mathcal{O}}_{S,s}$ such that $f^{\prime}\circ
g=1_{\mathcal{O}_{S,s}}$.

Consider the extension

$0\to\mathfrak{m}_{s}^{n}\to\mathcal{O}_{S,s}\to\mathcal{O}_{S,s}/\mathfrak{m}_{s}^{n}\to
0$

and note that $\mathcal{O}_{S,s}/\mathfrak{m}_{s}^{n}$ is a local, Artin
superalgebra. We have a map $E\to\mathcal{O}_{S,s}/\mathfrak{m}_{s}^{n}$ given
by the composition $E\to
B_{\lambda}\to\widehat{\mathcal{O}}_{S,s}\to\mathcal{O}_{S,s}/\mathfrak{m}_{s}^{n}$.
Since $E$ is an étale $B_{\lambda}$-superalgebra, the map
$E\to\mathcal{O}_{S,s}/\mathfrak{m}_{s}^{n}$ has a unique lift
$f^{\prime}:E\to\mathcal{O}_{S,s}$. In particular, $f^{\prime}\circ
g=1_{\mathcal{O}_{S,s}}$ is a section of $g$.

Now define $S^{\prime}$ to be the fiber product of the diagram

${S^{\prime}}$${\operatorname{Spec}\mathcal{O}_{S,s}}$${\operatorname{Spec}B_{\lambda}}$${\mathbb{A}_{\mathcal{O}_{S,s}}^{n|m}=\operatorname{Spec}E}$$\scriptstyle{i}$$\scriptstyle{f^{\prime}}$

and set $\xi=i^{*}\xi_{\lambda}$.

For (2): The map $S^{\prime}\to\operatorname{Spec}\mathcal{O}_{S,s}$ is étale
because the map
$\operatorname{Spec}B_{\lambda}\to\mathbb{A}_{\mathcal{O}_{S,s}}^{m|n}$ is
étale.

For (3): This is obvious since $\xi=i^{*}(\xi_{\lambda})$ and
$\xi_{\lambda}\in\mathcal{X}(\operatorname{Spec}B_{\lambda})$ was such that
$\xi_{\lambda}|_{\operatorname{Spec}(\mathcal{O}_{S,s}/\mathfrak{m}_{s}^{n})}\cong\widehat{\xi}|_{\operatorname{Spec}(\widehat{\mathcal{O}}_{S,s}/\mathfrak{m}_{s}^{n})}$
as objects in $\mathcal{X}$ over
$\operatorname{Spec}(\mathcal{O}_{S,s}/\mathfrak{m}_{s}^{n})\cong\operatorname{Spec}(\widehat{\mathcal{O}}_{S,s}/\mathfrak{m}_{s}^{n})$.
∎

##  6 Artin algebraization

In this section, we will prove super Artin algebraization following the ideas
of [CdJ02], except that since we have access already to the bosonic result, we
may replace the difficult arguments involving associated gradeds by induction
on the odd nilpotence degree.

In [CdJ02, Theorem 3.2], Conrad and de Jong formulate a more general version
of Artin approximation which requires only that the initial data is an object
over a complete, local, Noetherian ring. They also proved more; indeed the
most subtle and difficult part of their theorem establishes an isomorphism
$\operatorname{gr}_{\mathfrak{m}}(R)\cong\operatorname{gr}_{\mathfrak{m}_{u}}(A)$
of graded algebras, which was essential for their proof of Artin
algebraization. We will not require such a statement to prove super Artin
algebraization (given the bosonic Artin approximation result).

######  Theorem 6.1.

Let $\mathcal{X}$ be a limit preserving superstack. Let $(R,\mathfrak{m})$ be
a complete, local, Noetherian superalgebra, and let
$\widehat{\xi}\in\mathcal{X}(\operatorname{Spec}R)$. Then for every integer
$n\geq 1$, there exists

  * _(1)_

a finite type affine superscheme $\operatorname{Spec}A$ and a closed point
$u\in\operatorname{Spec}A$,

  * _(2)_

an object $\xi_{A}$ of $\mathcal{X}$ over $\operatorname{Spec}A$,

  * _(3)_

an isomorphism $\alpha_{n}:R/\mathfrak{m}^{n}\cong A/\mathfrak{m}_{u}^{n}$,
and

  * _(4)_

an isomorphism
$\widehat{\xi}|_{\operatorname{Spec}(R/\mathfrak{m}^{n})}\cong\xi_{A}|_{\operatorname{Spec}(A/\mathfrak{m}_{u}^{n})}$
of objects in $\mathcal{X}$ over
$\operatorname{Spec}(R/\mathfrak{m}^{n})\cong\operatorname{Spec}(A/\mathfrak{m}_{u}^{n})$.

###### Proof.

We follow [CdJ02], drawing also from the exposition [Alp15]. We recall that
any superalgebra is the direct limit of its finitely generated sub-
superalgebras. In particular, $R=\varinjlim B_{\lambda}$ where $B_{\lambda}$
are finitely generated (over $k$) sub-superalgebras of $R$.

Since $\mathcal{X}$ is limit preserving there exists large enough $\lambda$,
and an object $\xi_{\lambda}\in\mathcal{X}(\operatorname{Spec}B_{\lambda})$
such that $\widehat{\xi}\to\xi_{\lambda}$ is an arrow in $\mathcal{X}$ over
the natural map $\operatorname{Spec}R\to\operatorname{Spec}B_{\lambda}$. Let
$B=B_{\lambda}$, $V=\operatorname{Spec}B$, let $v\in\operatorname{Spec}B$ be
the image of the unique closed point in $\operatorname{Spec}R$. With this
notation, let $f$ denote the natural map $\mathcal{O}_{V,v}\to R$, and let
$\widehat{f}$ denote the induced map $\widehat{\mathcal{O}}_{V,v}\to R$ on the
completions. We may add enough generators to $B_{\lambda}$ so that the induced
map $\widehat{f}_{*}:t_{\mathcal{O}_{V,v}}^{*}\to t_{R}^{*}$ on Zariski
cotangent spaces is surjective. The map $\widehat{f}$ is then surjective by
Lemma A.1

Our next goal is to construct $(\operatorname{Spec}A,u)$ and $\xi_{A}$. Choose
a resolution

$(\widehat{\alpha},\widehat{\beta}):\mathcal{O}_{V,v}^{\oplus
r|r^{\prime}}\overset{\widehat{\alpha}}{\longrightarrow}\mathcal{O}_{V,v}^{\oplus
s|s^{\prime}}\overset{\widehat{\beta}}{\longrightarrow}\mathcal{O}_{V,v}\to
R\to 0$ (19)

of $R$ and consider the (limit-preserving) functor

$\displaystyle F$
$\displaystyle:(\operatorname{sAffSch}/V)\to\operatorname{Sets}$
$\displaystyle(T\to V)\mapsto\\{\text{complexes}\
(\alpha,\beta):\mathcal{O}_{T}^{\oplus
r|r^{\prime}}\overset{\alpha}{\longrightarrow}\mathcal{O}_{T}^{\oplus
s|s^{\prime}}\overset{\beta}{\longrightarrow}\mathcal{O}_{T}\\}$

Observe that (19) is an element of $F(\operatorname{Spec}\mathcal{O}_{V,v})$.
Since $F$ is limit preserving, we can apply Artin approximation, Theorem 5.
This gives for each $n\geq 1$, a superscheme
$(V^{\prime}=\operatorname{Spec}B^{\prime},v^{\prime})$, an étale morphism
$(V^{\prime},v^{\prime})\to(V,v):v^{\prime}\to v$, and an object

$(\alpha^{\prime},\beta^{\prime}):(B^{\prime})^{\oplus
r|r^{\prime}}\overset{\alpha^{\prime}}{\longrightarrow}(B^{\prime})^{\oplus
s|s^{\prime}}\overset{\beta^{\prime}}{\longrightarrow}B^{\prime}$

in $\xi^{\prime}\in F(V^{\prime})$ with the following property: Let
$(\alpha_{n}^{\prime},\beta_{n}^{\prime})$ denote the resolution modulo
$\mathfrak{m}_{B^{\prime}}^{n}$ and let
$(\widehat{\alpha}_{n},\widehat{\beta}_{n})$ denote the resolution in (19)
modulo $\mathfrak{m}_{R}^{n}$ . Then Artin approximation says that
$(\widehat{\alpha}_{n},\widehat{\beta}_{n})\cong(\alpha_{n}^{\prime},\beta_{n}^{\prime})$
as objects in $F$. This, moreover, implies that

$R/\mathfrak{m}_{R}^{n}=\operatorname{coker}(\widehat{\beta}_{n})\cong\operatorname{coker}(\beta_{n}^{\prime}).$

Now let
$U=\mathbb{V}(\operatorname{im}(\beta^{\prime}))\subset\operatorname{Spec}B^{\prime}$
be the closed sub-superscheme cut out by $\operatorname{im}(\beta^{\prime})$
and set $u=v^{\prime}\in U$. Let $A$ be the superalgebra for which
$U=\operatorname{Spec}A$. This gives a resolution

$(B^{\prime})^{\oplus
r|r^{\prime}}\overset{\alpha^{\prime}}{\longrightarrow}(B^{\prime})^{\oplus
s|s^{\prime}}\overset{\beta^{\prime}}{\longrightarrow}B^{\prime}\to A\to 0.$
(20)

The composition

$\xi_{A}:U\hookrightarrow V^{\prime}\to
V\overset{\xi_{V}}{\longrightarrow}\mathcal{X}$

then defines an object $\xi_{A}\in\mathcal{X}(U)$. Since the resolution in
(20) is equivalent to the resolution in (19), we have that
$R/\mathfrak{m}_{R}^{n}=\operatorname{coker}(\widehat{\beta}_{n})\cong\operatorname{coker}(\beta_{n}^{\prime})=A/\mathfrak{m}_{u}^{n}$
and
$\xi_{A}|_{\operatorname{Spec}(A/\mathfrak{m}_{u}^{n}}\cong\widehat{\xi}|_{\operatorname{Spec}(R/\mathfrak{m}_{R}^{n})}$.

∎

######  Theorem 6.2 (Artin algebraization).

Let $\mathcal{X}$ be a limit-preserving superstack and let
$(R,\mathfrak{m}_{R})$ be a local, complete, Noetherian superalgebra. Let
$\widehat{\xi}\in\mathcal{X}(\operatorname{Spec}R)$ be formally versal. Then
there exists

  1. 1.

an affine finite type superscheme $\operatorname{Spec}A$ and a closed point
$u\in\operatorname{Spec}A$,

  2. 2.

an object $\xi_{A}\in\mathcal{X}(\operatorname{Spec}A)$,

  3. 3.

an isomorphism
$\alpha:R\overset{\sim}{\longrightarrow}\widehat{A}_{\mathfrak{m}_{u}}$ of
superalgebras, and

  4. 4.

a compatible family of isomorphisms
$\widehat{\xi}|_{\operatorname{Spec}(R/\mathfrak{m}_{R}^{n})}\cong\xi_{A}|_{\operatorname{Spec}(A_{\mathfrak{m}_{u}}/\mathfrak{m}_{u}^{n})}$
in $\mathcal{X}$ over $R/\mathfrak{m}_{R}^{n}\cong
A_{\mathfrak{m}_{u}}/\mathfrak{m}_{u}^{n}$ for all $n\geq 1$.

###### Proof.

Let $r$ be the smallest integer for which $\mathcal{J}_{R}^{r+1}=0$. We can
apply Conrad-de Jong approximation at $N=r$ to obtain a finite type pointed
superscheme $(\operatorname{Spec}A,u)$, an object
$\xi_{A}\in\mathcal{X}(\operatorname{Spec}A)$, and an isomorphism

$\iota_{i}:\widehat{\xi}|_{\operatorname{Spec}(R/\mathfrak{m}^{i})}\overset{\sim}{\longrightarrow}\xi_{A}|_{\operatorname{Spec}(A/\mathfrak{m}_{u}^{i})}$

over

$\alpha_{i}:\operatorname{Spec}A/\mathfrak{m}_{u}^{i}\overset{\sim}{\longrightarrow}\operatorname{Spec}R/\mathfrak{m}_{R}^{i}$

for all $i=1,\dots,r$.

Since $\widehat{\xi}$ is formally versal, we have for each $n\geq 1$ an arrow
completing the diagram,

${\operatorname{Spec}A/\mathfrak{m}_{u}^{n}}$${\operatorname{Spec}R}$${\operatorname{Spec}A/\mathfrak{m}_{u}^{n+1}}$${\mathcal{X}}$$\scriptstyle{\alpha_{n}}$$\scriptstyle{\widehat{\xi}}$$\scriptstyle{\xi_{A}}$$\scriptstyle{\alpha_{n+1}}$

Taking the projective limit we obtain a morphism
$\widehat{\alpha}:R\to\widehat{A}_{\mathfrak{m}_{u}}$.

We claim that the morphism $\widehat{\alpha}$ is surjective. Indeed, the
morphism $\widehat{\alpha}$ descends to an isomorphism
$R/\mathfrak{m}^{r}\cong\widehat{A}_{\mathfrak{m}_{u}}/\mathfrak{m}_{u}^{r}$,
hence induces an isomorphism
$\mathfrak{m}_{R}/\mathfrak{m}_{R}^{2}\to\mathfrak{m}_{u}/\mathfrak{m}_{u}^{2}$
of Zariski cotangent spaces. In particular, the map $\widehat{\alpha}$ induces
a surjection between Zariski cotangent spaces. The claim that
$\widehat{\alpha}$ is surjective now follows from Lemma A.1.

We now claim that $\widehat{\alpha}$ is injective. Let
$M=\operatorname{ker}(\widehat{\alpha})$. By applying the classical Artin
algebraization theorem to the bosonic reductions of the objects appearing in
the statement of the Theorem, we get an isomorphism

$R_{\operatorname{bos}}\cong(\widehat{A}_{\mathfrak{m}_{u}})_{\operatorname{bos}}.$

Thus $M/\mathcal{J}_{R}=(0)$ and so $M\subset\mathcal{J}_{R}$. Let $m\in M$
and let $r(m)$ largest integer for which $m=m\ \ \operatorname{mod}\ \
\mathfrak{m}_{R}^{r(m)+1}$. We know such an integer $r(m)$ exists because
$m\in\mathcal{J}_{R}$. Moreover, we must have that $r(m)\leq r$. Thus,
$\alpha_{r(m)}(m)=0$ and so $m=0$ since the $\alpha_{i}$ are isomorphisms.

∎

##  7 Flenner’s criterion for openness of formal versality

###  7.1 Extension functor

Here we introduce Flenner’s $\mathrm{Ex}$ functors, following [Fle81] (and
somewhat expanding upon the exposition). Throughout this section, let
$T=\operatorname{Spec}R$ be a finite type affine superscheme.

######  Definition 7.1.

For every $v\in\mathcal{X}(T)$, and for every coherent
$\mathcal{O}_{T}$-module $\mathcal{M}$, we define
$\boldsymbol{\operatorname{Ex}}(v,\mathcal{M})$ to be the groupoid of pairs
$(i:T\hookrightarrow T^{\prime},v\to v^{\prime})$ with $i:T\hookrightarrow
T^{\prime}$ an extension of $T$ by $\mathcal{M}$ and $v\to v^{\prime}$ an
arrow in $\mathcal{X}$ over $i$.

Let

$\operatorname{Ex}(v,-):\operatorname{Coh}(T)\to\operatorname{Set}$ (21)

be the functor sending a coherent $\mathcal{O}_{T}$-module $\mathcal{M}$ to
the set of isomorphism classes in
$\boldsymbol{\operatorname{Ex}}(v,\mathcal{M})$.

In general, $\operatorname{Ex}(v,\mathcal{M})$ will _not_ have a
$\Gamma(\mathcal{O}_{T})$-module structure. However, there does exist in
$\operatorname{Ex}(v,\mathcal{M})$ a natural “zero object”, the pair
$(T[\mathcal{M}],v[\mathcal{M}])$, where
$v[\mathcal{M}]:=v\times_{T}T[\mathcal{M}]$ is the trivial
$\mathcal{M}$-extension of $v$ over $T[\mathcal{M}]$. An object
$(i:T\hookrightarrow T^{\prime},v\to v^{\prime})$ in
$\operatorname{Ex}(v,\mathcal{M})$ is trivial if there exists a section
$s:T^{\prime}\to T$ such that $is=1_{T}$ and an arrow $v^{\prime}\to v$ over
$s$.

We have the following natural morphisms of pointed sets (the point is the zero
object in each term):

${{\operatorname{Der}(\mathcal{O}_{T},\mathcal{M})}}$${D_{v}(\mathcal{M})}$${{\operatorname{Ex}(v,\mathcal{M})}}$${{\operatorname{Ex}(T,\mathcal{M})}.}$$\scriptstyle{c}$$\scriptstyle{i}$$\scriptstyle{\pi}$
(22)

We remind the reader that
$D_{v}(\mathcal{M})=\overline{\mathcal{X}}_{v}(T[\mathcal{M}])$ and that
$\operatorname{Ex}(T,\mathcal{M})$ is the set of isomorphism classes of
extensions of $T$ by $\mathcal{M}$.

  * •

The map $c$ sends a derivation $d$ in
$\operatorname{Der}(\mathcal{O}_{T},\mathcal{M})$ to the pullback of
$v[\mathcal{M}]$ under the automorphism $1+d:T[\mathcal{M}]\to
T[\mathcal{M}]$, that is

$d\mapsto c(d)=(1+d)^{*}(v[\mathcal{M}]).$

  * •

The map $i$ takes $v\to v^{\prime}$ in $D_{v}(\mathcal{M})$ to the pair
$(i:T\hookrightarrow T[\mathcal{M}],v\to v^{\prime})$ in
$\operatorname{Ex}(v,\mathcal{M})$. Note that
$\operatorname{Der}(\Gamma(\mathcal{O}_{T}),\mathcal{M})$ acts trivially on
$\operatorname{Ex}(v,\mathcal{M})$ but non-trivially on $D_{v}(\mathcal{M})$.

  * •

The map $\pi$ takes the pair $(i:T\hookrightarrow T^{\prime},v\to
v^{\prime})\in\operatorname{Ex}(v,\mathcal{M})$ to $i:T\hookrightarrow
T^{\prime}$ in $\operatorname{Ex}(T,\mathcal{M})$.

######  Lemma 7.2 (Compare to [Fle81, Lemma 2.2]).

Using the notation above and assuming that _(S1)_ and _(S2)_ hold for
$\mathcal{X}$, we have:

  1. _1_.

$\pi(\operatorname{Ex}(v,\mathcal{M}))\subset\operatorname{Ex}(T,\mathcal{M})$
is an $\Gamma(\mathcal{O}_{T})$-submodule

  2. _2_.

$\pi\circ i=0,i\circ c=0$, and

  3. _3_.

$\operatorname{Coker}(c)\to\pi^{-1}(0)$ is bijective.

###### Proof.

For (1): The fact that $\pi(\operatorname{Ex}(v,\mathcal{M}))$ is closed under
addition can be seen from the diagram

${\operatorname{Ex}(v,\mathcal{M})\times\operatorname{Ex}(v,\mathcal{M})}$${\operatorname{Ex}(v,\mathcal{M}\times\mathcal{M})}$${\operatorname{Ex}(v,\mathcal{M})}$${\operatorname{Ex}(T,\mathcal{M})\times\operatorname{Ex}(T,\mathcal{M})}$${\operatorname{Ex}(T,\mathcal{M}\times\mathcal{M})}$${\operatorname{Ex}(T,\mathcal{M})}$$\scriptstyle{p}$$\scriptstyle{\operatorname{add}}$$\scriptstyle{\sim}$$\scriptstyle{\operatorname{add}}$

where $p$ is surjective by (S1)a.

For (2): Let $v^{\prime}\in D_{v}(\mathcal{M})$, then $i(v^{\prime})$ is the
pair $(i:T\hookrightarrow T[\mathcal{M}],v\to v^{\prime})$ in
$\operatorname{Ex}(v,\mathcal{M})$ and $\pi(i(v^{\prime}))$ is the object
$i:T\hookrightarrow T[\mathcal{M}]$ in $\operatorname{Ex}(T,\mathcal{M})$.
Thus, $\pi\circ i=0$. Let
$d\in\operatorname{Der}(\mathcal{O}_{T},\mathcal{M})$, then
$c(d)=(1+d)^{*}v[\mathcal{M}]$ over $(1+d)(T[\mathcal{M}])$. The image
$i(c(d))$ is the pair

$(i:T\hookrightarrow(1+d)\left(T[\mathcal{M}]\right),v\to(1+d)^{*}v[\mathcal{M}])$

in $\operatorname{Ex}(v,\mathcal{M})$. In $\operatorname{Ex}(v,\mathcal{M})$,
we have that $(1+d)\left(T[\mathcal{M}]\right)\cong T[\mathcal{M}]$ and
$(1+d)^{*}v[\mathcal{M}]\cong v[\mathcal{M}]$. Therefore, $i\circ c=0$.

For (3): Define $D_{v}(\mathcal{M})\to\pi^{-1}(0)$ by $(v\to
v^{\prime})\mapsto(i:T\hookrightarrow T[\mathcal{M}],v\to v^{\prime})$. This
map is clearly surjective. The “kernel” of this map are arrows
$v\to(1+d)^{*}v[\mathcal{M}]$ for all
$d\in\operatorname{Der}(\mathcal{O}_{T},\mathcal{M})$. Therefore, we get a
bijection $\operatorname{coker}(c)\to\pi^{-1}(0)$. ∎

Coherence properties of $\operatorname{Der}(\mathcal{O}_{T},\mathcal{M})$ and
$\operatorname{Ex}(T,\mathcal{M})$ are ensured by the following relationship
with the cotangent complex $L_{T}^{\bullet}$ (see e.g. [FS92, Definition 3.6].
For any $\mathcal{O}_{T}$-module $\mathcal{M}$, we have the _cotangent
functors_

$T_{T}^{i}(\mathcal{M}):=\operatorname{Ext}_{T}^{i}(L_{T}^{\bullet},\mathcal{M}).$

######  Theorem 7.3 ([FS92, Theorem 4.6]).

There are canonical isomorphisms

$\operatorname{Ex}(T,\mathcal{M})\to T_{T}^{1}(\mathcal{M})_{ev},\ \ \
\operatorname{Der}(\mathcal{O}_{T},\mathcal{M})\to
T_{T}^{0}(\mathcal{M})_{ev}$

where $T_{T}^{i}(\mathcal{M})_{ev}$ denotes the set of even elements in
$T_{T}^{i}(\mathcal{M})$ for $i=0,1$

We consider the following sheaves:

  1. 1.

Let $\mathcal{E}x(v,\mathcal{M})$ be the sheaf of pointed sets over $T$
associated to the presheaf $T\supset
U\mapsto\operatorname{Ex}(v|_{U},\mathcal{M}|_{U})$.

  2. 2.

Let $\mathcal{D}_{v}(\mathcal{M})$ be the sheaf over $T$ associated to the
module $D_{v}(\mathcal{M})$.

  3. 3.

Let $\mathcal{D}er(\mathcal{O}_{T},\mathcal{M})$ be the sheaf of over $T$
associated to the module $\operatorname{Der}(\mathcal{O}_{T},\mathcal{M})$.

$\mathcal{E}x(v,\mathcal{M})_{t}$, etc. will denote the stalk at a closed
point $t\in T$. Note that Lemma 7.2 holds for the sheaves
$\mathcal{D}er(\mathcal{O}_{T},\mathcal{M})$, $\mathcal{E}x(v,\mathcal{M})$
and $\mathcal{D}_{v}(\mathcal{M})$.

The main result we need about the Ex functors is Flenner’s criterion for
formal versality:

######  Theorem 7.4 (Compare [Fle81, Theorem 3.2]).

Let $v$ be an object in $\mathcal{X}$ over an affine superscheme $T$. Assume
that _(S1)_ and _(S2)_ are satisfied. Then the following statements are
equivalent.

  1. _(1)_

$v$ is formally versal,

  2. _(2)_

$\mathcal{E}x(v,k(t))_{t}=\mathcal{E}x(v,\Pi k(t))_{t}=0$.

  3. _(3)_

$\mathcal{E}x(v,\mathcal{M})_{t}=0$ for every coherent
$\mathcal{O}_{T_{\operatorname{red}}}$ module $\mathcal{M}$.

We prove this result in Appendix B. The proof follows Flenner’s original
argument except at the (important) lemma B.6, where we are able to reduce to
the bosonic case rather than recapitulating the proof.

###  7.2 Criteria for openness of formal versality

We turn to the proof of openness of formal versality. Given a topological
space $X$, we say that a subset $E\subseteq X$ is is _ind-constructible_ in
$X$ if, for all $x\in X$, there exists an open neighborhood $U\ni x$, such
that $E\cap U$ is a union of constructible sets. A constructible set is a
finite disjoint union of locally closed sets. We will need the following
constructibility criterion (see e.g., [Gro65, Proposition 1.9.10] for the
ordinary version).

######  Proposition 7.5.

Let $X$ be a locally Noetherian superscheme and let $E$ be a subset of $X$.
For $E$ to be ind-constructible it is necessary and sufficient that for all
$x\in X$, the intersection $E\cap\overline{\\{x\\}}$ is a neighborhood of $x$
contained in $\overline{\\{x\\}}$.

######  Lemma 7.6 (Compare to [Fle81, Lemma 4.1]).

For every $T\in\operatorname{sAffSch}$, and $v\in\mathcal{X}(T)$, suppose the
following conditions are satisfied

  1. _1._

Let $T^{\prime}$ be a closed subspace of $T$. Suppose also that $T^{\prime}$
is integral. Then there exists a dense, open Zariski subset
$V^{\prime}\subseteq T^{\prime}$ such that

$\operatorname{Supp}(\mathcal{E}x(v,\mathcal{O}_{T^{\prime}}))\cap
V^{\prime}=\\{t\in V^{\prime}:\mathcal{E}x(v,k(t))_{t}\neq 0\\}$
$\operatorname{Supp}(\mathcal{E}x(v,\Pi\mathcal{O}_{T^{\prime}}))\cap
V^{\prime}=\\{t\in V^{\prime}:\mathcal{E}x(v,\Pi k(t))_{t}\neq 0\\}$

In other words, $V^{\prime}$ as the special property that
$\mathcal{E}x(v,\mathcal{M})_{t}=0$ (with $t\in V^{\prime}$) only when
$\mathcal{M}$ is $\mathcal{O}_{T^{\prime}}$ or when $\mathcal{M}$ is
$\Pi\mathcal{O}_{T^{\prime}}$.

  2. _2._

For every coherent $\mathcal{O}_{T_{\operatorname{red}}}$-module
$\mathcal{M}$, the support of $\mathcal{E}x(v,\mathcal{M})$ is Zariski closed
in $T$.

Then the set of points $t\in T$ at which $v$ is formally versal over
$\mathcal{X}$ is a Zariski open subset of $T$.

###### Proof.

Let

$V=\\{t\in T\ |\ \mathcal{E}x(v,k(t))_{t}=\mathcal{E}x(v,\Pi k)_{t}=0\\}.$

From Theorem 7.4, we have that $V$ is the set of closed points in $T$ at which
$v$ is formally versal and that

$V=\bigcap_{\mathcal{M}\in\operatorname{Coh}(T)}\\{t\in T\ |\
\mathcal{E}x(v,\mathcal{M})_{t}=0\\}.$

The goal is to prove that $V$ is open. We will do this by applying the
following theorem from EGA, [Gro65, Theorem 1.10.1]: If a set is ind-
constructible and stable under generalization, then it is open.

We first prove that $V$ is stable under generalization. Let $t\in V$, and
suppose that $t\in\overline{\\{t^{\prime}\\}}$ for some $t^{\prime}\in T$. We
claim that $t^{\prime}\in V$. Fix some $\mathcal{M}$. Since
$\mathcal{E}x(v,\mathcal{M})$ is closed (by assumption (2)), the set

$V_{M}:=T-\operatorname{Supp}(\mathcal{E}x(v,\mathcal{M}))=\\{t\in T\ |\
\mathcal{E}x(v,\mathcal{M})_{t}=0\\}$

is open. Since $V_{M}$ is open, it is stable under generalization, and so
$t^{\prime}\in V_{M}$. Repeating this argument for every $\mathcal{M}$, we
find that $t^{\prime}\in V$.

We now prove that $V$ is ind-constructible using Proposition 7.5. Let
$T^{\prime}$ be an integral closed subscheme of $T$ which meets $V$. Let

$\displaystyle V^{\prime\prime}$
$\displaystyle=T^{\prime}-\left(\operatorname{Supp}(\mathcal{E}x(v,\mathcal{O}_{T^{\prime}}))\cap\operatorname{Supp}(\mathcal{E}x(v,\Pi\mathcal{O}_{T^{\prime}}))\right)$
$\displaystyle=\\{t\in T^{\prime}\ |\
\mathcal{E}x(v,\mathcal{O}_{T^{\prime}})_{t}=\mathcal{E}x(v,\Pi\mathcal{O}_{T^{\prime}})_{t}=0\\}.$

$V^{\prime\prime}$ is open by condition (2). $V^{\prime\prime}\neq\emptyset$
since $V\cap T^{\prime}\neq\emptyset$ and so by Proposition 7.4 there exists
at least one $t\in T^{\prime}$ such that
$\operatorname{Ex}(v,\mathcal{M})_{t}=0$ for all $\mathcal{M}$. Set

$U^{\prime}=V^{\prime\prime}\cap V^{\prime}.$

It follows from the definition of $V^{\prime}$, that the set $U^{\prime}$ is
equal to the subset $\\{t\in V^{\prime}\ |\
\mathcal{E}x(v,k(t))=\mathcal{E}x(v,\Pi k(t))=0\\}$ of $V\cap T$. Since
$V^{\prime}$ is Zariski-dense in $T^{\prime}$, so is $U^{\prime}$, and, thus,
$V$ is ind-constructible.

∎

For $v$ an object in $\mathcal{X}$ over $T$, and $\mathcal{M}$ a coherent
$\mathcal{O}_{T_{\operatorname{red}}}$-module, define
$\mathcal{O}b_{v}(\mathcal{M})$ to be the sheaf over $T$ associated to the
presheaf sending $U\subseteq T$ to
$\operatorname{Ex}(U,\mathcal{M}|_{U})/\pi(\operatorname{Ex}(v|_{U},\mathcal{M}|_{U})$.
$\mathcal{O}b_{v}(-)$ is clearly functorial in $\mathcal{M}$.

We now introduce the following conditions. 666The labels (A3) and (A4) are
chosen to match Theorem 1.9.

Condition (A3). Coherence of $\mathcal{O}b$: For all $v,T,\mathcal{M}$ as
above, $\mathcal{O}b_{v}(\mathcal{M})$ is a coherent $\mathcal{O}_{T}$-module,
and

Condition (A4). Constructibility: Let $v,T$ be as above and suppose that
$T^{\prime}\subseteq T$ is an irreducible reduced subspace of $T$. Then there
is a Zariski open dense subset $V^{\prime}\subseteq T^{\prime}$ such that for
each $t\in V^{\prime}$, the canonical morphisms

  * (a)

$\mathcal{D}_{v}(\mathcal{O}_{T^{\prime}})\otimes
k(t)\to\mathcal{D}_{v}(\mathcal{O}_{T^{\prime}}\otimes k(t))_{t}$ and
$\mathcal{D}_{v}(\Pi\mathcal{O}_{T^{\prime}})\otimes
k(t)\to\mathcal{D}_{v}(\Pi\mathcal{O}_{T^{\prime}}\otimes k(t))_{t}$ are
bijective, and

  * (b)

$\mathcal{O}b_{v}(\mathcal{O}_{T^{\prime}})\otimes
k(t)\to\mathcal{O}b_{v}(\mathcal{O}_{T^{\prime}}\otimes k(t))_{t}$ and
$\mathcal{O}b_{v}(\Pi\mathcal{O}_{T^{\prime}})\otimes
k(t)\to\mathcal{O}b_{v}(\Pi\mathcal{O}_{T^{\prime}}\otimes k(t))_{t}$ are
injective.

######  Theorem 7.7 (Compare [Fle81, Theorem 4.3]).

Let $\mathcal{X}$ be a superstack satisfying conditions _(S1), (S2), (A3)_ and
_(A4)_. Then for every superscheme $T$ and every $v\in\mathcal{X}(T)$, the
subset of points $t$ in $T$ at which $v$ is formally versal form a Zariski-
open subset of $T$.

###### Proof.

The goal is to prove that the two conditions of Lemma 7.6 are satisfied.

Let $T^{\prime}$ and $V^{\prime}$ be as in condition (A2). We first prove that
the subset $V^{\prime}\subset T^{\prime}$ is equal to the subset $V^{\prime}$
defined in Lemma 7.6. We begin by proving that $V^{\prime}$ has the following
properties

  * (i).

$\mathcal{O}b_{v}(\mathcal{O}_{T^{\prime}})|_{V^{\prime}}$ and
$\mathcal{O}b_{v}(\Pi\mathcal{O}_{T^{\prime}})|_{V^{\prime}}$ are locally
free, and

  * (ii).

the natural maps

$\mathcal{D}er(\mathcal{O}_{T},\mathcal{O}_{T^{\prime}})\otimes
k(t)\to\mathcal{D}er(\mathcal{O}_{T},k(t))_{t},$ (23)

and

$\mathcal{E}x(T,\mathcal{O}_{T^{\prime}})\otimes
k(t)\to\mathcal{E}x(T,k(t))_{t}$ (24)

are bijective for all $t\in V^{\prime}$, and analogously for
$\mathcal{D}er(\mathcal{O}_{T},\Pi\mathcal{O}_{T^{\prime}})$ and
$\mathcal{E}x(T,\Pi\mathcal{O}_{T^{\prime}})$.

Henceforth, all statements we make in terms of $\mathcal{O}_{T^{\prime}}$ hold
for $\Pi\mathcal{O}_{T^{\prime}}$ using the exact same proof.

Since $\mathcal{O}b_{v}(\mathcal{O}_{T^{\prime}})$ is a coherent
$\mathcal{O}_{T^{\prime}}$-module by (A1), we can pass to a dense open subset
$W\subset V^{\prime}$, on which
$\mathcal{O}b_{v}(\mathcal{O}_{T^{\prime}})|_{W}$ is locally free. Clearly,
condition (A2) holds for $V^{\prime}\cap W$ and so we can just redefine
$V^{\prime}=V^{\prime}\cap W$. This proves the first claim

For the second claim, identify $\mathcal{D}er(\mathcal{O}_{T},-)$ and
$\mathcal{E}x(\mathcal{O}_{T},-)$ with
$\mathcal{E}xt_{\mathcal{O}_{T}}^{0}(L^{\bullet},-)_{ev}$ and
$\mathcal{E}xt_{\mathcal{O}_{T}}^{1}(L^{\bullet},-)_{ev}$, respectively. Under
these identifications, the maps (23) and (24) are well-known to be bijections.
This proves the second claim.

We now consider the diagram

${{\mathcal{D}er(\mathcal{O}_{T},\mathcal{O}_{T^{\prime}})}}$${\mathcal{D}_{v}(\mathcal{O}_{T^{\prime}})}$${{\mathcal{E}x(v,\mathcal{O}_{T^{\prime}})}}$${{\mathcal{E}x(T,\mathcal{O}_{T^{\prime}})}}$${\mathcal{O}b_{v}(\mathcal{O}_{T^{\prime}})}$${{\mathcal{D}er(\mathcal{O}_{T},k(t))}}$${\mathcal{D}_{v}(k(t))}$${{\mathcal{E}x(v,k(t))}}$${{\mathcal{E}x(T,k(t))}}$${\mathcal{O}b_{v}(k(t))}$$\scriptstyle{c}$$\scriptstyle{\pi}$$\scriptstyle{c_{t}}$$\scriptstyle{\pi_{t}}$
(25)

where all terms are assumed to be restricted to $V^{\prime}$. We will now
prove that the canonical morphisms

$\displaystyle\operatorname{coker}(c)\otimes k(t)$
$\displaystyle\to\operatorname{coker}(c_{t})_{t}$
$\displaystyle\operatorname{Im}(\pi)\otimes k(t)$
$\displaystyle\to\operatorname{Im}(\pi_{t})_{t}$

are bijections for all $t\in V^{\prime}$.

All terms in the top row of diagram (25) (other than
$\mathcal{E}x(v,\mathcal{O}_{T^{\prime}})$) are coherent modules over the
Noetherian domain $V^{\prime}$ and so we can apply generic flatness to all the
objects in the top row, as well as to all their images and kernels and get an
open subset $W\subset V^{\prime}$ on which the functor
$-\otimes_{\mathcal{O}_{V^{\prime}}}k(t)$ is exact. Redefine
$V^{\prime}=V^{\prime}\cap W$.

For the remainder of the proof, we set
$A=\mathcal{D}er(\mathcal{O}_{T},\mathcal{O}_{T^{\prime}})$,
$B=\mathcal{D}_{v}(\mathcal{O}_{T^{\prime}})$,
$C=\mathcal{E}x(v,\mathcal{O}_{T^{\prime}})$,
$D=\mathcal{E}x(T,\mathcal{O}_{T^{\prime}})$ and
$E=\mathcal{O}b_{v}(\mathcal{O}_{T^{\prime}})$. We also set
$A_{k}=\operatorname{Der}(\mathcal{O}_{T},k(t))$,
$B_{k}=\mathcal{D}_{v}(k(t))$, etc..

We begin by proving that $\operatorname{coker}(c)\otimes
k(t)\to\operatorname{coker}(c_{t})_{t}$ is bijective for all $t\in
V^{\prime}$. Consider the short exact sequence

$0\to A\overset{c}{\to}B\to B/\operatorname{im}(c)\to 0.$ (26)

Since $A,B$ are flat, we can tensor (26) with $k(t)$, and get a short exact
sequence

$0\to A\otimes k(t)\overset{c\otimes 1}{\longrightarrow}B\otimes
k(t)\to\left(B/\operatorname{im}(c)\right)\otimes k(t)\to 0.$ (27)

From which it follows that $\operatorname{coker}(c)\otimes
k(t)=\operatorname{coker}(c\otimes 1)$. We have that (1) $B\otimes
k(t)=(B_{k})_{t}$ by condition (A2)a, and (2) that $A\otimes k(t)=(A_{k})_{t}$
by bijectivity of the map (23). From these identifications, we get the
following diagram

${0}$${A\otimes k(t)}$${B\otimes k(t)}$${\operatorname{coker}(c\otimes
1)}$${0}$${0}$${(A_{k})_{t}}$${(B_{k})_{t}}$${(B_{k})_{t}/(A_{k})_{t}}$${0.}$$\scriptstyle{c\otimes
1}$ (28)

Here $(B_{k}/A_{k})_{t}=(B_{k})_{t}/(A_{k})_{t}$ by flatness of localization,
and so $(B_{k})_{t}/(A_{k})_{t}=\operatorname{coker}(c_{t})_{t}$ since
$B_{k}/A_{k}=\operatorname{coker}(c_{t})$. This proves that
$\operatorname{coker}(c)\otimes k(t)=\operatorname{coker}(c_{t})_{t}$.

We will now prove that the canonical morphism $\operatorname{im}(\pi)\otimes
k(t)\to\operatorname{im}(\pi_{t})_{t}$ is a bijection.

Consider the short exact sequence

$0\to\operatorname{im}(\pi)\to D\to D/\operatorname{im}(\pi)\to 0$ (29)

where $E=D/\operatorname{im}(\pi)$ by definition. The term
$\operatorname{im}(\pi)$ is a $\mathcal{O}_{V^{\prime}}$-module by Lemma 7.2.
All terms in (29) are flat (by generic flatness) and coherent (by condition
(A1)). Since all terms in (29) are flat, we can tensor with $k(t)$ and get a
short exact sequence

$0\to\operatorname{im}(\pi)\otimes k(t)\to D\otimes k(t)\to E\otimes k(t)\to
0$ (30)

from which it follows that $E\otimes k(t)=\left(D\otimes
k(t)\right)/\left(\operatorname{im}(\pi)\otimes k(t)\right)$. We have that (1)
$D\otimes k(t)=(D_{k})_{t}$ by bijectivity of (24), and (2) that $E\otimes
k(t)\subset(E_{k})_{t}$ by (A2)b.

Using the above identifications, we get the following diagram

${0}$${\operatorname{im}(\pi)\ \otimes k(t)}$${D\otimes k(t)}$${E\otimes
k(t)}$${0}$${0}$${(\operatorname{im}(\pi_{t}))_{t}}$${(D_{k})_{t}}$${(E_{k})_{t}}$${0.}$
(31)

The map $\operatorname{im}(\pi)\otimes k(t)\to\operatorname{im}(\pi_{t})_{t}$
is bijective by the 5 lemma.

We will now prove that $V^{\prime}$ is the from Lemma 7.6. Recall that the
subset $V^{\prime}$ in Lemma 7.6 is defined by the following property: For all
$t\in V^{\prime}$, $C=0$ implies that $(C_{k})_{t}=0$. From (25) we get the
following diagram of pointed sets:

${0}$${\operatorname{coker}(c)}$${C}$${\operatorname{im}(\pi)}$${0}$${0}$${(\operatorname{coker}(c_{t})_{t}}$${(C_{k})_{t}}$${\operatorname{im}(\pi_{t})_{t}}$${0}$
(32)

If $C=0$, then $\operatorname{coker}(c)=0$ and $\operatorname{im}(\pi)=0$.
Tensoring $\operatorname{coker}(c)$ and $\operatorname{im}(\pi)$ with $k(t)$,
we find that $0=\operatorname{coker}(c)\otimes
k(t)=\operatorname{coker}(c_{t})_{t}$ and that
$0=\operatorname{im}(\pi)\otimes k(t)=\operatorname{im}(\pi_{t})_{t}$. Thus,
$(C_{k})_{t}=0$. This proves that $V^{\prime}$ is a subset of $T^{\prime}$ as
in Lemma 7.6.

We will now prove that condition two of Lemma 7.6 is satisfied: For every
coherent $\mathcal{O}_{T_{\operatorname{red}}}$-module $\mathcal{M}$ the
support of $\operatorname{Ex}(v,\mathcal{M})$ is a closed subset of $T$. For
each such $\mathcal{M}$, consider the sequence of pointed sets:

$0\to\mathcal{D}er(\mathcal{O}_{T},\mathcal{M})\overset{c}{\to}\mathcal{D}_{v}(\mathcal{M})\to\mathcal{E}x(v,\mathcal{M})\overset{\pi}{\to}\mathcal{E}x(T,\mathcal{M})\to\mathcal{O}b_{v}(\mathcal{M}).$
(33)

We then have a sequence of pointed sets,

$0\to\operatorname{coker}(c)\to\mathcal{E}x(v,\mathcal{M})\to\operatorname{im}(\pi)\to
0$ (34)

and so we have that

$\operatorname{Supp}(\mathcal{E}x(v,\mathcal{M}))=\operatorname{Supp}(\operatorname{coker}(c))\cup\operatorname{Supp}(\operatorname{im}(\pi)).$

Here $\operatorname{coker}(c)$ is a coherent $\mathcal{O}_{T}$-module by (S1)b
and $\operatorname{im}(\pi)$ is coherent because it is a submodule of
$\mathcal{E}x(T,\mathcal{M})$. Therefore,
$\operatorname{Supp}(\mathcal{E}x(v,\mathcal{M}))$ since it is a union of the
supports of coherent modules. ∎

##  8 Formal smoothness from formal versality

Let $C^{*}$ denote the category of local, Noetherian, Henselian superalgebras
with residue field at the unique closed point all equal to $k$. Throughout
this section, let $R$ be a finite type superalgebra in $C^{*}$.

######  Definition 8.1 (Versal).

An object $v$ in $\mathcal{X}$ over $\operatorname{Spec}R$ is called versal if
for every surjection $A^{\prime}\to A$ in $C^{*}$ there exists an arrow
completing the diagram

${\operatorname{Spec}A}$${\operatorname{Spec}R}$${\operatorname{Spec}A^{\prime}}$${\mathcal{X}}$$\scriptstyle{v}$
(35)

The difference between the lifting condition in (35) and formal versality is
that versality is tested against all Henselian superalgebras, rather than just
Artin superalgebras.

######  Theorem 8.2 (Compare [Art74, Theorem 3.3]).

Let $v$ be a formally versal object in $\mathcal{X}$ over
$\operatorname{Spec}R$. Suppose the following conditions hold over $C^{*}$.

  * _(i)_

_(S1)_ and _(S2)_ hold.

  * _(ii)_

($D$ is compatible with completion): Let $A^{\prime}\to A\to A_{0},\
\operatorname{ker}(A^{\prime}\to A)=M$ be as in (10) with $A_{0}$ of finite
type. Let $a_{0}\in\mathcal{X}(\operatorname{Spec}A_{0})$, then

$D_{a_{0}}(M)\otimes\widehat{A}_{0}\overset{\sim}{\longrightarrow}\varprojlim
D_{a_{0}}(M/\mathfrak{m}^{n}M)$ (36)

  * _(iii)_

Let $A$ be a finite type (ordinary) algebra with an ideal $I$ and let
$\overline{A}$ be its $I$-adic completion. Let
$a,b\in\mathcal{X}(\operatorname{Spec}\overline{A})$. If there exists a
compatible sequence of isomorphisms
$a_{n}\overset{\sim}{\longrightarrow}b_{n}$ between the truncations in
$\mathcal{X}(\operatorname{Spec}A/I^{n+1})$, then there is an isomorphism
$a\overset{\sim}{\longrightarrow}b$ compatible with the given isomorphism
$a_{0}\overset{\sim}{\longrightarrow}b_{0}$.

If $(i),(ii)$ hold, then the lifting condition (35) holds for all
infinitesimal extensions $A^{\prime}\to A$. If $(i)-(iii)$ hold, then $v$ is
versal.

###### Proof.

We follow the first part of Artin’s proof of [Art74, Theorem 3.3]), concerning
infinitesimal extensions, but afterwards avoid some difficult limit arguments
by then reducing to the bosonic case.

Consider the lifting problem for an infinitesimal extension $A^{\prime}\to
A\to A_{0},\operatorname{ker}(A^{\prime}\to A)=M$. Since $\mathcal{X}$ is
limit preserving, we may assume that $A^{\prime}$ is finite type. We may also
assume that $M$ is a finitely generated $A_{0}$-module with $M^{2}=0$. By
adding an appropriate number of generators to $R$, we may further assume that
the map $R\to A$ is surjective so that $A$ is a finite type $R$-module.

The goal is to prove that for every map $f:R\to A$ there exists an arrow
completing the diagram

${\operatorname{Spec}A}$${\operatorname{Spec}R}$${\operatorname{Spec}A^{\prime}}$${\mathcal{X}}$$\scriptstyle{v}$
(37)

######  Lemma 8.3.

There exist an arrow completing the diagram

${R}$${A^{\prime}}$${A}$$\scriptstyle{f}$ (38)

Set $A_{n}^{\prime}=A^{\prime}/\mathfrak{m}_{A^{\prime}}^{n}$,
$A_{n}=A/\mathfrak{m}_{A}^{n},\ \operatorname{ker}(A_{n}^{\prime}\to
A_{n})=M(n)$. Since $v$ is formally versal, there exists for each $n$ an arrow
$R\to A_{n}^{\prime}$ completing the diagram

${R}$${A_{n}^{\prime}}$${A_{n}.}$$\scriptstyle{f_{n}}$ (39)

The set of arrows completing (39) can be (non-canonically) identified with the
set $\operatorname{Der}(R,M(n))$. To prove there exists an arrow completing
diagram (38) it suffices to prove that the inverse system
$\\{\operatorname{Der}(R,M(n),\phi_{m,n}:\operatorname{Der}(R,M(m))\to\operatorname{Der}(R,M(n))\\}$
for $m\geq n$ satisfies the Mittag-Leffler (ML) condition. 777 For each $n$,
there exists some $n_{0}\geq n$, such that
$\phi_{n^{\prime},n}\left(\operatorname{Der}(R,M(n^{\prime}))\right)=\phi_{n^{\prime\prime},n}\left(\operatorname{Der}(R,M(n))\right)\subset\operatorname{Der}(R,M(n))$
for all $n^{\prime},n^{\prime\prime}\geq n_{0}$. Since
$\operatorname{Der}(R,M(n))$ can be identified with the set of arrows
completing (39), the _stability_ of the system $\phi_{m,n}$ implies that the
set of arrows completing (39) is also stable. In particular, the set of arrows
does not go to the empty set in the limit since it is non-empty for each $n$.
To prove that the inverse system $\\{\operatorname{Der}(R,M(n),\phi_{m,n}\\}$
satisfies the ML condition, we will prove that $\operatorname{Der}(R,M(n)$ is
a finite length $A_{0}$-module and that the maps $\phi_{m,n}$ are
$A_{0}$-linear since such a system automatically satisfies the ML condition
(see e.g, [Har13, Example 9.1.2]).

We begin by proving that each $M(n)$ is finite length over $A_{0}$. We find
that $M(n)=M/\mathfrak{m}_{A^{\prime}}^{n}\cap M$ by applying the $9$-lemma to
the diagram

${0}$${0}$${0}$${0}$${\mathfrak{m}_{A^{\prime}}^{n}\cap
M}$${\mathfrak{m}_{A^{\prime}}^{n}}$${\mathfrak{m}_{A}^{n}}$${0}$${0}$${M}$${A^{\prime}}$${A}$${0}$${0}$${M(n)}$${A_{n}^{\prime}}$${A_{n}}$${0}$${0}$${0}$${0}$

In particular, $M(n)$ is naturally a finite $A_{0}$-module since $M$ is finite
over $A_{0}$. To prove that $M(n)$ has finite length over $A_{0}$, we will
construct for it a finite length composition series. Consider

$0=\mathfrak{m}_{A^{\prime}}^{n}M(n)\subset\mathfrak{m}_{A^{\prime}}^{n-1}M(n)\subset\mathfrak{m}_{A^{\prime}}^{m-2}M(n)\subset\cdots\subset\mathfrak{m}_{A^{\prime}}M(n)\subset
M(n).$ (40)

The quotients
$\mathfrak{m}_{A^{\prime}}^{i}M(n)/\mathfrak{m}_{A^{\prime}}^{i+1}M(n)$ are
finite dimensional super vector spaces over
$k=A^{\prime}/\mathfrak{m}_{A^{\prime}}$ and so we can take refinement of (40)
and get a finite length composition series for $M(n)$. This proves that $M(n)$
has finite length over $A_{0}$.

We will now prove that $\operatorname{Der}(R,M(n))$ is a finite length
$A_{0}$-module. The $A_{0}$-module structure on $\operatorname{Der}(R,M(n))$
is as follows: For each $a\in A_{0}$, and $d\in\operatorname{Der}(R,M(n))$,
define $(a\cdot d)(r)=ad(r)$. It is easy to check that $(a\cdot d)(r)$ is a
derivation. To see that $\operatorname{Der}(R,M(n))$ is of finite length over
$A_{0}$, just replace $M(n)$ in (40) with $\operatorname{Der}(R,M(n))$.

To prove that the maps
$\phi_{m,n}:\operatorname{Der}(R,M(m))\to\operatorname{Der}(R,M(n))$ are
$A_{0}$-linear, note that the composition $f_{0}:R\overset{f}{\to}A\to A_{0}$
gives $M(n)$ an $R$-module structure, $r\cdot m=f_{0}(r)\cdot m$. In
particular,
$\operatorname{Der}(R,M(n))\cong\operatorname{Hom}_{R}(\Omega_{R},M(n))$ and

$\phi_{m,n}:\operatorname{Hom}_{R}(\Omega_{R},M(m))\to\operatorname{Hom}_{R}(\Omega_{R},M(n)):(R\to
M(m))\mapsto(R\to M(m)\overset{\overline{\phi}_{m,n}}{\longrightarrow}M(n)$

where $\overline{\phi}_{m,n}$ is the quotient map

$\overline{\phi}_{m,n}:M(m)=M/\mathfrak{m}_{A^{\prime}}^{m}\cap M\to
M(n)=M/\mathfrak{m}_{A^{\prime}}^{n}\cap
M=M(m)/\mathfrak{m}_{A^{\prime}}^{n}\cap M(m).$

The map $\phi_{m,n}$ is $A_{0}$-linear, because $\overline{\phi}_{m,n}$ is
$A_{0}$-linear. This proves that $\\{\operatorname{Der}(R,M(n),\phi_{m,n}\\}$
is an inverse system of finite length $A_{0}$-modules and thus satisfies the
Mittag-Leffler condition. In particular, we get a map
$R\to\widehat{A}^{\prime}$ compatible with $f:R\to A$ which can be
approximated by a map $\phi:R\to A^{\prime}$ by Theorem 5.

We will now describe how the lifting problem for $R$ can be reduced to solving
the same lifting problem when $A^{\prime}=A[M]$. Let
$b^{\prime}=\phi^{*}(v)\in\mathcal{X}_{a}(\operatorname{Spec}A^{\prime})$ and
note that $b^{\prime}$ need not be isomorphic to
$a^{\prime}\in\mathcal{X}_{a}(\operatorname{Spec}A^{\prime})$. The lifting
problem for $R$ is thus equivalent to proving there exists
$d\in\operatorname{Der}(R,M)$ such that

$(\phi+d)^{*}(v)\cong a^{\prime}$ (41)

in $\mathcal{X}_{a}(\operatorname{Spec}A^{\prime})$. By (S1), the group
$D_{a_{0}}(M)$ acts transitively on the set
$\overline{\mathcal{X}}_{a}(\operatorname{Spec}A^{\prime})$ (see Lemma 4.2)
and so there exists $e\in D_{a_{0}}(M)$ such that $a^{\prime}\cong
b^{\prime}+e$. In particular, to prove (41) it suffices to show there exists
$d\in\operatorname{Der}(R,M)$ such that $d^{*}(v)\cong e$. To prove such a $d$
exists it suffices to prove that the map $c:\operatorname{Der}(R,M)\to
D_{a_{0}}(M)$ in (22) is surjective. Since $D_{a_{0}}(M)=D_{a}(M)$ by (S1)b
and since $\operatorname{Der}(R,M)=\operatorname{Hom}_{/f}(R,M)$, proving that
the map $c$ is surjective is equivalent to solving the lifting problem for $R$
when $A^{\prime}=A[M]$.

If $M$ has length $(1|0)$ or $(0|1)$, then the map $A_{0}\times_{k}k[M]\to
A_{0}[M]:(a,(s(a),m))\mapsto(a,m)$, where $s:A_{0}\to
A_{0}/\mathfrak{m}_{A_{0}}=k$, is an isomorphism of superalgebras. By (S1)b,
this implies that
$\overline{\mathcal{X}}_{a}(\operatorname{Spec}A[M])=\overline{\mathcal{X}}_{a_{0}}(\operatorname{Spec}A_{0}[M])=\overline{\mathcal{X}}(\operatorname{Spec}k[M])$.

Since $k[M]$ is Artin, the lifting condition for $R$ holds by formal versality
of $v$. By induction, the lifting property will hold whenever $M$ has finite
length. Let $M_{n}:=M/\mathfrak{m}_{A}^{n+1}M$, then we have just proved that
$c_{n}:\operatorname{Der}(R,M_{n})\to D_{a_{0}}(M_{n})$ is surjective for each
$n$. We are left to prove that it remains surjective in the limit. Both
$M_{n}$ and $R$ are finite length $R$-modules and so
$\operatorname{Der}(R,M_{n})$ and $D_{a_{0}}(R)$ are finite-length
$R$-modules. In particular the ML condition is satisfied and,

$\widehat{c}:\varprojlim_{n}\operatorname{Der}(R,M_{n})\to\varprojlim_{n}D_{a_{0}}(M_{n})$

is surjective. Thus,

$\varprojlim\operatorname{Der}(R,M_{n})=\varprojlim\operatorname{Hom}_{R}(\Omega_{R},M_{n})=\operatorname{Hom}_{R}(\Omega,M)\otimes\widehat{R}$

and

$\varprojlim
D_{a_{0}}(M_{n})=D_{a_{0}}(M)\otimes_{A_{0}}\widehat{A_{0}}=D_{a_{0}}(M)\otimes_{R}\widehat{R}$

by assumptions (ii) of the theorem. Thus $\operatorname{Der}(R,M(n))\to
D_{a_{0}}(M_{n})$ is surjective since its tensor product with $\widehat{R}$ is
surjective.

This completes the proof that the desired lifting holds for infinitesimal
extensions. We now use this fact, together with Artin approximation for
bosonic schemes, to deduce our result.

Let $A^{\prime}\to A$ be a surjection in $C^{*}$ and let
$f:\operatorname{Spec}A\to\operatorname{Spec}R$. By [Art74, Theorem 3.3], we
can find a lift on the bosonic parts.

${\operatorname{Spec}A_{\operatorname{bos}}}$${\operatorname{Spec}R}$${\operatorname{Spec}A^{\prime}_{\operatorname{bos}}}$${\mathcal{X}}$$\scriptstyle{f_{\operatorname{bos}}}$$\scriptstyle{\xi_{A^{\prime}_{\operatorname{bos}}}}$$\scriptstyle{g_{\operatorname{bos}}}$

Since $A\to A_{\operatorname{bos}}$ is an infinitesimal extension by
$\mathcal{J}_{A^{\prime}}$, our arguments above imply the existence of $g$
completing the following diagram:

${\operatorname{Spec}A_{\operatorname{bos}}^{\prime}}$${\operatorname{Spec}R}$${\operatorname{Spec}A^{\prime}}$${\mathcal{X}}$$\scriptstyle{g_{\operatorname{bos}}}$$\scriptstyle{\xi_{A^{\prime}}}$$\scriptstyle{g}$

However, such $g$ has no reason to satisfy $g|_{\operatorname{Spec}A}=f$, i.e.
to make the upper triangle of the following diagram commute (although the
lower triangle would):

${\operatorname{Spec}A}$${\operatorname{Spec}R}$${\operatorname{Spec}A^{\prime}}$${\mathcal{X}.}$$\scriptstyle{f}$$\scriptstyle{\xi_{A}}$$\scriptstyle{g}$
(42)

Instead, note we have a map

$g_{\operatorname{bos}}\sqcup
f:\operatorname{Spec}A^{\prime}_{\operatorname{bos}}\sqcup_{\operatorname{Spec}A_{\operatorname{bos}}}\operatorname{Spec}A\to\operatorname{Spec}R$

The source is equivalently
$\operatorname{Spec}(A^{\prime}_{\operatorname{bos}}\times_{A_{\operatorname{bos}}}A)$.
Observe that $A^{\prime}\to
A^{\prime}_{\operatorname{bos}}\times_{A_{\operatorname{bos}}}A$ is
surjective, since
$A^{\prime}_{\operatorname{bos}}\times_{A_{\operatorname{bos}}}A$ is generated
over $A^{\prime}_{\operatorname{bos}}$ by the odd part of $A$. The kernel is
evidently nilpotent. Thus by the above discussion we may lift:

${\operatorname{Spec}(A^{\prime}_{\operatorname{bos}}\times_{A_{\operatorname{bos}}}A)}$${\operatorname{Spec}R}$${\operatorname{Spec}A^{\prime}}$${\mathcal{X}.}$$\scriptstyle{g_{\operatorname{bos}}\sqcup
f}$$\scriptstyle{\xi_{A}}$$\scriptstyle{g}$ (43)

This $g$ will satisfy $g|_{\operatorname{Spec}A}=f$, as desired.

∎

######  Remark 8.4.

Let $S$ be a finite type superscheme and let $s\in S$ be a closed point. We
recall that the henselization of $\mathcal{O}_{S,s}$ at $s$ is defined to be
$\mathcal{O}_{S,s}^{h}=\varinjlim B_{i}$ where the $B_{i}$ are a directed
system of étale $\mathcal{O}_{S,s}$-superalgebras. The fact that $\mathcal{X}$
is limit preserving, means that

$\varinjlim_{i}\mathcal{X}(\operatorname{Spec}B_{i}))=\mathcal{X}(\operatorname{Spec}\mathcal{O}_{S,s}^{h}).$

Let $v\in\mathcal{X}(\mathcal{O}_{T,t}^{h})$ and
$a\in\mathcal{X}(\mathcal{O}_{S,s}^{h})$, let
$f:\mathcal{O}_{T,t}^{h}\to\mathcal{O}_{S,s}^{h}$ be a morphism in $C^{*}$,
and let $a\to v$ be an arrow in $\mathcal{X}$ over $f$. Since $\mathcal{X}$ is
limit preserving, there exists $i$ large enough such that there exists a
factorization $\mathcal{O}_{T,t}^{h}\to B_{i}\to\mathcal{O}_{S,s}^{h}$ and an
object $a_{i}\in\mathcal{X}(B_{i})$ with an arrow $a\to a_{i}\to v$ over
$\mathcal{O}_{T,t}^{h}\to B_{i}\to\mathcal{O}_{S,s}^{h}$.

Following Artin [Art74], we deduce formal smoothness from formal versality and
the following additional conditions.

Condition (A5). The sheaf $\mathcal{D}$ is compatible with completion: If
$T_{0}$ is a reduced finite type scheme, $a_{0}\in\mathcal{X}(T_{0})$, $t\in
T_{0}$ and $\mathcal{M}$ is a coherent $\mathcal{O}_{T_{0}}$-module, then

$\mathcal{D}_{a_{0}}(\mathcal{M})\otimes_{\mathcal{O}_{T_{0}}}\widehat{\mathcal{O}}_{T_{0},t}\overset{\sim}{\longrightarrow}\varprojlim\mathcal{D}_{a_{0}}(\mathcal{M}/\mathfrak{m}_{t}^{n}\mathcal{M})_{t}$
(44)

Condition (A6). The sheaf $\mathcal{D}$ is compatible with étale localization:
If $e:S_{0}\to T_{0}$ is étale, where $e^{*}(a_{0})=b_{0}$, and $\mathcal{M}$
a coherent $\mathcal{O}_{T_{0}}$-module, then

$\mathcal{D}_{b_{0}}(\mathcal{M}\otimes_{\mathcal{O}_{T_{0}}}\mathcal{O}_{S_{0}})\cong\mathcal{D}_{a_{0}}(\mathcal{M})\otimes_{\mathcal{O}_{T_{0}}}\mathcal{O}_{S_{0}}$
(45)

######  Theorem 8.5.

Suppose that _(S1)_ , _(S2)_ , _(A5)_ and _(A6)_ hold for $\mathcal{X}$. An
object $v\in\mathcal{X}(T)$ is formally versal at every closed point $t\in T$
if and only if it is formally smooth over $\mathcal{X}$.

###### Proof.

Let $T=\operatorname{Spec}R$ and assume that $v$ is formally versal at every
closed point $t\in T$ (the other implication is trivial). Consider the lifting
problem in (5) for the infinitesimal extension $A^{\prime}\to A,\
\operatorname{ker}(A^{\prime}\to A)=M$ and a given map $f:R\to A$. We may
assume that $M^{2}=0$ and that $M$ is a finitely generated $A_{0}$-module.

Since $D$ is compatible with completion, we can apply Theorem 8.2 to conclude
that $T$ satisfies the lifting condition in (4) étale locally around each
closed point $t\in T$; apply Theorem 8.2 to the henselizations of
$A,A^{\prime}$ and $R$ at each closed point in $T$ and use the fact that
$\mathcal{X}$ is limit preserving, see Remark 8.4.

We therefore have étale covers, $e_{0,i}:A_{0}\to\Pi C_{i}$,
$e_{i}^{\prime}:A^{\prime}\to\Pi B_{i}^{\prime}$ and $e_{i}:A\to\Pi B_{i}$ and
a family of dotted arrows in $\mathcal{X}$ completing the diagrams

${v}$${b_{i}^{\prime}}$${b_{i}}$ (46)

where $b_{i}=e_{i}^{*}(a)$, $b_{i}^{\prime}=(e_{i}^{\prime})^{*}(a^{\prime})$.
We also have an étale cover $e_{0,i}:A_{0}\to\Pi C_{i}$ with
$b_{0,i}=(e_{0,i}^{*})(a_{0})$.

The obstruction to the existence of an arrow completing the diagram

${R}$${A^{\prime}}$${A}$$\scriptstyle{f}$ (47)

is a class $\mathfrak{o}$ in
$\operatorname{Ext}^{1}(\phi_{*}L_{R}^{\bullet},M)_{ev}$, which is local for
the étale topology. Thus $\mathfrak{o}=0$, and so there is some arrow
$f^{\prime}:R\to A^{\prime}$ determined up to a derivation
$d\in\operatorname{Der}(R,M)$. As in the proof of Theorem 8.2, we have reduced
the problem to proving that the map $c:\operatorname{Der}(R,M)\to
D_{a_{0}}(M)$ is surjective.

We can repeat the proof of Theorem 8.2 and find that the map
$\operatorname{Der}(R,M\otimes_{A_{0}}C_{i})\to
D_{b_{0,i}}(M\otimes_{A_{0}}C_{i})$ is surjective for each $i$. Both terms are
compatible with étale localization and so we get a surjection
$\operatorname{Der}(R,M)\otimes_{A_{0}}C_{i}\to
D_{a_{0}}(M)\otimes_{A_{0}}C_{i}$ by (A6). Thus, the map
$\operatorname{Der}(R,M)\to D_{a_{0}}(M)$ is surjective because it is
surjective after étale base change. 888 More precisely, for each
$t\in\operatorname{Spec}A_{0}$, we have
$\operatorname{Der}(R,M)_{t}\otimes_{(A_{0})_{t}}C_{i}\to
D(M)_{t}\otimes_{(A_{0})_{t}}C_{i}$ is surjective, where $C_{i}$ is an étale
$(A_{0})_{t}$-superalgebra, and $\operatorname{Der}(R,M)_{t},D(M)_{t}$ is the
localization at the maximal ideal $\mathfrak{m}_{t}$ of $t$. Thus
$\operatorname{Der}(R,M)_{t}\to D(M)_{t}$ is surjective since it is surjective
after étale base change. Therefore, $\operatorname{Der}(R,M)\to D(M)$ is
surjective since it is surjective after localization at every maximal ideal.

∎

##  Appendix A Proof of Schlessinger’s theorem

Here we follow [Sch68] very closely (except in the proof of Lemma A.1).

######  Lemma A.1 (Compare [Sch68, Lemma 1.1]).

A morphism $B\to A$ of local, complete, Noetherian superalgebras is surjective
if and only if the induced map on Zariski cotangent spaces $t_{B}^{*}\to
t_{A}^{*}$ is surjective.

###### Proof.

For the forwards implication, note that the map $B/\mathfrak{m}_{B}^{2}\to
A/\mathfrak{m}_{A}^{2}$ induced by $B\to A$ is surjective, and that

$B/\mathfrak{m}_{B}^{2}=k\oplus\mathfrak{m}_{B}/\mathfrak{m}_{B}^{2},\ \
\text{and}\ \
A/\mathfrak{m}_{A}^{2}=k\oplus\mathfrak{m}_{A}/\mathfrak{m}_{A}^{2}$

as super vector spaces. The map $B/\mathfrak{m}_{B}^{2}\to
A/\mathfrak{m}_{A}^{2}$ clearly descends to a surjection
$t_{B}^{*}=\mathfrak{m}_{B}/\mathfrak{m}_{B}^{2}\to\mathfrak{m}_{A}/\mathfrak{m}_{A}^{2}=t_{A}^{*}$
of super vector spaces.

For the converse, set $A_{n}=A/\mathfrak{m}_{B}^{n}$,
$B_{n}=B/\mathfrak{m}_{B}^{n}$ and consider the inverse systems
$\\{A_{n},f_{m,n}:A_{m}\to A_{n}\\}$ and $\\{B_{n},g_{m,n}:B_{m}\to B_{n}\\}$
for $m\geq n$. Let $\\{\phi_{n}:B_{n}\to A_{n}\\}$ be the morphism of inverse
systems induced by $B\to A$. We will now prove that $\phi_{n}$ is surjective
for each $n$. For $n=2$, $\phi_{n}$ is surjective by assumption. Apply
$\operatorname{Sym}_{k}^{n-1}(-)$ to the surjection
$\mathfrak{m}_{B}/\mathfrak{m}_{B}^{2}\to\mathfrak{m}_{A}/\mathfrak{m}_{A}^{2}$.
The resulting map

$\operatorname{Sym}_{k}^{n-1}(\mathfrak{m}_{B}/\mathfrak{m}_{B}^{2})\to\operatorname{Sym}_{k}^{n-1}(\mathfrak{m}_{A}/\mathfrak{m}_{A}^{2})$

is surjective by right-exactness of $\operatorname{Sym}_{k}^{n-1}(-)$. The
maps
$\operatorname{Sym}_{k}^{n-1}(\mathfrak{m}_{B}/\mathfrak{m}_{B}^{2})\to\mathfrak{m}_{B}^{n-1}/\mathfrak{m}_{B}^{n}$
and
$\operatorname{Sym}_{k}^{n-1}(\mathfrak{m}_{A}/\mathfrak{m}_{A}^{2})\to\mathfrak{m}_{A}^{n-1}/\mathfrak{m}_{A}^{n}$
are clearly surjective. The surjectivity of
$\phi_{n}:\mathfrak{m}_{B}^{n-1}/\mathfrak{m}_{B}^{n}\to\mathfrak{m}_{A}^{n+1}/\mathfrak{m}_{A}^{n}$
now follows from commutativitity of the diagram

${\operatorname{Sym}_{k}^{n-1}(\mathfrak{m}_{B}/\mathfrak{m}_{B}^{2})}$${\operatorname{Sym}_{k}^{n-1}(\mathfrak{m}_{A}/\mathfrak{m}_{A}^{2})}$${\mathfrak{m}_{B}^{n-1}/\mathfrak{m}_{B}^{n}}$${\mathfrak{m}_{A}^{n+1}/\mathfrak{m}_{A}^{n}}$$\scriptstyle{\phi_{n}}$

Observe that $\varprojlim\\{\phi_{n}\\}_{n}$ converges to $B\to A$ since $A$
and $B$ are complete. Thus to conclude that $B\to A$ is surjective it suffices
to note that the inverse system $\\{A_{n},f_{m,n}\\}$ satisfies the Mittag-
Leffler (ML) condition. Indeed, $\\{A_{n},f_{m,n}\\}$ is an inverse system of
finite dimensional super vector spaces and is thus easily seen to satisfy ML
(see e.g, [Har13, Example 9.1.3]).

∎

Henceforth, let $p:B\to A$ be a surjection of local, Artin, superalgebras.
Recall that we call $p$ a small extension if the kernel of $p$ is a non-zero
principal ideal such that $\mathfrak{m}_{B}t=(0)$.

######  Definition A.2 (Compare [Sch68, Definition 1.3]).

A small extension $p:A\to B$ of local, Artin superalgebras is called essential
if for any other morphism $q:C\to A$ of local, Artin superalgebras with $pq$
surjective implies that $q$ is surjective.

######  Lemma A.3 (Compare [Sch68, Lemma 1.4]).

Let $p:A\to B$ be a small extension of local, Artin superalgebras. Then

  * (a)

$p$ is essential if and only if the induced map $p_{*}:t_{A}^{*}\to t_{B}^{*}$
on Zariski cotangent spaces is an isomorphism.

  * (b)

$p$ is not essential if and only if $p$ has a section $s:B\to A$ such that
$ps=\operatorname{id}_{B}$.

###### Proof.

(a). Suppose $p_{*}$ is an isomorphism and let $q:C\to A$ be any other map
with $C\to A\to B$ surjective. Then the induced map $t_{C}^{*}\to
t_{A}^{*}\overset{\sim}{\longrightarrow}t_{B}^{*}$ is surjective, by Lemma
A.1. In particular, $q_{*}:t_{C}^{*}\to t_{A}^{*}$ is surjective since
$t_{A}^{*}\cong t_{B}^{*}$. Thus, $q$ is surjective, and $p$ is essential.

Conversely, suppose that $p$ is essential. Since $p$ is surjective
$p_{*}:t_{A}^{*}\to t_{B}^{*}$ is surjective and
$\operatorname{sdim}_{k}t_{B}^{*}\leq\operatorname{sdim}_{k}t_{A}^{*}$.

Let
$t_{1},\dots,t_{r},\alpha_{1},\dots,\alpha_{r^{\prime}},|t_{i}|=0,|\alpha_{i}|=1$
be a basis for $t_{B}^{*}$ and let
$\overline{t}_{1},\dots,\overline{t}_{r},\overline{\alpha}_{1},\dots,\overline{\alpha}_{s}\in
A$ be a choice of lift of each $t_{i},\alpha_{i}$ to $A$. Let

$C=k[\overline{t}_{1},\dots,\overline{t}_{r},\overline{\alpha}_{1},\dots,\overline{\alpha}_{r^{\prime}}]\subseteq
A.$

be the sub-superalgebra of $A$ generated by these lifts and consider the
associated map $C\hookrightarrow A\to B$. Since $t_{C}^{*}\cong t_{B}^{*}$,
$C\hookrightarrow A\to B$ is surjective. Thus $C\hookrightarrow A$ is
surjective, since $p$ is is essential. This means that
$\operatorname{sdim}_{k}t_{A}^{*}\leq\operatorname{sdim}_{k}t_{C}^{*}=\operatorname{sdim}_{k}t_{B}^{*}$
and so $\operatorname{sdim}_{k}t_{A}^{*}=\operatorname{sdim}_{k}t_{B}^{*}$. In
particular, $p_{*}:t_{A}^{*}\to t_{B}^{*}$ is an isomorphism since it is a
surjection of super vector space of the same dimension.

(b). Suppose $p$ has a section $s:B\to A$ and observe that if $p$ were
essential, $s$ would have to be surjective, which is impossible.

Conversely, suppose $p$ is not essential. By part (a), we have a strict
inequality $\operatorname{sdim}_{k}t_{B}^{*}<\operatorname{sdim}_{k}t_{A}^{*}$
and so $C\subset A$ is strictly contained in $A$. Recall that $p$ maps $C$
onto $B$. We will now prove that $C\to B$ is also injective as this will
automatically give us a section. Let $J=\operatorname{ker}(C\hookrightarrow
A\to B)$ and suppose that $J$ is non-zero. Observe that either $J=(t)$ or
$J\subset(t)$ where $(t)=\operatorname{ker}(A\to B$).

Suppose $J=(t)$. We then have the following morphism of small extensions of
$B$ by $(t)$,

${0}$${(t)}$${C}$${B}$${0}$${0}$${(t)}$${A}$${B}$${0}$$\scriptstyle{\phi}$

where $\phi:C\to B$ is the composition $C\hookrightarrow A\overset{p}{\to}B$.
Since any morphism of small extensions is an isomorphism, we must have that
$C=A$. However, $C$ is strictly contained in $A$ and we have arrived at a
contradiction. Thus, we are left to consider the case $J\subset(t)$.

If $J\subset(t)$, then $J=(ct)$ for some non-zero $c$ in $k$. However, every
non-zero element in $k$ is invertible and so $J=(ct)=(t)$, which we already
saw is impossible. Thus $J=(0)$ and $C=B$.

∎

We arrive now at the super version Schlessinger’s theorem. Fix
$x\in\mathcal{X}(\operatorname{Spec}k)$ and restrict the functor
$\overline{\mathcal{X}}_{x}$ to the category $C$ of Artin superschemes.

######  Theorem A.4 (Compare [Sch68, Theorem 2.11]).

Suppose _(S1)_ and _(S2)_ hold for $\overline{\mathcal{X}}_{x}$. Then there
exists a complete, local, noetherian superalgebra $A$ and a formally versal,
formal object
$\widehat{\xi}\in\widehat{\overline{\mathcal{X}}}_{x}(\operatorname{Spf}A)$.

###### Proof.

Set $\overline{\mathcal{X}}:=\overline{\mathcal{X}}_{x}$. The goal is to
construct simultaneously a formal superscheme $\operatorname{Spf}A$ and a
formal object $\widehat{\xi}$ over $\operatorname{Spf}A$ such that
$\widehat{\xi}$ has the infinitesimal lifting property in (1).

We recall that $t_{\overline{\mathcal{X}}}$ is a finite dimensional super
vector space by (S1)b and (S2) and let
$\operatorname{dim}_{k}t_{\overline{\mathcal{X}}}^{+}=r$ and
$\operatorname{dim}_{k}t_{\overline{\mathcal{X}}}^{-}=s$. Choose a dual basis
$x_{1},\dots,x_{r},\theta_{1},\dots,\theta_{s}$ for
$t_{\overline{\mathcal{X}}}$ and let

$S=k[[x_{1},\dots,x_{r},\theta_{1},\dots,\theta_{s}]],\ \
|x_{i}|=0,|\theta_{i}|=1,$

We will now prove that $\widehat{\xi}$ and $\operatorname{Spf}A$ can be
constructed as the projective limit of successive quotients of $S$. Let

  * •

$A_{1}=S/\mathfrak{m}_{S}\cong k$

  * •

$A_{2}=S/\mathfrak{m}_{S}^{2}S$

and observe that

$\operatorname{Spec}(A_{2})\cong\operatorname{Spec}(k[\epsilon])\sqcup_{k}\cdots\sqcup_{k}\operatorname{Spec}(k[\epsilon])\sqcup_{k}\operatorname{Spec}(k[\eta])\sqcup_{k}\cdots\sqcup_{k}\operatorname{Spec}(k[\eta]),\
(r\text{-times},s\text{-times})$ (48)

where $k[\epsilon]$ are the ordinary dual numbers and $|\eta|=1$. Recall that
the set
$\operatorname{Hom}(\operatorname{Spec}k[\epsilon,\eta],\operatorname{Spec}A_{2})$
is the Zariski tangent space $t_{A_{2}}$ of $\operatorname{Spec}A_{2}$. Let
$\bigsqcup_{i=1}^{r}\operatorname{Spec}(k[\epsilon])_{i}$ and
$\bigsqcup_{i=1}^{s}\operatorname{Spec}(k[\eta])_{i}$ denote the factors in
(48). Then, since $\operatorname{Hom}$ is grading preserving, we have that

$t_{A_{2}}=\operatorname{Hom}(\operatorname{Spec}(k[\epsilon]),\bigsqcup_{i=1}^{r}\operatorname{Spec}(k[\epsilon])_{i})\oplus\operatorname{Hom}_{k}(\operatorname{Spec}(k[\eta]),\bigsqcup_{i=1}^{s}\operatorname{Spec}(k[\eta])_{i})$

is a decomposition of $t_{A_{2}}$ into its even and odd components.

We will now prove that there exists an object $\widehat{\xi}_{2}$ in
$\overline{\mathcal{X}}$ over $\operatorname{Spec}A_{2}$ inducing a bijection
$t_{A_{2}}\overset{\sim}{\longrightarrow}t_{\overline{\mathcal{X}}}$. By (S1)b
and (48),

$\overline{\mathcal{X}}(\operatorname{Spec}A_{2})=(t_{\overline{\mathcal{X}}}^{+})^{r}\times(t_{\overline{\mathcal{X}}}^{-})^{s}.$
(49)

Now let $X_{1},\dots,X_{r},\Theta_{1},\dots,\Theta_{s}$ be a basis for
$t_{\overline{\mathcal{X}}}$ and set

$\widehat{\xi}_{2}=(X_{1},\dots,X_{r},\Theta_{1},\dots,\Theta_{s}).$

Observe that $\widehat{\xi}_{2}$ is an object in
$\overline{\mathcal{X}}(\operatorname{Spec}A_{2})=(t_{\overline{\mathcal{X}}}^{+})^{r}\times(t_{\overline{\mathcal{X}}}^{-})^{s}$,
that we have a natural map

$t_{A_{2}}\to
t_{\overline{\mathcal{X}}}:(d:\operatorname{Spec}k[\epsilon,\eta]\to\operatorname{Spec}A_{2})\to
d^{*}\widehat{\xi}_{2},$

and that the inclusions

$\\{p_{i}:\operatorname{Spec}(k[\epsilon])\hookrightarrow\bigsqcup_{i=1}^{r}\operatorname{Spec}(k[\epsilon])_{i}\\}_{i},\
\
\\{q_{i}:\operatorname{Spec}(k[\eta])\hookrightarrow\bigsqcup_{i=1}^{s}\operatorname{Spec}(k[\eta])_{i}\\}_{i}$
(50)

form a basis for $t_{A_{2}}$. Observe that
$(p_{i})^{*}\widehat{\xi}_{2}=X_{i}$ and
$(q_{i})^{*}\widehat{\xi}_{2}=\Theta_{i}.$ Therefore, $t_{A_{2}}\to
t_{\overline{\mathcal{X}}}$ is an isomorphism since any morphism
$k[\epsilon,\eta]\to\operatorname{Spec}A_{2}$ is linear combination of the
inclusions $p_{i}$ and $q_{i}$ and since the $X_{i}$ and $\Theta_{i}$ form a
basis for $t_{\overline{\mathcal{X}}}$.

We now proceed by induction. Suppose we have found a pair
$(\widehat{\xi}_{q},A_{q}=S/J_{q})$ formally versal up to order $q$. We claim
that there exists an ideal $J_{q+1}\subset S$ which is minimal amongst ideals
$J$ satisfying the following conditions

  * (a)

$\mathfrak{m}_{S}J_{q}\subseteq J\subseteq J_{q}$

  * (b)

$\widehat{\xi}_{q}$ lifts to $S/J$. 101010At first glance it seems like we
should take $J_{q+1}=\mathfrak{m}_{S}^{q+1}$; However,
$\mathfrak{m}_{S}^{q+1}$ may not satisfy condition (b) and so we need to be
more careful.

Let $\mathscr{J}$ denote the set of ideals in $S$ satisfying the above
conditions and note that each $J\in\mathscr{J}$ is a sub super vector space of
$J_{q}/\mathfrak{m}_{S}J_{q}$. Thus it suffices to prove that $\mathscr{J}$ is
stable under pairwise intersection, that is if $J,K\in\mathscr{J}$ then $J\cap
K\in\mathscr{J}$. Clearly, given any $J,K\in\mathscr{J}$ we may enlarge $J$ so
that $J+K=J_{q}$ without changing the intersection of $J\cap K$. From this we
obtain that

$S/J\times_{S/J_{q}}S/K\cong S/(J\cap K).$

From condition (S1)a we have a surjection

$\mathcal{X}(S/(J\cap
K))=\mathcal{X}(S/J\times_{S/J_{q}}S/K)\to\mathcal{X}(S/J)\times_{\mathcal{X}(S/J_{q})}\mathcal{X}(S/K)$

from which it follows that $\widehat{\xi}_{q}$ lifts to $S/J\cap K$ (since
$v_{q}$ lifts to $S/J$ and $S/K$). We now define $J_{q+1}$ to be the
intersection of all members in $\mathscr{J}$ and take $A_{q+1}=S/J_{q+1}$ and
let $v_{q+1}\in\mathcal{X}(A_{q+1})$ be any extension of $\widehat{\xi}_{q}$
to $A_{q+1}$. Now define

$J=\bigcap_{q=2,3,\cdots}J_{q}$

and set $A=S/J=\varprojlim_{q}S/J_{q}$, and
$\widehat{\xi}=\varprojlim\widehat{\xi}_{j}\in\widehat{\mathcal{X}}(\operatorname{Spf}A)$.

We now claim that the pair $(\widehat{\xi},A)$ is formally versal. Let
$(B^{\prime},b^{\prime})\to(B,b)\to(k,x)$ be an infinitesimal extension with
kernel $I$ (a finite dimensional $k$ super vector space) where $b^{\prime},b$
and $x$ are the respective objects in $\mathcal{X}$. Suppose we are given a
map $u=p(\widehat{\xi}\to b):A\to B$. Since $\mathfrak{m}_{B}\cdot I=0$, we
get an isomorphism

$B^{\prime}\times_{B}B^{\prime}\overset{\sim}{\longrightarrow}B^{\prime}\times_{k}k[I].$

Now applying condition (S1).b, we get a surjection

$\mathcal{X}_{b}(B^{\prime})\times(t_{\overline{\mathcal{X}}}\otimes
I)\to\mathcal{X}_{b}(B^{\prime})\times\mathcal{X}_{b}(B^{\prime})$

which says that the space $t_{\overline{\mathcal{X}}}\otimes I$ acts
transitively on $\mathcal{X}_{b}(B^{\prime})$. Therefore, if we can show that
there exists a map $A\to B^{\prime}$ compatible with the given map $u:A\to B$,
then the problem reduces to proving that there is a surjection between the set
of maps $A\to B^{\prime}$ compatible with $u:A\to B$ and
$t_{\overline{\mathcal{X}}}\otimes I$.

We first show that there exists a map $A\to B^{\prime}$ compatible with
$u:A\to B$. Since $A=\varprojlim_{q}S/J_{q}$, we can factor $u$ as
$(A,\widehat{\xi})\to(A_{q},\widehat{\xi}_{q})\to(B,b)$ for some $q$. Thus it
suffices to prove that there exists an arrow completing the diagram

${A_{q+1}}$${B^{\prime}}$${A_{q}}$${B}$

This is equivalent to the existence of an arrow filling in the diagram

${S}$${A_{q}\times_{B}B^{\prime}}$${B^{\prime}}$${A_{q+1}}$${A_{q}}$${B}$$\scriptstyle{w}$$\scriptstyle{p}$$\scriptstyle{u^{\prime}}$

where $w$ was chosen to make the diagram commute. To prove that
$\widehat{\xi}$ exists we will now show that the map $p$ has a section.
Suppose $p$ does not have a section, then by Lemma A.3 $p$ is essential and
$w$ is a surjection. Thus,
$A_{q}\times_{B}B^{\prime}=S/\operatorname{ker}(w)$. Furthermore, from
condition (S1)a, applied to $A_{q}\times_{B}B^{\prime}$, we find that
$\widehat{\xi}_{q}\in\mathcal{X}(A_{q})$ must lift back to an object in
$\mathcal{X}(A_{q}\times_{B}B^{\prime})=\mathcal{X}(S/\operatorname{ker}(w))$.
By construction, this means that $\operatorname{ker}(w)\subseteq J_{q+1}$.
Thus $w$ factors through $A_{q+1}$, and so we obtain the map $u^{\prime}:A\to
B^{\prime}$ compatible with $u:A\to B$.

To finish the proof, recall that the set of maps $A\to B^{\prime}$ compatible
with $A\to B$ is a torsor over $\operatorname{Der}_{k}(A,I)=t_{A}\otimes I$
where $I=\operatorname{ker}(B^{\prime}\to B)$. We already proved that
$t_{A}\cong t_{\overline{\mathcal{X}}}$ and so $t_{A}\otimes_{k}I\cong
t_{\overline{\mathcal{X}}}\otimes I$.

∎

##  Appendix B Proof of Flenner’s criterion

Here we prove Theorem 7.4, following [Fle81] except in the proof of Lemma B.6,
where we can reduced to the bosonic case rather than recapitulating the
original proof.

###  B.1 Half-Exact functors

######  Definition B.1.

We say that a functor $F:\operatorname{Coh}(R)\to\operatorname{Coh}(R)$ is
_half-exact_ if for every exact sequence

$0\to M^{\prime}\to M\to M^{\prime\prime}\to 0$

in $\operatorname{Coh}(R)$, the sequence

$F(M^{\prime})\to F(M)\to F(M^{\prime\prime})$

is exact.

######  Lemma B.2 (Compare to [Fle81, Lemma 2.1]).

Suppose _(S1)a_ holds. Then the functor $D_{v_{0}}(-)$ is half-exact.

###### Proof.

Let

$0\to M^{\prime}\overset{f}{\to}M\overset{g}{\to}M^{\prime\prime}\to 0$ (51)

be an exact sequence of of finitely generated modules of a reduced ring
$R_{0}$. Let $v_{0}\in\mathcal{X}(\operatorname{Spec}R_{0})$ and consider the
sequence

$D_{v_{0}}(M^{\prime})\overset{j}{\to}D_{v_{0}}(M)\overset{p}{\to}D_{v_{0}}(M^{\prime\prime}):v_{0}\mapsto
f^{*}v_{0}\mapsto g^{*}(f^{*}v_{0}).$

We will first prove that $p\circ j=0$, _i.e._ , that there is a section
$s:\operatorname{Spec}R_{0}[M^{\prime\prime}]\to\operatorname{Spec}R_{0}$ with
an arrow $g^{*}(f^{*}v)\to v_{0}$ in $\mathcal{X}$ over it. The sequence in
(51) induces a sequence of trivial extensions of $R_{0}$,

$R_{0}[M^{\prime}]\to R_{0}[M]\to
R_{0}[M^{\prime\prime}]:(r,m^{\prime})\mapsto(r,f(m^{\prime}))\mapsto(r,gf(m^{\prime}))=(r,0)$
(52)

sending $R_{0}[M^{\prime}]$ to a copy of $R_{0}$ sitting inside of
$R_{0}[M^{\prime\prime}]$. In particular, (52) determines a section
$s:R_{0}\hookrightarrow R_{0}[M^{\prime\prime}]$. The maps in (52) induce maps
$\operatorname{Spec}R_{0}[M^{\prime\prime}]\overset{s}{\to}\operatorname{Spec}R_{0}\to\operatorname{Spec}R_{0}[M]\to\operatorname{Spec}R_{0}[M^{\prime}]$.
In particular, the pullback of $v_{0}$ to
$\operatorname{Spec}R_{0}[M^{\prime\prime}]$ factors through
$\operatorname{Spec}R_{0}$ and so there is an arrow $g^{*}(f^{*}v)\to v_{0}$
in $\mathcal{X}$ over $s$.

We will now prove that $p^{-1}(0)=\operatorname{im}(j)$. Take $v$ be an object
in $D_{v_{0}}(M)$ such that $p(v)=0$. Recall that $0$ is the trivial extension
of $v_{0}$ over $\operatorname{Spec}R_{0}[M^{\prime\prime}]$. Consider the
diagram

${0}$${v}$${v_{0}}$${b}$

in $\mathcal{X}$ over
$\operatorname{Spec}R_{0}\to\operatorname{Spec}R_{0}[M]\leftarrow\operatorname{Spec}R_{0}[M^{\prime\prime}]$.
By (S1)a, there exists an object $b$ over
$\operatorname{Spec}(R_{0}\times_{R_{0}[M^{\prime\prime}]}R_{0}[M])$ filling
in the dotted arrows. Since
$R_{0}\times_{R_{0}[M^{\prime\prime}]}R_{0}[M]=R_{0}[M^{\prime}]$, the object
$b$ is in $\mathcal{X}(\operatorname{Spec}R_{0}[M^{\prime}])$. Thus,
$v\in\operatorname{im}(j)$.

∎

######  Remark B.3.

Lemma B.2 holds for the functor
$\operatorname{Ex}(v,-):\operatorname{Coh}(R)\to\operatorname{Set}$ in the
following sense. For every exact sequence of finitely generated $R$-modules,
$0\to M^{\prime}\to M\to M^{\prime\prime}\to 0$, the sequence

${\operatorname{Ex}(v,M^{\prime})}$${\operatorname{Ex}(v,M)}$${\operatorname{Ex}(v,M^{\prime\prime})}$$\scriptstyle{j}$$\scriptstyle{p}$

is such that $p^{-1}(0)=\operatorname{im}(j)$.

###  B.2 Vergleichsatz (Comparison Theorem)

In this section we closely follow [Fle81, Section 6].

######  Lemma B.4 (Compare with [Fle81, Lemma 6.2]).

Let $R$ be a Noetherian superalgebra and let

$\tilde{R}=R[T_{1},\dots,T_{k},T_{k+1},\dots,T_{k+s}],\ \ |T_{i}|=0,\ 1\leq
i\leq r,\ \ |T_{k+i}|=1,\ 1\leq i\leq s,$

. Let $M$ be a ($\mathbb{Z}_{\geq 0}\times\mathbb{Z}_{2}$)-graded
$\tilde{R}$-module. Then

  * _(i)_

Let

$M^{\prime}\to M\to M^{\prime\prime}$

be an exact sequence of graded $\tilde{R}$-modules. Then $M$ is finitely
generated over $\tilde{R}$ if $M^{\prime},M^{\prime\prime}$ are finitely
generated over $\tilde{R}$,

  * (ii)

If $M/T_{i}M$ is a finitely generated $\tilde{R}$-module, then so is $M$.

###### Proof.

Part (i) is obvious. For part (ii), suppose, by contradiction, that $M$ has
infinite rank. Then multiplication by $T_{i}$ must annihilate an infinite
subset of the set of generators of $M$. Therefore, $T_{i}M$ must be finitely
generated. Now apply part (i) to the sequence

$T_{k}M\to M\to M/T_{k}M,$

to conclude that $M$ is finitely generated over $\tilde{R}$

∎

Now let $\mathfrak{a}\subset R$ be an ideal and let $M_{i},i\in\mathbb{Z}$ be
finitely generated $R$-modules. Define

$M:=\bigsqcup_{i\geq 0}M_{i}=M_{0}\oplus M_{1}\oplus\cdots.$

Let $\mathcal{J}\subset R$ denote the ideal of odd nilpotents and consider the
following graded $R$-algebras,

$\displaystyle B_{\mathfrak{a}}(R)$ $\displaystyle:=\bigsqcup_{i\geq
0}\mathfrak{a}^{i}R=R\oplus\mathfrak{a}\oplus\mathfrak{a}^{2}\oplus\mathfrak{a}^{3}\oplus\cdots$
$\displaystyle B_{\mathfrak{a}}(\mathcal{J})$ $\displaystyle:=\bigsqcup_{i\geq
0}\mathcal{J}\mathfrak{a}^{i}=\mathcal{J}\oplus\mathcal{J}\mathfrak{a}\oplus\cdots$
$\displaystyle B_{\mathfrak{a}}(R)/\mathcal{J}B_{\mathfrak{a}}(R)$
$\displaystyle:=\bigsqcup_{i\geq
0}\mathfrak{a}^{i}/\mathcal{J}\mathfrak{a}^{i}=R/\mathcal{J}\oplus\mathfrak{a}/\mathcal{J}\mathfrak{a}\oplus\cdots$

######  Lemma B.5.

Let $R$ be a Noetherian superalgebra and let $M$ be an $R$-module. Then $M$ is
finitely generated over $R$ if and only if
$M_{\operatorname{bos}}=M/\mathcal{J}M$ is finitely generated over
$R_{\operatorname{bos}}=R/\mathcal{J}$.

######  Lemma B.6.

Let $F:\operatorname{Coh}(R)\to\operatorname{Coh}(R)$ be a half-exact functor.
If $\bigsqcup_{i\geq 0}M_{i}$ is finitely generated over
$B_{\mathfrak{a}}(R)$, then $F(M)=\bigsqcup_{i\in\mathbb{Z}}F(M_{i})$ is
finitely generated over $B_{\mathfrak{a}}(R)$.

###### Proof.

The bosonic case is proved in [Fle81, Satz 6.1]. Let

$M/\mathcal{J}M=\bigsqcup_{i\geq 0}M_{i}/\mathcal{J}M_{i}.$

Observe that $M/\mathcal{J}M$ is a finitely generated
$B_{\mathfrak{a}}(R)/\mathcal{J}B_{\mathfrak{a}}(R)$-module by Lemma B.5 and
that $F(M/\mathcal{J}M)$ is a finitely-generated
$B_{\mathfrak{a}}(R)/\mathcal{J}B_{\mathfrak{a}}(R)$-module by [Fle81, Satz
6.1]. Furthermore, $F(M/\mathcal{J}M)$ is a finitely-generated
$B_{\mathfrak{a}}(R)$-module since the map $B_{\mathfrak{a}}(R)\to
B_{\mathfrak{a}}(R)/\mathcal{J}B_{\mathfrak{a}}(R)$ is surjective.

Now apply the functor $F$ to the exact sequence of
$B_{\mathfrak{a}}(R)$-modules

$0\to\mathcal{J}M\to M\to M/\mathcal{J}M\to 0$

to get an exact sequence of $B_{\mathfrak{a}}(R)$-modules

$F(\mathcal{J}M)\to F(M)\to F(M/\mathcal{J}M).$

We now claim that $F(\mathcal{J}M)$ is finite. We want to prove this claim
because it automatically implies that $F(M)$ is finite; from the above
sequence we get an injective map $F(M)/F(\mathcal{J}M)\to F(M/\mathcal{J}M)$,
with $F(M/\mathcal{M}J)$ is finitely generated, and so if $F(\mathcal{J}M)$ is
finite so must be $F(M)$, otherwise the quotient $F(M)/F(\mathcal{J}M)$ would
be infinite and we would have a contradiction.

We prove the claim by induction on $\mathcal{J}$. Let $s$ be the smallest
integer for which $\mathcal{J}^{s+1}=0$. We can apply [Fle81, Satz 6.1] to the
finite $B_{\mathfrak{a}}(R)/\mathcal{J}B_{\mathfrak{a}}(R)$-module
$\mathcal{J}M/\mathcal{J}^{2}M$ to conclude that
$F(\mathcal{J}M/\mathcal{J}^{2}M)$ is finite over
$B_{\mathfrak{a}}(R)/\mathcal{J}B_{\mathfrak{a}}(R)$ and thereby also over
$B_{\mathfrak{a}}(R)$. Now $\mathcal{J}M/\mathcal{J}^{3}M$ is finite over
$\mathcal{J}M/\mathcal{J}^{2}M$ and thus also over
$B_{\mathfrak{a}}(R)/\mathcal{J}B_{\mathfrak{a}}(R)$. Thus
$F(\mathcal{J}M/\mathcal{J}^{3}M)$ is finite over
$B_{\mathfrak{a}}(R)/\mathcal{J}B_{\mathfrak{a}}(R)$ and thereby also over
$B_{\mathfrak{a}}(R)$. By induction on $k=1,\dots,s$ we find that
$F(\mathcal{J}M)$ is finite over
$B_{\mathfrak{a}}(R)/\mathcal{J}B_{\mathfrak{a}}(R)$ and thereby also over
$B_{\mathfrak{a}}(R)$. Thus $F(M)$ is finite and this concludes the proof.

∎

######  Corollary B.7 (Compare with [Fle81, Corollary 6.3]).

Let $F:\operatorname{Coh}(R)\to\operatorname{Coh}(R)$ be a half-exact functor
and let $\mathfrak{a}\subset R$ be an ideal. Then

  * (1)

$\bigsqcup_{i\geq 0}F(\mathfrak{a}^{i}M)$ is a finite $\bigsqcup_{i\geq
0}\mathfrak{a}^{i}$-module, and

  * (2)

The canonical map

$\\{F(M)/\mathfrak{a}^{i}F(M)\\}_{i\geq
0}\to\\{F(M/\mathfrak{a}^{i}M)\\}_{i\geq 0}$

is injective.

###### Proof.

(2) follow from Lemma B.6, because $\bigsqcup\mathfrak{a}^{i}M$ is a finite
$\bigsqcup\mathfrak{a}^{i}R$-module

∎

######  Corollary B.8 (Compare with [Fle81, Corollary 6.4] ).

The module
$\bigsqcup\operatorname{Ext}^{i}(L_{R}^{\bullet},\mathfrak{a}^{n}M)$ is a
coherent $\bigsqcup\mathfrak{a}^{n}$-modules. In particular, the natural maps

$\phi_{n}:\operatorname{Ext}^{i}(L_{R}^{\bullet},M)/\mathfrak{a}^{n}\operatorname{Ext}^{i}(L_{R}^{\bullet},M)\to\operatorname{Ext}^{i}(L_{R}^{\bullet},M/\mathfrak{a}^{n}M)$

are bijective for all $n$.

######  Lemma B.9 (Compare with [Fle81, Lemma 3.1]).

Let $R$ be a local, noetherian $k$-superalgebra and let
$v\in\mathcal{X}(T=\operatorname{Spec}R)$. Let $\mathfrak{a}\subseteq R$ be an
ideal and let $M$ be a finitely generated $R_{0}$-module. Let $p$ denote the
canonical map

$p:\operatorname{Ex}(v,M)\to\varprojlim_{n}\operatorname{Ex}(v,M/\mathfrak{a}^{n}M).$

Then $p^{-1}(0)=0$.

###### Proof.

Let the subscript $n$ denote the residue class modulo $\mathfrak{a}^{n}$ and
consider the diagram:

${{\operatorname{Der}(R,M)}}$${D_{v}(M)}$${{\operatorname{Ex}(v,M)}}$${{\operatorname{Ex}(T,M)}}$${{\operatorname{Der}(R,M_{n})}}$${D_{v}(M_{n})}$${{\operatorname{Ex}(v,M_{n})}}$${{\operatorname{Ex}(T,M_{n})}}$$\scriptstyle{c}$$\scriptstyle{i}$$\scriptstyle{p}$$\scriptstyle{\pi}$$\scriptstyle{c_{n}}$$\scriptstyle{i_{n}}$$\scriptstyle{\pi_{n}}$

We begin by proving that the natural map
$q:\operatorname{Ex}(T,M)\to\varprojlim\operatorname{Ex}(T,M_{n})$ is
injective. We have an identification of $\operatorname{Ex}(T,-)$ with the
half-exact functor
$T_{R}^{1}(-)_{ev}=\operatorname{Ext}(L_{R}^{\bullet},-)_{ev}$. Applying
Corollary B.8 we obtain a bijection

$\operatorname{Ex}(T,M)_{n}=\operatorname{Ex}(T,M_{n})$

for all $n$. Consider the commutative diagram

${{\operatorname{Ex}(T,M)}}$${{\varprojlim\operatorname{Ex}(T,M_{n})}}$${{\operatorname{Ex}(T,M_{n})}}$${{\operatorname{Ex}(T,M)_{n}}.}$$\scriptstyle{q}$$\scriptstyle{q_{n}}$

Suppose $f\in\operatorname{Ex}(T,M)$ such that $q(f)=0$. Then $q_{n}(f)=0$ for
all $n$ and so $f\in\bigcap\mathfrak{a}^{n}\operatorname{Ex}(T,M)=0$, by the
Krull intersection theorem. In particular, we find that

$\operatorname{ker}(q\circ\pi)=\operatorname{ker}(\pi).$

Thus if $g\in\operatorname{Ex}(v,M)$ is such that $p(g)=0$, then
$g\in\pi^{-1}(0)$. This reduces the problem to proving that the map

$\pi^{-1}(0)\to\varprojlim\pi_{n}^{-1}(0)$

is injective. Using the identification between $\pi^{-1}(0)$ and
$\operatorname{coker}(c)$, this is equivalent to proving that the map

$\operatorname{coker}(c)\to\varprojlim\operatorname{coker}(c_{n})$

is injective.

Since
$\operatorname{Der}(T,-)=\operatorname{Ext}^{0}(L_{R}^{\bullet},-)_{ev}$, we
can apply Corollary B.8 to conclude that the map

$\operatorname{Der}(T,M)_{n}\to\operatorname{Der}(T,M_{m})$

is bijective. Since $D_{v}(-)$ is half-exact, we can apply Corollary B.7 to
conclude that the map

$D_{v}(M)_{n}\to D_{v}(M_{n})$

is injective. The fact that
$\operatorname{coker}(c)\to\varprojlim\operatorname{coker}(c_{n})$ is
injective the follows from a standard diagram chase.

∎

###  B.3 Formal versality: Vanishing of the extension functor

We now arrive at Flenner’s criterion.

######  Theorem B.10.

Let $v$ be an object in $\mathcal{X}$ over an affine superscheme $T$. Assume
that _(S1)_ and _(S2)_ are satisfied. Then the following statements are
equivalent.

  1. (1)

$v$ is formally versal,

  2. (2)

$\mathcal{E}x(v,k(t))_{t}=\mathcal{E}x(v,\Pi k(t))_{t}=0$.

  3. (3)

$\mathcal{E}x(v,\mathcal{M})_{t}=0$ for every coherent
$\mathcal{O}_{T_{\operatorname{red}}}$-module $\mathcal{M}$.

###### Proof.

Fix $t\in T$ and set $\mathcal{E}x(v,-)=\mathcal{E}x(v,-)_{t}$. Since
everything is affine, we can go ahead and replace $\mathcal{M}$ with its
associated module $M$, $\mathcal{O}_{T}$ with $R$, etc. Let
$\mathfrak{m}=\mathfrak{m}_{t}\subset R$ denote the maximal ideal of $t$.

We will first prove that (2) and (3) are equivalent. That condition (3)
implies (2) is obvious since $k(t)$ is a finitely generated
$\mathcal{O}_{T}$-module, so assume that