concentration fluctuations of model NiCoCr CSAs using the Farkas-Caro
potential function. Figure S1(a-f) illustrates that the the Warren–Cowley SRO
parameters $p_{ab}$ associated with the annealed CSAs are statistically
indistinguishable from $p_{ab}^{\text{rsa}}$ including the six (distinct)
elemental pairs at $T_{a}=400$ K. Therefore, the former can be essentially
treated as RSAs with random distributions of the chemical compositions and
associated fluctuations that exhibit the expected scale-dependence due to
randomness (see Fig. 8 (b,d)).

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Figure S1: Short range ordering in annealed NiCoCr CSAs based on the Farkas-
Caro potential. Warren–Cowley SRO parameters including a) $p_{\text{NiNi}}$ b)
$p_{\text{NiCo}}$ c) $p_{\text{NiCr}}$ d) $p_{\text{CoCo}}$ e)
$p_{\text{CoCr}}$ f) $p_{\text{CrCr}}$ plotted against distance $r$ at
$T_{a}=400$ K. The base (red) dashdotted line indicates the random
concentration.


e^{\prime}(a,v),g\cdot e^{\prime}(a^{\prime},v^{\prime})]=\prod_{1\leq i,k\leq
t}[e^{\prime}(r_{i}a,v_{i}),e^{\prime}(r^{\prime}_{k}a^{\prime},v_{k})]=$
$\prod_{1\leq i,k\leq
t}f^{\prime}(-r_{i}ar^{\prime}_{k}a^{\prime},[v_{i},v_{k}])=f^{\prime}(-aa^{\prime},g\cdot[v,v^{\prime}])=g\cdot
f^{\prime}(-aa^{\prime},[v,v^{\prime}]).$

The identities (3) and (4) can be derived similarly. The identity $(2)$
follows from $g\cdot[x,v]=[\mathrm{Ad}(g)(x),g\cdot v]$, which is obtained by
using the identity at the beginning of the proof of Lemma 7.5. The lemma is
proven. ∎

The map $f^{\prime}(b,x)\mapsto f(b,x)$ induces a morphism of group functors
$i:\mathsf{F}^{\prime}\to\mathsf{F}$. In fact, it is the composition of the
above morphism from $\mathsf{F}^{\prime}$ to the group functor
$R\mapsto\mathsf{Gpl}(\mathsf{U}(\mathsf{g})\otimes R_{0})$ and the natural
morphism from the latter to $\mathsf{F}$ , induced by the Hopf algebra
morphism $\mathsf{U}(\mathsf{g})\to\mathrm{hyp}(\mathbb{G}_{ev})$.

It is clear that
$({\bf\Sigma}^{\prime}(R),\mathsf{F}^{\prime}(R),\mathsf{G}(R),i(R),\alpha(R))$
is a quintuple in the sense of [11], with the only difference that $\alpha(R)$
is a group homomorphism. We define a group ${\bf\Gamma}(R)$ as a factor-group
of $\mathsf{G}(R)\ltimes{\bf\Sigma}^{\prime}(R)$ by a normal subgroup
${\bf\Xi}(R)=\\{(i(R)(f),f^{-1})\mid f\in\mathsf{F}^{\prime}(R)\\}$.

######  Remark 11.8.

${\bf\Gamma}(R)$ is isomorphic to the factor-group of the amalgamated free
product $\mathsf{G}(R)*_{\mathsf{F}^{\prime}(R)}{\bf\Sigma}^{\prime}(R)$
modulo the relations $s^{g}=(\alpha(R)(g))(s)$ for $g\in\mathsf{G}(R)$ and
$s\in{\bf\Sigma}^{\prime}(R)$.

Lemma 2.1 from [11] implies that every element of ${\bf\Gamma}(R)$ can be
uniquely expressed as $ge^{\prime}(a_{1},v_{1})\ldots
e^{\prime}(a_{t},v_{t})$, where $g\in\mathsf{G}(R)$ and $a_{i}\in R_{1}$ for
$1\leq i\leq t$.

The following lemma is now apparent.

######  Lemma 11.9.

$R\mapsto{\bf\Gamma}(R)$ is a group functor, hence a locally algebraic group
superscheme. Moreover, $\Psi:(\mathsf{G},\mathsf{V})\to{\bf\Gamma}$ is a
functor from $\mathsf{HCP}$ to $\mathcal{SFG}_{la}$.

######  Theorem 11.10.

The functors $\Phi$ and $\Psi$ are quasi-inverse to each other.

###### Proof.

Since the group structure of ${\bf\Gamma}(R)$ is defined by the relations
similar to $(1)-(5)$, Theorem 11.5 implies
$\Psi\circ\Phi\simeq\mathrm{id}_{\mathcal{SFG}_{la}}$. Further, the odd
component of the Lie superalgebra of $\bf\Gamma$ can be identified with the
(commutative) subgroup of ${\bf\Gamma}(K[\epsilon_{0},\epsilon_{1}])$
consisting of all elements $e^{\prime}(c_{1}\epsilon_{1},v_{1})\ldots
e^{\prime}(c_{t}\epsilon_{1},v_{t})$, where $c_{i}\in K$ for $1\leq i\leq t$,
which can be identified with $\mathsf{V}$. Moreover, the adjoint action of
$\mathsf{G}$ is identified with the original action of $\mathsf{G}$. Thus
$\Phi\circ\Psi\simeq\mathrm{id}_{\mathsf{HCP}}$. The theorem is proven. ∎

In what follows, let $\mathsf{E}$ denote $\mathbb{E}_{ev}$.

Let $\mathbb{G}$ be a locally algebraic group superscheme. If
$\Phi(\mathbb{G})\simeq(\mathsf{G},\mathsf{V})$ or
$\Psi((\mathsf{G},\mathsf{V}))\simeq\mathbb{G}$, then we say that $\mathbb{G}$
is represented by $(\mathsf{G},\mathsf{V})$.

A sequence
$(\mathsf{R},\mathsf{W})\to(\mathsf{G},\mathsf{V})\to(\mathsf{H},\mathsf{U})$
in $\mathsf{HCP}$ is called _exact_ whenever the following conditions hold:

  1. (1)

The sequences $0\to\mathsf{W}\to\mathsf{V}\to\mathsf{U}\to 0$ and
$\mathsf{E}\to\mathsf{R}\to\mathsf{G}\to\mathsf{H}\to\mathsf{E}$ are exact in
the categories of superspaces and group schemes, respectively.

  2. (2a)

$\mathsf{W}$ is a $\mathsf{G}$-submodule of $\mathsf{V}$.

  3. (2b)

$\mathsf{R}$ acts trivially on $\mathsf{V}/\mathsf{W}$.

  4. (2c)

$[\mathsf{V},\mathsf{W}]\subseteq\mathrm{Lie}(\mathsf{R})$.

######  Theorem 11.11.

A sequence of group superschemes $\mathbb{R}\to\mathbb{G}\to\mathbb{H}$ is
exact if and only if the sequence
$\Phi(\mathbb{R})\to\Phi(\mathbb{G})\to\Phi(\mathbb{H})$ is exact.

###### Proof.

Suppose that $\mathbb{R},\mathbb{G}$ and $\mathbb{H}$ are represented by the
pairs $(\mathsf{R},\mathsf{W}),(\mathsf{G},\mathsf{V})$, and
$(\mathsf{H},\mathsf{U})$, respectively. Without loss of a generality, one can
replace $\mathbb{R}\to\mathbb{G}\to\mathbb{H}$ by the sequence
$\Psi((\mathsf{R},\mathsf{W}))\to\Psi((\mathsf{G},\mathsf{V}))\to\Psi((\mathsf{H},\mathsf{U}))$.
Then $\mathbb{R}$ coincides with $\ker(\mathbb{G}\to\mathbb{H})$ if and only
if
$\mathsf{R}=\ker(\mathsf{G}\to\mathsf{H}),\mathsf{W}=\ker(\mathsf{V}\to\mathsf{U})$
and $\mathsf{W}$ satisfies the conditions $(2a)-(2c)$. For example, since
$e^{\prime}(a,v)he^{\prime}(a,v)^{-1}\in\mathbb{R}(A)$ for every
$h\in\mathbb{R}(A),a\in A_{1}$ and $v\in\mathsf{V}$, then

$e^{\prime}(a,v)he^{\prime}(a,v)^{-1}h^{-1}=e^{\prime}(a,v)e^{\prime}(a,-h\cdot
v)=e^{\prime}(a,v-h\cdot v)\in\mathbb{R}(A)$

implies $v-h\cdot v\in\mathsf{W}$, hence $(2b)$. The converse statement is
obvious.

Suppose that $\mathbb{G}\to\mathbb{H}$ is surjective in the Grothendieck
topology. Consider a couple $h\in\mathsf{H}(A_{0})=\mathbb{H}(A_{0})$ and
$e^{\prime}(a,u)\in\mathbb{H}(A)$. There is an fppf covering $A^{\prime}$ of
$A$, such that $e^{\prime}(\iota(a),u)=e^{\prime}(a^{\prime},\overline{v}),$
where $v\in\mathsf{V}$, and $\overline{v}$ is the image of $v$ in
$\mathsf{U}$. That is, $\mathsf{V}\to\mathsf{U}$ is a surjective map. Finally,
let $\iota:A_{0}\to B$ be an fppf covering of $A_{0}$ , such that
$\mathbb{H}(\iota)(h)$ belongs to the image of
$\mathbb{G}_{ev}(B)=\mathsf{G}(B_{0})$. Then $B_{0}$ is an fppf covering of
$A_{0}$. Conversely, if $h\in\mathsf{H}(A_{0})$ belongs to the image of
$\mathsf{G}$ up to an fppf covering $B$ of $A_{0}$ (in the category of
algebras), then $h$ belongs to the image of $\mathbb{G}$ up to an fppf
covering $B\otimes_{A_{0}}A$ of $A$. The theorem is proven. ∎

##  12\. Applications

###  12.1. Radicals

Let $\mathbb{G}$ be an algebraic group superscheme, represented by a Harish-
Chandra pair $(\mathsf{G},\mathsf{V})$, where
$\mathsf{V}=\mathrm{Lie}(\mathbb{G})_{1}$, and $\mathsf{G}=\mathbb{G}_{ev}$ is
regarded as an algebraic group scheme.

Let $\mathsf{R}$ be a normal group subscheme of $\mathsf{G}$. The largest
normal group super-subscheme $\mathbb{H}$ of $\mathbb{G}$ such that
$\mathsf{H}=\mathbb{H}_{ev}\leq\mathsf{R}$, is called the $\mathsf{R}$-radical
of $\mathbb{G}$. If the $\mathsf{R}$-radical of $\mathbb{G}$ is trivial, then
$\mathbb{G}$ is called _$\mathsf{R}$ -semisimple_.

Let $\mathsf{W}$ be a $\mathsf{G}$-submodule of $\mathsf{V}$ such that
$[\mathsf{W},\mathsf{V}]\subseteq\mathrm{Lie}(\mathsf{R})$ and
$[\mathsf{W},\mathsf{V},\mathsf{V}]\subseteq\mathsf{W}$. Such a submodule is
called _$\mathsf{R}$ -subordinated_. The sum of two $\mathsf{R}$-subordinated
submodules is again $\mathsf{R}$-subordinated. Thus, there is the largest
$\mathsf{R}$-subordinated submodule, denoted by $\mathsf{W}_{\mathsf{R}}$.

Set
$\mathsf{H}_{\mathsf{R}}=\ker(\mathsf{R}\to\mathrm{GL}(\mathsf{V}/\mathsf{W}_{\mathsf{R}}))$.
Since $\mathsf{W}_{\mathsf{R}}$ is a $\mathsf{G}$-submodule of $\mathsf{V}$,
we have $\mathsf{H}_{\mathsf{R}}\unlhd\mathsf{G}$.

######  Lemma 12.1.

The Harish-Chandra sub-pair
$(\mathsf{H}_{\mathsf{R}},\mathsf{W}_{\mathsf{R}})$ represents the
$\mathsf{R}$-radical of $\mathbb{G}$.

###### Proof.

Let $(\mathsf{H},\mathsf{W})$ be a Harish-Chandra sub-pair of
$(\mathsf{G},\mathsf{V})$ that represents a normal group super-subscheme of
$\mathbb{G}$ with $\mathsf{H}\leq\mathsf{R}$. Since $\mathsf{H}$ acts
trivially on $\mathsf{V}/\mathsf{W}$, we have

$[\mathsf{W},\mathsf{V},\mathsf{V}]\subseteq[\mathrm{Lie}(\mathsf{H}),\mathsf{V}]\subseteq\mathsf{W},$

that is, $\mathsf{W}$ is $\mathsf{R}$-subordinated. Thus
$\mathsf{W}\subseteq\mathsf{W}_{\mathsf{R}}$ and
$\mathsf{H}\leq\mathsf{H}_{\mathsf{R}}$. To complete the proof, it remains to
show that
$[\mathsf{W}_{\mathsf{R}},\mathsf{V}]\subseteq\mathrm{Lie}(\mathsf{H_{R}})$.
Recall that

$\mathrm{Lie}(\mathsf{H}_{\mathsf{R}})=\\{x\in\mathrm{Lie}(\mathsf{R})\mid[x,\mathsf{V}]\subseteq\mathsf{W}_{\mathsf{R}}\\},$

hence
$[\mathsf{W}_{\mathsf{R}},\mathsf{V}]\subseteq\mathrm{Lie}(\mathsf{H_{R}})$ if
and only if
$[\mathsf{W}_{\mathsf{R}},\mathsf{V},\mathsf{V}]\subseteq\mathsf{W}_{\mathsf{R}}$.
The lemma is proven. ∎

######  Remark 12.2.

The submodule $\mathsf{W}_{\mathsf{R}}$ can be also defined as

$\mathsf{W}_{\mathsf{R}}=\\{w\in\mathsf{V}\mid\mbox{for any odd positive
integer}\ n,\mbox{ there is }\
[w,\underbrace{\mathsf{V},\ldots,\mathsf{V}}_{n-\mbox{times}}]\subseteq\mathrm{Lie}(\mathsf{R})\\}.$

######  Corollary 12.3.

Every algebraic group superscheme $\mathbb{G}$ contains the largest connected
normal affine group super-subscheme, denoted by $\mathbb{G}_{aff}$. The group
super-subschemes $\mathbb{G}_{aff}$ can be characterized by the property that
$\mathbb{G}/\mathbb{G}_{aff}$ does not contain non-trivial normal connected
affine group super-subschemes.

###### Proof.

Let $\mathsf{G}_{aff}$ denote the largest connected normal affine group
subscheme of $\mathsf{G}$ (cf. [9], Proposition 8.1). Then the sub-pair
$(\mathsf{H}^{0}_{\mathsf{G}_{Aff}},\mathsf{W}_{\mathsf{G}_{Aff}})$ represents
$\mathbb{G}_{aff}$, i.e., $\mathbb{G}_{aff}$ is nothing else but the connected
component of the $\mathsf{G}_{aff}$-radical of $\mathbb{G}$. The second
statement is now apparent. ∎

Let $\mathbb{G}$ be an algebraic group superscheme, represented by a Harish-
Chandra pair $(\mathsf{G},\mathsf{V})$. For every group subscheme $\mathsf{S}$
of $\mathsf{G}$, let $\mathsf{V}_{\mathsf{S}}$ denote the smallest
$\mathsf{G}$-submodule of $\mathsf{V}$ such that
$\mathsf{S}\leq\ker(\mathsf{G}\to\mathrm{GL}(\mathsf{V}/\mathsf{V}_{\mathsf{S}}))$.

######  Lemma 12.4.

Let $\mathsf{R}$ be a normal group subscheme of $\mathsf{G}=\mathbb{G}_{ev}$.
Then $\mathbb{G}$ is $\mathsf{R}$-semisimple if and only if the following
conditions hold:

  1. (1)

For any normal group subscheme $\mathsf{S}$ of $\mathbb{G}$ such that
$\mathsf{S}\leq\mathsf{R}$, there is
$[\mathsf{V},\mathsf{V}_{\mathsf{S}}]\not\subseteq\mathrm{Lie}(\mathsf{S})$;

  2. (2)

There is no non-zero $\mathsf{G}$-submodule $\mathsf{W}$ of $\mathsf{V}$ such
that $[\mathsf{V},\mathsf{W}]=0$.

A connected algebraic group superscheme $\mathbb{G}$ is called _pseudoabelian_
if $\mathbb{G}_{aff}=\mathbb{E}$.

Let $\mathbb{G}$ be a pseudoabelian group superscheme and
$\mathbb{G}\neq\mathbb{E}$. Then $\mathbb{G}_{ev}=\mathsf{G}$ is not affine.
Recall that $\mathsf{G}^{aff}$ denotes the largest affine quotient of
$\mathsf{G}$.

Set $\mathsf{N}=\ker(\mathsf{G}\to\mathsf{G}^{aff})$. Then $\mathsf{N}$ is a
nontrivial anti-affine algebraic group, hence smooth and connected normal
group subscheme of positive dimension (cf. [9], Corollary 8.14 and Proposition
8.37). Since the natural morphism $\mathsf{G}\to\mathrm{GL}(\mathsf{V})$
factors through $\mathsf{G}\to\mathsf{G}^{aff}$, $\mathsf{N}$ acts trivially
on $\mathsf{V}$. Besides, by Corollary 8.13, [9], $\mathsf{N}$ is central in
$\mathsf{G}$.

######  Lemma 12.5.

The group scheme $\mathsf{N}$ is an abelian group variety such that
$\mathsf{G}=\mathsf{G}_{aff}\mathsf{N}$.

###### Proof.

The sub-pair $(\mathsf{N}_{aff},0)$ represents a connected normal affine group
super-subscheme of $\mathbb{G}$. Thus, $\mathsf{N}_{aff}=\mathsf{E}$, and by
Theorem 8.28, [9], $\mathsf{N}$ is an abelian group variety. The group scheme
$\mathsf{G}/\mathsf{G}_{aff}\mathsf{N}$ is a quotient of both
$\mathsf{G}/\mathsf{G}_{aff}$ and $\mathsf{G}/\mathsf{N}$. Recall that
$\mathsf{G}/\mathsf{G}_{aff}$ is an abelian variety by the same Theorem 8.28,
[9]. Therefore, $\mathsf{G}/\mathsf{G}_{aff}\mathsf{N}$ is a complete and
affine connected scheme simultaneously, hence trivial (cf. [9], A.75(g)). ∎

The following example shows that the class of pseudoabelian group superschemes
is extensive.

######  Example 12.6.

For every connected algebraic group scheme $\mathsf{G}$ such that
$\mathsf{N}=\ker(\mathsf{G}\to\mathsf{G}^{aff})$ is an abelian variety, and
$\mathsf{G}=\mathsf{G}_{aff}\mathsf{N}$, there exists a pseudoabelian group
superscheme $\mathbb{G}$ with $\mathbb{G}_{ev}=\mathsf{G}$. Let $\mathsf{W}$
be a faithful $\mathsf{G}_{aff}/(\mathsf{G}_{aff}\cap\mathsf{N})$-module and
$\mathsf{V}=\mathsf{W}\oplus\mathsf{W}^{*}$. Then $\mathsf{V}$ has the natural
structure of $\mathsf{G}$-module for the diagonal action of $\mathsf{G}_{aff}$
and the trivial action of $\mathsf{N}$. Furthermore, we have a bilinear
symmetric $\mathsf{G}$-equivariant map
$\mathsf{V}\times\mathsf{V}\to\mathrm{Lie}(\mathsf{G})$ defined by the rule

$[v,w]=[\phi,\psi]=0,[\phi,v]=\phi(v)x\text{ for }v,w\in\mathsf{W}\text{ and
}\phi,\psi\in\mathsf{W}^{*},$

where $x\in\mathrm{Lie}(\mathsf{N})\setminus 0$.

Then $(\mathsf{G},\mathsf{V})$ is a Harish-Chandra pair representing a group
superscheme $\mathbb{G}$. By Lemma 12.1, the group superscheme $\mathbb{G}$ is
pseudoabelian.

The following theorem is a super-version of well-known Barsotti-Chevalley
theorem (cf. [9], Theorem 8.27).

######  Theorem 12.7.

Let $\mathbb{G}$ be a connected algebraic group superscheme. Then there are
normal group super-subschemes $\mathbb{G}_{1}$ and $\mathbb{G}_{2}$ of
$\mathbb{G}$ such that $\mathbb{G}_{1}$ is affine and connected,
$\mathbb{G}_{2}/\mathbb{G}_{1}$ is a (purely-even) abelian group variety, and
$\mathbb{G}/\mathbb{G}_{2}$ is affine.

###### Proof.

Set $\mathbb{G}_{1}=\mathbb{G}_{aff}$. Then
$\mathbb{H}=\mathbb{G}/\mathbb{G}_{1}$ is a pseudoabelian group superscheme
represented by a Harish-Chandra pair $(\mathsf{H},\mathsf{V})$. As above, let
$\mathsf{N}$ denote the kernel $\ker(\mathsf{H}\to\mathsf{H}^{aff})$. Then
$(\mathsf{N},0)$ represents a normal (purely-even) group subvariety
$\mathbb{N}$ of $\mathbb{H}$. Moreover, Lemma 12.5 implies that
$\mathbb{H}/\mathbb{N}$ is an affine group superscheme. We define
$\mathbb{G}_{2}$ as $\pi^{-1}(\mathbb{N})$, where
$\pi:\mathbb{G}\to\mathbb{G}/\mathbb{G}_{1}$ is the quotient morphism. ∎

######  Corollary 12.8.

Let $\mathbb{G}$ be an abelian group supervariety. Then

  1. (1)

$\mathbb{G}_{ev}\unlhd\mathbb{G}$, and $\mathbb{G}_{ev}$ is an abelian group
supervariety;

  2. (2)

$\mathbb{G}_{aff}\cap\mathbb{G}_{ev}=\mathbb{E}$, hence $\mathbb{G}_{aff}$ is
a (purely-odd) affine unipotent group superscheme;

  3. (3)

$\mathbb{G}/(\mathbb{G}_{aff}\times\mathbb{G}_{ev})$ is a (purely-odd) affine
unipotent group superscheme.

###### Proof.

We have seen that $\mathbb{G}$ is an abelian group supervariety if and only if
$\mathsf{G}=\mathbb{G}_{res}$ is an abelian group variety. Thus $\mathsf{G}$
acts trivially on $\mathsf{V}=\mathrm{Lie}(\mathbb{G})_{1}$ and
$(\mathbb{G}_{1})_{ev}$ is trivial, which implies the first and second
statements. The third statement is clear. ∎

######  Remark 12.9.

If $\mathbb{G}$ is an abelian group supervariety, then the group super-
subscheme $\mathbb{G}_{aff}$ is represented by the pair
$(\mathsf{E},\mathsf{W})$, where
$\mathsf{W}=\\{w\in\mathsf{V}\mid[w,\mathsf{V}]=0\\}$.

###  12.2. Anti-affine group superschemes

An algebraic group superscheme $\mathbb{G}$ is called _anti-affine_ whenever
$\mathcal{O}(G)$ is a Grassman-like superalgebra. Set
$\Phi(\mathbb{G})=(\mathsf{G},\mathsf{V})$. Theorem 11.5 implies that
$\mathbb{G}$ is anti-affine if and only if the algebraic group scheme
$\mathsf{G}$ is (cf. [9], page 39). If
$\mathsf{N}=\ker(\mathsf{G}\to\mathsf{G}^{aff})$, then $(\mathsf{N},0)$
represents a central anti-affine group super-subscheme $\mathbb{N}$ of
$\mathbb{G}$ such that $\mathbb{G}/\mathbb{N}\simeq\mathbb{G}^{aff}$. But an
anti-affine group super-subscheme of $\mathbb{G}$ is not necessarily central
in $\mathbb{G}$, contrary to the purely-even case (cf. [9], Corollary 8.13).
For example, if $\mathbb{G}$ is a psedoabelian group superscheme from Example
12.6, then its group super-subscheme, represented by the pair
$(\mathsf{N},\mathsf{V})$, is anti-affine but not central.

Similarly, an anti-affine group superscheme is no longer commutative, but it
is always nilpotent. If $\mathbb{G}$ is anti-affine, then $\mathbb{G}_{ev}$ is
central, and $\mathbb{G}/\mathbb{G}_{ev}$ is a purely-odd unipotent group
superscheme.

##  13\. Sheaf quotients are superschemes

###  13.1. A reduction

Let $\mathbb{G}$ be an algebraic group superscheme, and $\mathbb{H}$ its group
super-subscheme. They are represented by Harish-Chandra pairs
$(\mathsf{G},\mathsf{V})$ and $(\mathsf{H},\mathsf{W})$, respectively.

Recall that if $\mathbb{H}_{1}$ and $\mathbb{H}_{2}$ are group super-
subschemes of $\mathbb{G}$ such that $\mathbb{H}_{1}\leq\mathbb{H}_{2}$,
$\mathbb{G}/\mathbb{H}_{2}$ is a superscheme, and
$\mathbb{H}_{2}/\mathbb{H}_{1}$ is an affine superscheme, then
$\mathbb{G}/\mathbb{H}_{1}$ is a superscheme as well (see Remark 9.11, [14]).
As above, let $\mathsf{N}$ denote $\ker(\mathsf{G}\to\mathsf{G}^{aff})$.

######  Proposition 13.1.

The following statements are equivalent:

  1. (1)

$\mathbb{G}/\mathbb{H}$ is a superscheme for arbitrary $\mathbb{H}$.

  2. (2)

$\mathbb{G}/\mathbb{H}$ is a superscheme, provided
$\mathsf{G}=\mathsf{M}\times\mathsf{N}$, where $\mathsf{M}$ is an affine group
subscheme of $\mathsf{G}$ such that $\mathsf{H}\leq\mathsf{M}$.

###### Proof.

All we need is to show that $(2)$ implies $(1)$. Let us start with two
elementary observations.

First, suppose $\mathsf{S}$ is a group sub-scheme of $\mathsf{N}$. In that
case, the pair $(\mathsf{S},0)$ represents a central group super-subscheme of
$\mathbb{G}$. Besides, the group superscheme $\mathbb{G}/\mathbb{S}$ is
represented by the pair $(\mathsf{G}/\mathsf{S},\mathsf{V})$ such that
$\ker(\mathsf{G}/\mathsf{S}\to(\mathsf{G}/\mathsf{S})^{aff})=\mathsf{N}/\mathsf{S}$,
i.e., $(\mathsf{G}/\mathsf{S})^{aff}$ is naturally isomorphic to
$\mathsf{G}^{aff}$.

Second, if $\mathsf{R}$ is an affine normal group subscheme of $\mathsf{G}$,
then $(\mathsf{G}/\mathsf{R})_{aff}=\mathsf{G}_{aff}\mathsf{R}/\mathsf{R}$.
Every normal affine connected group subscheme of $\mathsf{G}/\mathsf{R}$ has
the form $\mathsf{L}/\mathsf{R}$, where $\mathsf{L}$ is a normal affine group
subscheme of $\mathbb{G}$ (use Proposition 8.1, [9]). By Proposition 1.52,
[9], $\mathsf{L}^{0}$ is normal, hence $\mathsf{L}^{0}\leq\mathsf{G}_{aff}$.
Since
$(\mathsf{L}/\mathsf{R})/(\mathsf{L}^{0}\mathsf{R}/\mathsf{R})\simeq\mathsf{L}/\mathsf{L}^{0}\mathsf{R}$
is etale and $\mathsf{L}^{0}\mathsf{R}/\mathsf{R}$ is connected, we have
$\mathsf{L}/\mathsf{R}=\mathsf{L}^{0}\mathsf{R}/\mathsf{R}\leq\mathsf{G}_{aff}\mathsf{R}/\mathsf{R}$.

By the first observation, $(\mathsf{H}\cap\mathsf{N},0)$ represents a central
group super-subscheme $\mathbb{L}$ of $\mathbb{G}$ such that
$\mathbb{L}\leq\mathbb{H}$. Since
$\mathbb{G}/\mathbb{H}\simeq(\mathbb{G}/\mathbb{L})/(\mathbb{H}/\mathbb{L})$,
without loss of generality, one can assume that
$\mathsf{H}\cap\mathsf{N}=\mathsf{E}$. In particular, $\mathsf{H}$ is a group
subscheme of $\mathsf{G}^{aff}$, hence affine.

Let $\mathsf{R}$ denote the group scheme
$\mathsf{G}_{aff}\mathsf{H}\cap\mathsf{N}$. The second observation implies
that $(\mathsf{G}/\mathsf{R})_{aff}=\mathsf{G}_{aff}\mathsf{R}/\mathsf{R}$ and
$(\mathsf{G}_{aff}\mathsf{H})/\mathsf{R}\cap\mathsf{N}/\mathsf{R}=\mathsf{E}$.
Since $\mathbb{H}\mathbb{R}/\mathbb{H}\simeq\mathbb{R}$ is affine,
$\mathbb{G}/\mathbb{H}$ is a superscheme provided
$\mathbb{G}/\mathbb{H}\mathbb{R}\simeq(\mathbb{G}/\mathbb{R})/(\mathbb{H}\mathbb{R}/\mathbb{R})$
is. Therefore, one can assume that
$\mathsf{G}_{aff}\mathsf{H}\cap\mathsf{N}=\mathsf{E}$. Arguing as in Lemma
12.5, one can show that $\mathsf{N}$ is an abelian variety and
$\mathsf{G}^{0}=\mathsf{G}_{aff}\mathsf{N}$. In particular, $\mathsf{N}$ can
be identified with $(\mathsf{G}/\mathsf{G}_{aff}\mathsf{H})^{0}$. By Theorem
1.1, [2], there is a finite, hence affine, group subscheme $\mathsf{F}$ of
$\mathsf{G}/\mathsf{G}_{aff}\mathsf{H}$ such that
$\mathsf{F}\mathsf{N}=\mathsf{G}/\mathsf{G}_{aff}\mathsf{H}$. The inverse
image of $\mathsf{F}$ in $\mathsf{G}$ is an affine group subscheme
$\mathsf{M}$ such that $\mathsf{G}=\mathsf{M}\mathsf{N}$. Repeating the above
arguments with $\mathsf{R}=\mathsf{M}\cap\mathsf{N}$, one can reduce the
general case to $\mathsf{M}\cap\mathsf{N}=\mathsf{E}$. The proposition is
proven. ∎

###  13.2. Sheaf quotient

Let $\mathbb{G}$ be an algebraic group superscheme, represented by a Harish-
Chandra pair $(\mathsf{G},\mathsf{V})$. Let $\mathbb{H}$ be its closed group
super-subscheme, represented by a sub-pair $(\mathsf{H},\mathsf{W})$. In this
section, we prove the following theorem.

######  Theorem 13.2.

A sheaf quotient $\mathbb{G}/\mathbb{H}$ is a superscheme of finite type.

The proof will be given in the series of lemmas. Without loss of generality,
one can assume that $\mathbb{G}$ and $\mathbb{H}$ satisfy the condition $(2)$
in Proposition 13.1.

Since $\mathsf{X}=\mathsf{G}/\mathsf{H}$ is a scheme of finite type (see
Theorem 7.35 and Remark 7.36 from [9]), any open affine subscheme
$\mathsf{U}^{\prime}$ of $\mathsf{G}/\mathsf{H}$ has a form
$\mathsf{U}/\mathsf{H}$, where $\mathsf{U}$ is an open affine
$\mathsf{H}$-saturated subscheme of $\mathsf{G}$ (cf. [8], I.5.7.1(1)).

Set $\mathrm{Sp}(A)=\mathsf{U}$ and $\mathrm{Sp}(D)=\mathsf{H}$. Then
$A^{\prime}=\mathcal{O}(\mathsf{U}^{\prime})\simeq A^{D}$.

Recall that the superschemes $\mathbb{G}$ and $\mathbb{H}$ can be represented
as $\mathsf{G}{\bf E}$ and $\mathsf{H}{\bf E}^{\prime}$, respectively.

######  Lemma 13.3.

The subfunctor $\mathbb{U}=\mathsf{U}{\bf E}={\bf E}\mathsf{U}$ is an open
affine $\mathbb{H}$-saturated super-subscheme of $\mathbb{G}$.

###### Proof.

It is clear that $\mathsf{U}{\bf E}$ is open in $\mathbb{G}$. Further, we have
$\mathbb{U}=\mathsf{U}{\bf E}=\mathsf{U}\mathsf{S}{\bf
E}=\mathsf{U}\mathbb{S}$. Since $\mathbb{S}$ is a normal group subfunctor, we
also have $\mathbb{U}=\mathbb{S}\mathsf{U}={\bf E}\mathsf{S}\mathsf{U}={\bf
E}\mathsf{U}$. Finally, for every $hx\in\mathbb{H}$, where $h\in\mathsf{H}$
and $x\in{\bf E}^{\prime}$, there is

$\mathbb{U}(hx)=\mathsf{U}\mathbb{S}(hx)=(\mathsf{U}h)(\mathbb{S}^{h^{-1}}x)\subseteq\mathsf{U}\mathbb{S}=\mathbb{U}.$

The lemma is proven. ∎

Set $\mathrm{SSp}(\mathbb{A})=\mathbb{U}$ and
$\mathrm{SSp}(\mathbb{D})=\mathbb{H}$. Let
$\tau:\mathbb{A}\to\mathbb{A}\otimes\mathbb{D}$ be the corresponding
$\mathbb{D}$-coideal superalgebra map.

Recall that $\mathsf{G}=\mathsf{M}\times\mathsf{N}$, where
$\mathsf{H}\leq\mathsf{M}$ and $\mathsf{M}$ is affine. Thus $\mathsf{U}$ can
be chosen so that $\mathsf{U}=\mathsf{U}_{aff}\times\mathsf{U}_{ab}$, where
$\mathsf{U}_{aff}$ is an open affine subscheme of $\mathsf{M}$ and
$\mathsf{U}_{ab}$ is an open affine subscheme of $\mathsf{N}$. Moreover, the
algebra $A$ is isomorphic to $A_{aff}\otimes A_{ab}$ as a $D$-comodule, where
$A_{aff}\simeq K[\mathsf{U}_{aff}],A_{ab}\simeq K[\mathsf{U}_{ab}]$, and
$A_{ab}$ is a trivial $D$-comodule.

Following notations from [13], one can define a morphism of
$\mathbb{D}$-coideal superalgebras

$\omega_{\theta}:\mathbb{A}\stackrel{{\scriptstyle\tau^{\prime}}}{{\to}}(\wedge(\mathsf{V^{*}})\otimes
A)\square_{D}\mathbb{D}\stackrel{{\scriptstyle\wedge(\theta^{\prime})\square_{D}\mathrm{id}_{\mathbb{D}}}}{{\to}}(A\otimes\wedge(\mathsf{Z}))\square_{D}\mathbb{D},$

where $\mathsf{Z}=\ker(\mathsf{V^{*}}\to\mathsf{W^{*}})$. The first morphism
makes sense since $\tau$ factors through the natural embedding
$\mathbb{A}\square_{D}\mathbb{D}\subseteq\mathbb{A}\otimes\mathbb{D}$. The
proofs of Lemma 4.1, Proposition 4.2, and Lemma 4.4 in [13] can be repeated
verbatim. Only the definition of $\theta^{\prime}$ and the proof of Lemma 4.6,
[13] needs a commentary. The structure of the right $\mathsf{G}$-module of
$\mathsf{V}^{*}$ is completely defined by the structure of $\mathsf{M}$-module
because $\mathsf{N}$ acts trivially on $\mathsf{V}^{*}$. Therefore, one can
define the map

$\kappa=\kappa_{A}:A\otimes\mathsf{V}^{*}\to\mathsf{V}^{*}\otimes A$

by

$a_{1}\otimes a_{2}\otimes v^{*}\mapsto v^{*}_{(0)}\otimes
a_{1}j(v^{*}_{(1)})\otimes a_{2},$

where $a_{1}\in A_{aff},a_{2}\in A_{ab}$, $v^{*}\mapsto v_{(0)}\otimes
v^{*}_{(1)}$ is the corresponding comodule map
$\mathsf{V}^{*}\to\mathsf{V}^{*}\otimes K[\mathsf{M}]$, and
$j:K[\mathsf{M}]\to A_{aff}$ is dual to the open immersion
$\mathsf{U}_{aff}\to\mathsf{M}$.

We will show that $\omega_{\theta}$ is an isomorphism.

######  Remark 13.4.

The definition of $\omega_{\theta}$ is consistent with a base change, and once
needed, $K$ can be replaced by a suitable field extension $L\supseteq K$.

It has been proven that
$\mathsf{gr}(\mathbb{G})\simeq\mathsf{G}\ltimes\mathbb{G}_{odd}$ and
$\mathsf{gr}(\mathbb{H})\simeq\mathsf{H}\ltimes\mathbb{H}_{odd}$, where
$\mathbb{G}_{odd}\simeq\mathrm{SSp}(\Lambda(\mathsf{V}^{*}))$,
$\mathbb{H}_{odd}\simeq\mathrm{SSp}(\Lambda(\mathsf{W}^{*}))$ and
$\mathsf{V}^{*}$ and $\mathsf{W}^{*}$ consist of primitive elements.

Using Proposition 9.3 and arguing as in Proposition 9.5, one sees that
$\mathsf{gr}(\mathbb{U})\simeq\mathrm{SSp}(\mathsf{gr}(\mathbb{A}))$ is an
open $\mathsf{gr}(\mathbb{H})$-saturated super-subscheme of
$\mathsf{gr}(\mathbb{G})$.

######  Proposition 13.5.

The following statements hold:

  1. (1)

$\mathsf{gr}(\mathbb{U})\simeq\mathsf{U}\mathbb{G}_{odd}=\mathbb{G}_{odd}\mathsf{U}$.

  2. (2)

As the right and left $D$-coideal superalgebras, $\mathsf{gr}(\mathbb{A})$ and
$\mathsf{gr}(\mathbb{D})$ are naturally isomorphic to $\mathbb{A}$ and
$\mathbb{D}$, respectively.

  3. (3)

$\mathsf{gr}(\tau^{\prime})=\mathsf{gr}(\tau)^{\prime}$.

###### Proof.

The geometric counterpart of $\mathsf{U}\mathbb{G}_{odd}$ is the open super-
subscheme $q_{G}^{-1}(U_{ev})\simeq U_{ev}\times G_{odd}$, where $U$ is the
geometric counterpart of $\mathbb{U}$ in $G$. Since
$\mathsf{gr}(U)^{e}=U^{e}=U_{ev}^{e}=(q_{G}^{-1}(U_{ev}))^{e}$, the condition
$(1)$ follows.

Further, the right $\mathsf{gr}(\mathbb{D})$-coideal superalgebra structure on
$\mathsf{gr}(\mathbb{A})$ is defined by a superalgebra morphism
$\mathsf{gr}(\mathbb{A})\stackrel{{\scriptstyle\mathsf{gr}(\tau)}}{{\to}}\mathsf{gr}(\mathbb{A}\otimes\mathbb{D})\simeq\mathsf{gr}(\mathbb{A})\otimes\mathsf{gr}(\mathbb{D})$.
More precisely, for an element $a\in I_{\mathbb{A}}^{n}$, let
$\tau(a)=\sum_{0\leq k\leq n}a_{(0),k}\otimes a_{(1),n-k}$ be adapted to the
$I_{\mathbb{A}\otimes\mathbb{D}}$-adic filtration. That is, $a_{(0),k}\in
I^{k}_{\mathbb{A}}$ and $a_{(1),n-k}\in I^{n-k}_{\mathbb{D}}$ for every $0\leq
k\leq n$. Then $\mathsf{gr}(\tau)$ is defined by

$a+I_{\mathbb{A}}^{n+1}\mapsto\sum_{0\leq k\leq
n}(a_{(0),k}+I_{\mathbb{A}}^{k+1})\otimes(a_{(1),n-k}+I_{\mathbb{D}}^{n-k+1}).$

It follows that the right $D$-coideal superalgebra structure of
$\mathsf{gr}(\mathbb{A})$ is defined as

$a+I_{\mathbb{A}}^{n+1}\mapsto(a_{(0),n}+I_{\mathbb{A}}^{n+1})\otimes(a_{(1),0}+I_{\mathbb{D}}),$

where $a\mapsto a_{(0),n}\otimes(a_{(1),0}+I_{\mathbb{D}})$ is the right
$D$-coideal superalgebra map of $\mathbb{A}$.

On the other hand, if we identify $\mathbb{A}$ with
$\Lambda(\mathsf{V}^{*})\otimes A$, then the latter map is $x\otimes a\mapsto
x\otimes a_{(0)}\otimes a_{(1)}$, where $x\in\Lambda(\mathsf{V}^{*}),a\in A$
and $a\mapsto a_{(0)}\otimes a_{(1)}$ is the $D$-coideal algebra map of $A$.
Moreover, we have $I_{\mathbb{A}}^{k}=(\oplus_{s\geq
k}\Lambda^{s}(\mathsf{V}^{*}))\otimes A$ for $0\leq k\leq\dim\mathsf{V}$. Then
$\mathbb{A}$ and $\mathsf{gr}(\mathbb{A})$ are isomorphic as $D$-coideal
superalgebras. The case of $\mathbb{D}$ is similar.

Furthermore, $\mathbb{A}\square_{D}\mathbb{D}$ is isomorphic to
$\Lambda(\mathsf{V}^{*})\otimes B\otimes\Lambda(\mathsf{W}^{*})$, where

$B=\\{a_{(0)}\otimes a_{(1)}\mid a\in A\\}\subseteq A\otimes D.$

In particular,
$I^{k}_{\mathbb{A}\square_{D}\mathbb{D}}=\mathbb{A}\square_{D}\mathbb{D}\cap
I^{k}_{\mathbb{A}\otimes\mathbb{D}}$ for $0\leq
k\leq\dim\mathsf{V}+\dim\mathsf{W}$, and we obtain a natural isomorphism
$\mathsf{gr}(\mathbb{A}\square_{D}\mathbb{D})\simeq\mathsf{gr}(\mathbb{A})\square_{D}\mathsf{gr}(\mathbb{D})$
that makes the diagram

$\begin{array}[]{ccccc}\mathsf{gr}(\mathbb{A})&\stackrel{{\scriptstyle\mathsf{gr}(\tau^{\prime})}}{{\to}}&\mathsf{gr}(\mathbb{A}\square_{D}\mathbb{D})&\to&\mathsf{gr}(\mathbb{A}\otimes\mathbb{D})\\\
\parallel&&\downarrow&&\downarrow\\\
\mathsf{gr}(\mathbb{A})&\stackrel{{\scriptstyle\mathsf{gr}(\tau)^{\prime}}}{{\to}}&\mathsf{gr}(\mathbb{A})\square_{D}\mathsf{gr}(\mathbb{D})&\to&\mathsf{gr}(\mathbb{A})\otimes\mathsf{gr}(\mathbb{D})\end{array}$

commutative. The proposition is proven. ∎

######  Lemma 13.6.

The morphism
$\mathsf{gr}(\wedge(\theta^{\prime})\square_{D}\mathrm{id}_{\mathbb{D}})$ is
identified with
$\wedge(\theta^{\prime})\square_{D}\mathrm{id}_{\mathsf{gr}(\mathbb{D})}$.

###### Proof.

Recall that the morphism $\wedge(\theta^{\prime})$ is defined as

$v^{*}_{1}\wedge\ldots\wedge v^{*}_{k}\otimes
a\mapsto\sum_{i_{1},\ldots,i_{k}}a_{i_{1}}\ldots a_{i_{k}}a\otimes
z_{i_{1}}\wedge\ldots\wedge z_{i_{k}},$

where $\theta^{\prime}(v^{*}_{s})=\sum_{i_{s}}a_{i_{s}}\otimes z_{i_{s}}$ for
$1\leq s\leq k$. Moreover, the morphism
$\wedge(\theta^{\prime})\square\mathrm{id}$ is defined as

$v^{*}_{1}\wedge\ldots\wedge v^{*}_{k}\otimes a_{(0)}\otimes
a_{(1)}\otimes\wedge w^{*}_{1}\wedge\ldots\wedge w^{*}_{l}\mapsto$
$\sum_{i_{1},\ldots,i_{k}}a_{i_{1}}\ldots a_{i_{k}}a_{(0)}\otimes
z_{i_{1}}\wedge\ldots\wedge z_{i_{k}}\otimes a_{(1)}\otimes\wedge
w^{*}_{1}\wedge\ldots\wedge w^{*}_{l},$

and is compatible with the corresponding filtrations of the superalgebras
$(\wedge(\mathsf{V}^{*})\otimes A)\square_{D}\mathbb{D}$ and
$(A\otimes\wedge(\mathsf{Z}))\square_{D}\mathbb{D}$. ∎

The following lemma is now obvious.

######  Lemma 13.7.

The morphism $\mathsf{gr}(\omega_{\theta})$ can be naturally identified with

$\mathsf{gr}(\mathbb{A})\stackrel{{\scriptstyle\mathsf{gr}(\tau)^{\prime}}}{{\to}}(\wedge(\mathsf{V}^{*})\otimes
A)\square_{D}\mathsf{gr}(\mathbb{D})\stackrel{{\scriptstyle\wedge(\theta^{\prime})\square_{D}\mathrm{id}_{\mathsf{gr}(\mathbb{D})}}}{{\to}}(A\otimes\wedge(\mathsf{Z}))\square_{D}\mathsf{gr}(\mathbb{D}).$

Let $\theta$ be a retract of the inclusion $A\otimes\mathsf{Z}\to
A\otimes\mathsf{V}^{*}$ from Lemma 4.4 , [13]. Let
$\mathsf{V}^{*}=\mathsf{Z}\oplus\mathsf{T}$, and elements $t_{j}$ form a basis
of $\mathsf{T}$ for $1\leq j\leq s=\dim\mathsf{W}^{*}$. The free $A$-module
$\ker\theta$ has a basis consisting of the elements $t^{\prime}_{j}$ such that
$t^{\prime}_{j}=t_{j}\pmod{A\otimes\mathsf{Z}}$.

Choose a closed point $x$ in $\mathsf{U}$. By Remark 13.4, one can assume that
$x\in\mathsf{U}(K)$. That is, $x$ is an algebra morphism $A\to K$.

The natural embedding
$\mathbb{H}_{odd}\to\mathsf{U}\times\mathbb{G}_{odd}\simeq\mathsf{gr}(\mathbb{U})$,
defined as $h\mapsto(x,h)$, is dual to the superalgebra morphism induced by
$x$ and $\mathsf{V}^{*}\to\mathsf{W}^{*}$. The natural (right-hand side)
action of group $\mathbb{H}_{odd}$ on $\mathsf{gr}(\mathbb{U})$ is defined by
a coideal superalgebra map

$z\mapsto z\otimes 1\text{ and }t^{\prime}_{j}\mapsto t^{\prime}_{j}\otimes
1+1\otimes\overline{t_{j}}\text{ for }z\in\mathsf{Z},$

where $\overline{t_{j}}$ is the image of $t_{j}$ in $\mathsf{W}^{*}$. Finally,
the $A$-superalgebra morphism
$\wedge(\theta):A\otimes\Lambda(\mathsf{V}^{*})\to
A\otimes\Lambda(\mathsf{Z})$ induces the embedding ${\bf
s}:\mathsf{U}\times\mathrm{SSp}(\Lambda(\mathsf{Z}))\to\mathsf{gr}(\mathbb{U})$.

######  Lemma 13.8.

There is a natural isomorphism

$(\mathsf{U}\times\mathrm{SSp}(\Lambda(\mathsf{Z})))\times\mathbb{H}_{odd}\simeq\mathsf{gr}(\mathbb{U}),$

induced by the above two embeddings and the multiplication map.

###### Proof.

Choose a basis $z_{1},\ldots,z_{t-s}$ of $\mathsf{Z}$. We use notations from
Proposition 10.6. Any couple
$(g,h)\in(\mathsf{U}\times\mathrm{SSp}(\Lambda(\mathsf{Z})))\times\mathbb{H}_{odd}$
is taken to a superalgebra morphism

$(g,h)(az_{I}t^{\prime}_{J})=\sum_{S\subseteq
J}g(\wedge(\theta)(az^{I}t^{\prime S}))h(\overline{t}^{J\setminus
S})=g(az_{I})h(\overline{t}_{J}).$

The proof concludes by the observation that
$A\otimes\Lambda(\mathsf{V}^{*})\simeq(A\otimes\Lambda(\mathsf{Z}))\otimes\Lambda(\mathsf{T}^{\prime})$,
where $\mathsf{T}^{\prime}$ is the $K$-span of $t^{\prime}_{j}$ for $1\leq
j\leq s$. ∎

######  Lemma 13.9.

The superalgebra morphism $\mathsf{gr}(\omega_{\theta})$ is an isomorphism,
and therefore $\omega_{\theta}$ is as well.

###### Proof.

Consider the following sequence of superscheme morphisms

$(\star)\ \mathsf{gr}(\mathbb{U})\stackrel{{\scriptstyle\bf
a}}{{\leftarrow}}\mathsf{gr}(\mathbb{U})\times^{\mathsf{H}}\mathsf{gr}(\mathbb{H})\stackrel{{\scriptstyle\bf
b}}{{\leftarrow}}(\mathsf{U}\times\mathbb{G}_{odd})\times^{\mathsf{H}}\mathsf{gr}(\mathbb{H})\stackrel{{\scriptstyle\bf
c}}{{\leftarrow}}(\mathsf{U}\times\mathrm{SSp}(\Lambda(\mathsf{Z})))\times^{\mathsf{H}}\mathsf{gr}(\mathbb{H})\simeq$
$\mathrm{SSp}((A\otimes\wedge(\mathsf{Z}))\square_{D}\mathsf{gr}(\mathbb{D})).$

The morphism $\bf a$ is induced by $(u,h)\mapsto uh$ for
$u\in\mathsf{gr}(\mathbb{U})$ and $h\in\mathsf{gr}(\mathbb{H})$. Since
$\mathsf{gr}(\mathbb{H})\simeq\mathsf{H}\times\mathbb{H}_{odd}$, the
superscheme
$\mathsf{gr}(\mathbb{U})\times^{\mathsf{H}}\mathsf{gr}(\mathbb{H})$ is
canonically isomorphic to $\mathsf{gr}(\mathbb{U})\times\mathbb{H}_{odd}$,
hence it is affine. Lemma 3.1 implies that
$\mathsf{gr}(\mathbb{U})\times^{\mathsf{H}}\mathsf{gr}(\mathbb{H})\simeq\mathrm{SSp}(\mathsf{gr}(\mathbb{A})\square_{D}\mathsf{gr}(\mathbb{D}))$
and the morphism $\bf a$ is dual to $\mathsf{gr}(\tau)^{\prime}$.

Similarly, we have

$(\mathsf{U}\times\mathbb{G}_{odd})\times^{\mathsf{H}}\mathsf{gr}(\mathbb{H})\simeq(\mathsf{U}\times\mathbb{G}_{odd})\times\mathbb{H}_{odd}\simeq\mathrm{SSp}((\Lambda(\mathsf{V}^{*})\otimes
A)\square_{D}\mathsf{gr}(\mathbb{D})),$

and the isomorphism $\bf b$ is dual to the superalgebra isomorphism induced by
$\kappa^{-1}$,

$(\mathsf{U}\times\mathrm{SSp}(\Lambda(\mathsf{Z})))\times^{\mathsf{H}}\mathsf{gr}(\mathbb{H})\simeq(\mathbb{G}_{odd}\times\mathsf{U})\times\mathbb{H}_{odd}\simeq\mathrm{SSp}((A\otimes\Lambda(\mathsf{Z}))\square_{D}\mathsf{gr}(\mathbb{D}))$

and the morphism ${\bf c}={\bf
s}\times^{\mathsf{H}}\mathrm{id}_{\mathsf{gr}(\mathbb{H})}$ is just
$\mathrm{SSp}(\wedge(\theta)\square\mathrm{id}_{\mathsf{gr}(\mathbb{D})})$.
Thus the composition ${\bf a}{\bf b}{\bf c}$ coincides with
$\mathrm{SSp}(\mathsf{gr}(\omega_{\theta}))$. The above isomorphisms allow us
to identify ${\bf a}{\bf b}{\bf c}$ with the isomorphism from Lemma 13.8.
Thus, the lemma follows. ∎

######  Lemma 13.10.

The sheaf quotient $\mathbb{U}/\mathbb{H}$ is an affine superscheme.

###### Proof.

By Theorem 3.1 (1), [13], we need to show that the superalgebra morphism
$\alpha:\mathbb{A}\otimes\mathbb{A}\to\mathbb{A}\otimes\mathbb{D}$ given by
$a\otimes a^{\prime}\mapsto\sum aa^{\prime}_{(0)}\otimes a^{\prime}_{(1)}$, is
surjective. Equivalently, the functor morphism
$\mathrm{SSp}(\alpha):\mathbb{U}\times\mathbb{H}\to\mathbb{U}\times\mathbb{U}$
given by $(u,h)\mapsto(u,uh),$ maps $\mathbb{U}\times\mathbb{H}$
isomorphically onto a closed super-subscheme of $\mathbb{U}\times\mathbb{U}$.
Define a functor morphism ${\bf p}:\mathbb{G}\times\mathbb{G}\to\mathbb{G}$ by
$(g,g^{\prime})\mapsto g^{-1}g^{\prime}$. Then
$\mathrm{SSp}(\alpha)(\mathbb{U}\times\mathbb{H})=(\mathbb{U}\times\mathbb{U})\cap{\bf
p}^{-1}(\mathbb{H})$, hence it is closed in $\mathbb{U}\times\mathbb{U}$. The
lemma follows. ∎

Let $\mathsf{V}^{\prime}$ be an open affine subscheme of
$\mathsf{U}^{\prime}$. As above, $\mathsf{V}^{\prime}=\mathsf{V}/\mathsf{H}$,
where $\mathsf{V}\simeq\mathrm{Sp}(B)$ is an affine subscheme of $\mathsf{U}$
and $B^{\prime}=\mathcal{O}(\mathsf{V}^{\prime})\simeq B^{D}$. Set
$\mathbb{V}=\mathsf{V}{\bf E}$. Then $\mathbb{V}$ is an open affine
$\mathbb{H}$-saturated super-subscheme of $\mathbb{U}$.

######  Lemma 13.11.

$\mathbb{V}/\mathbb{H}$ is an open affine super-subscheme of
$\mathbb{U}/\mathbb{H}$.

###### Proof.

The inclusion $\mathsf{V}^{\prime}\subseteq\mathsf{U}^{\prime}$ is defined by
a morphism $\phi:A\to B$ of $D$-coideal algebras. By Lemma 3.5, [14], there
are elements $a_{1},\ldots,a_{k}\in A^{\prime}$ such that $\sum_{1\leq i\leq
k}B^{\prime}\phi(a_{i})=B^{\prime}$, and for each $i$, the induced morphism
$A^{\prime}_{a_{i}}\to B^{\prime}_{\phi(a_{i})}$ is an isomorphism. Thus,
$B=\sum_{1\leq i\leq k}B\phi(a_{i})=B$ and $A_{a_{i}}\simeq B_{\phi(a_{i})}$
for $1\leq i\leq k$. There is
$\mathbb{B}=\mathcal{O}(\mathbb{V})\simeq(B\otimes\Lambda(\mathsf{Z}))\square_{D}\mathbb{D}$,
and the morphism of $\mathbb{D}$-coideal superalgebras
$\mathbb{A}\to\mathbb{B}$, dual to the inclusion
$\mathbb{V}\subseteq\mathbb{U}$, is
$(\phi\otimes\mathrm{id}_{\Lambda(\mathsf{Z})})\square_{D}\mathrm{id}_{\mathbb{D}}$.
Moreover, by Proposition 4.2(1), [13],
$\mathbb{V}/\mathbb{H}\simeq\mathrm{SSp}((B\otimes\Lambda(\mathsf{Z}))^{D})$
and
$\mathbb{U}/\mathbb{H}\simeq\mathrm{SSp}((A\otimes\Lambda(\mathsf{Z}))^{D})$.
Therefore, the inclusion $\mathbb{V}/\mathbb{H}\subseteq\mathbb{U}/\mathbb{H}$
is dual to $\phi\otimes\mathrm{id}_{\Lambda(\mathsf{Z})}$, and the
superalgebras $\mathcal{O}(\mathbb{U}/\mathbb{H})$ and
$\mathcal{O}(\mathbb{V}/\mathbb{H})$ satisfy the conditions of Lemma 3.5,
[14], for the elements $a_{i}\otimes 1$, where $1\leq i\leq k$. The lemma is
proven. ∎

######  Corollary 13.12.

The statement of Lemma 13.11 remains valid for an arbitrary open subscheme
$\mathsf{V}^{\prime}$ of $\mathsf{U}^{\prime}$.

###### Proof.

Indeed, $\mathsf{V}^{\prime}$ is a finite union of open affine subschemes. ∎

Fix a finite open covering of $\mathsf{G}/\mathsf{H}$ by affine subschemes
$\mathsf{U}_{i}^{\prime}$ for $1\leq i\leq t$. Then there is a collection of
affine superschemes $\mathbb{U}_{i}^{\prime}=\mathbb{U}_{i}/\mathbb{H}$ and
quotient morphisms $\mathbb{U}_{i}\to\mathbb{U}_{i}^{\prime}$ such that for
every pair of indices $1\leq i\neq j\leq t$ we have
$(\mathbb{U}_{i}\cap\mathbb{U}_{j})/\mathbb{H}=\mathbb{U}_{i}^{\prime}\cap\mathbb{U}_{j}^{\prime}$.

Consider the collection of corresponding geometric superschemes
$U_{i}^{\prime}$. Since the underlying topological space of each
$U^{\prime}_{i}$ coincides with $(\mathsf{U}_{i}^{\prime})^{e}$ (see
Proposition 4.2, [13]), one can construct a geometric superscheme $Z$ with
$Z^{e}=\mathsf{X}^{e}$ as follows. For any open subset $W\subseteq Z^{e}$, we
define

$\mathcal{O}_{Z}(W)=\ker(\prod_{1\leq i\leq
t}\mathcal{O}_{U^{\prime}_{i}}(W\cap(U_{i}^{\prime})^{e})\rightrightarrows\prod_{1\leq
i\neq j\leq t}\mathcal{O}_{U_{i}^{\prime}\cap
U_{j}^{\prime}}(W\cap(U^{\prime}_{i}\cap U^{\prime}_{j})^{e})),$

where $U^{\prime}_{i}\cap U^{\prime}_{j}$ denotes the geometric counterpart of
$\mathbb{U}_{i}^{\prime}\cap\mathbb{U}_{j}^{\prime}$. It is clear that
$\mathcal{O}_{Z}$ is a superalgebra sheaf such that
$\mathcal{O}_{Z}|_{(U^{\prime}_{i})^{e}}\simeq\mathcal{O}_{U^{\prime}_{i}}$
for $1\leq i\leq t$.

The following lemma concludes the proof of the main theorem.

######  Lemma 13.13.

There is the unique morphism $\mathbb{G}\to\mathbb{Z}$, the restriction of
which on each $\mathbb{U}_{i}$ coincides with
$\mathbb{U}_{i}\to\mathbb{U}_{i}^{\prime}\subseteq\mathbb{Z}$. In particular,
$\mathbb{Z}\simeq\mathbb{G}/\mathbb{H}$.

###### Proof.

The open immersions $U^{\prime}_{i}\to Z$ induce the open embeddings
$\mathbb{U}^{\prime}_{i}\to\mathbb{Z}$. Thus, $\mathbb{U}^{\prime}_{i}$ form
an open covering of $\mathbb{Z}$. Since $\mathbb{Z}$ is a local functor, the
collection of morphisms $\mathbb{U}_{i}\to\mathbb{U}_{i}^{\prime}$ uniquely
extends to a morphism $\mathbb{G}\to\mathbb{Z}$, which is constant on
$\mathbb{H}$-orbits. Moreover, any morphism $\mathbb{G}\to\mathbb{Y}$, where
$\mathbb{Y}$ is a faisceau, is uniquely defined by its restrictions on
$\mathbb{U}_{i}$ for $1\leq i\leq t$. Since such morphisms
$\mathbb{U}_{i}\to\mathbb{Y}$ factor through the morphisms
$\mathbb{U}_{i}^{\prime}\to\mathbb{Y}$, $\mathbb{G}\to\mathbb{Y}$ factors
through the unique morphism $\mathbb{Z}\to\mathbb{Y}$. Lemma is proven. ∎

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