Geometrical Construction of

Quotients G /H in Super-symmetry
Akira Masuoka from Tsukuba
Based on joint work with Yuta Takahashi
arXiv:1808.05753

Okayama Science Univ., September 22, 2018

Changing my attitude to students; G /H is ...

Changing my attitude to students; G /H is ...
To junior undergraduates: “G /H is easy to understand. Unless
your are able to understand it,...”

Changing my attitude to students; G /H is ...
To junior undergraduates: “G /H is easy to understand. Unless
your are able to understand it,...”To senior undergraduates: “G /H is diﬃcult to understand. You
have to study hard to understand it.”

Changing my attitude to students; G /H is ...
To junior undergraduates: “G /H is easy to understand. Unless
your are able to understand it,...”To senior undergraduates: “G /H is diﬃcult to understand. You
have to study hard to understand it.”
To graduates: “G /H is the simplest among the diﬃcult quotients
X /G , because the relevant group-action is free.”Changing my attitude to students; G /H is ...
To junior undergraduates: “G /H is easy to understand. Unless
your are able to understand it,...”To senior undergraduates: “G /H is diﬃcult to understand. You
have to study hard to understand it.”
To graduates: “G /H is the simplest among the diﬃcult quotients
X /G , because the relevant group-action is free.”In any way the following is known: given an aﬃne algebraic group
G and its closed subgroup H, there exits uniquely a scheme G /Hwhich is characterized by the co-equalizer diagram of schemes
G × H ⇒ G → G /H.

Changing my attitude to students; G /H is ...
To junior undergraduates: “G /H is easy to understand. Unless
your are able to understand it,...”To senior undergraduates: “G /H is diﬃcult to understand. You
have to study hard to understand it.”
To graduates: “G /H is the simplest among the diﬃcult quotients
X /G , because the relevant group-action is free.”In any way the following is known: given an aﬃne algebraic group
G and its closed subgroup H, there exits uniquely a scheme G /Hwhich is characterized by the co-equalizer diagram of schemes
G × H ⇒ G → G /H.
Question (Brundan 2006): Is this generalized to the super-context?

Changing my attitude to students; G /H is ...
To junior undergraduates: “G /H is easy to understand. Unless
your are able to understand it,...”To senior undergraduates: “G /H is diﬃcult to understand. You
have to study hard to understand it.”
To graduates: “G /H is the simplest among the diﬃcult quotients
X /G , because the relevant group-action is free.”In any way the following is known: given an aﬃne algebraic group
G and its closed subgroup H, there exits uniquely a scheme G /Hwhich is characterized by the co-equalizer diagram of schemes
G × H ⇒ G → G /H.
Question (Brundan 2006): Is this generalized to the super-context?
Answer: Yes (M-Zubkov 2011):

Changing my attitude to students; G /H is ...
To junior undergraduates: “G /H is easy to understand. Unless
your are able to understand it,...”To senior undergraduates: “G /H is diﬃcult to understand. You
have to study hard to understand it.”
To graduates: “G /H is the simplest among the diﬃcult quotients
X /G , because the relevant group-action is free.”In any way the following is known: given an aﬃne algebraic group
G and its closed subgroup H, there exits uniquely a scheme G /Hwhich is characterized by the co-equalizer diagram of schemes
G × H ⇒ G → G /H.
Question (Brundan 2006): Is this generalized to the super-context?
Answer: Yes (M-Zubkov 2011): Yes. Moreover, the structure sheafof the super-symmetric quotient G /H is...(M-Takahashi, this time)

Super-symmetry
Everything is over a ﬁxed ﬁeld k of characteristic ̸= 2.

Super-symmetry
Everything is over a ﬁxed ﬁeld k of characteristic ̸= 2.
To deﬁne objects over k, such as comm. algebra, Hopf algebra H,
Lie algebra g, you needSuper-symmetry
Everything is over a ﬁxed ﬁeld k of characteristic ̸= 2.
To deﬁne objects over k, such as comm. algebra, Hopf algebra H,
Lie algebra g, you need
▶ the tensor product ⊗, which is used to present their structures
[ , ]
ϵ
∆

so as k ←− H −→ H ⊗ H, g ⊗ g −→ g; andSuper-symmetry
Everything is over a ﬁxed ﬁeld k of characteristic ̸= 2.
To deﬁne objects over k, such as comm. algebra, Hopf algebra H,
Lie algebra g, you need
▶ the tensor product ⊗, which is used to present their structures
[ , ]
ϵ
∆

so as k ←− H −→ H ⊗ H, g ⊗ g −→ g; and▶ the obvious symmetry V ⊗ W → W ⊗ V , v ⊗ w 7→ w ⊗ v ,which is involved in their axioms.

Super-symmetry
Everything is over a ﬁxed ﬁeld k of characteristic ̸= 2.
To deﬁne objects over k, such as comm. algebra, Hopf algebra H,
Lie algebra g, you need
▶ the tensor product ⊗, which is used to present their structures
[ , ]
ϵ
∆

so as k ←− H −→ H ⊗ H, g ⊗ g −→ g; and▶ the obvious symmetry V ⊗ W → W ⊗ V , v ⊗ w 7→ w ⊗ v ,which is involved in their axioms.
Those ordinary objects are thus deﬁned based on the symmetrictensor category (vector space) of vector spaces.

Super-symmetry
Everything is over a ﬁxed ﬁeld k of characteristic ̸= 2.
To deﬁne objects over k, such as comm. algebra, Hopf algebra H,
Lie algebra g, you need
▶ the tensor product ⊗, which is used to present their structures
[ , ]
ϵ
∆

so as k ←− H −→ H ⊗ H, g ⊗ g −→ g; and▶ the obvious symmetry V ⊗ W → W ⊗ V , v ⊗ w 7→ w ⊗ v ,which is involved in their axioms.
Those ordinary objects are thus deﬁned based on the symmetrictensor category (vector space) of vector spaces.
As a generalized symmetric tensor category we have the category(super-vector space) of super-vector spaces,

Super-symmetry
Everything is over a ﬁxed ﬁeld k of characteristic ̸= 2.
To deﬁne objects over k, such as comm. algebra, Hopf algebra H,
Lie algebra g, you need
▶ the tensor product ⊗, which is used to present their structures
[ , ]
ϵ
∆

so as k ←− H −→ H ⊗ H, g ⊗ g −→ g; and▶ the obvious symmetry V ⊗ W → W ⊗ V , v ⊗ w 7→ w ⊗ v ,which is involved in their axioms.
Those ordinary objects are thus deﬁned based on the symmetrictensor category (vector space) of vector spaces.
As a generalized symmetric tensor category we have the category(super-vector space) of super-vector spaces, based on which aredeﬁned super-objects, such as super-commutative super-algebra,
super-Hopf algebra, super-Lie algebra.

“Super” = “graded by Z/(2) = {0, 1}”; 0 is called even, 1 odd

“Super” = “graded by Z/(2) = {0, 1}”; 0 is called even, 1 odd
Super-vector space = Z/(2)-graded vector space V = V0 ⊕ V1 .

“Super” = “graded by Z/(2) = {0, 1}”; 0 is called even, 1 odd
Super-vector space = Z/(2)-graded vector space V = V0 ⊕ V1 .
The super-symmetry V ⊗ W → W ⊗ V is given by{−w ⊗ v if v and w are odd,v ⊗ w 7→ (−1)|v ||w | w ⊗ v =w ⊗ v otherwise.

“Super” = “graded by Z/(2) = {0, 1}”; 0 is called even, 1 odd
Super-vector space = Z/(2)-graded vector space V = V0 ⊕ V1 .
The super-symmetry V ⊗ W → W ⊗ V is given by{−w ⊗ v if v and w are odd,v ⊗ w 7→ (−1)|v ||w | w ⊗ v =w ⊗ v otherwise.
Ex. If V = V1 , then ∧(V ) is a super-comm. super-algebra.

“Super” = “graded by Z/(2) = {0, 1}”; 0 is called even, 1 odd
Super-vector space = Z/(2)-graded vector space V = V0 ⊕ V1 .
The super-symmetry V ⊗ W → W ⊗ V is given by{−w ⊗ v if v and w are odd,v ⊗ w 7→ (−1)|v ||w | w ⊗ v =w ⊗ v otherwise.
Ex. If V = V1 , then ∧(V ) is a super-comm. super-algebra.
We have(vector space) ⊂ (super-vector space)so that ordinary objects A are precisely super-objects which are
purely even, A = A0 .

Thm (Deligne 2002). In characteristic 0, the super-symmetry
mostly exhausts all possible symmetries.

Thm (Deligne 2002). In characteristic 0, the super-symmetry
mostly exhausts all possible symmetries. To be more precise,Thm (Deligne 2002). In characteristic 0, the super-symmetry
mostly exhausts all possible symmetries. To be more precise,
if k is an algebraically closed ﬁeld of characteristic zero, any rigid,
abelian k-linear symmetric tensor category satisfying a certain mild
assumption is realized as the category of ﬁnite-dimensional
super-modules over some aﬃne super-algebraic group.

Geometrical vs. functorial viewpoints–GrothendieckGeometrical vs. functorial viewpoints–Grothendieck
In what follows a super-algebra A = A0 ⊕ A1 is assumed to besuper-commutative, so that A includes A0 as a central subalgebra,
and ab = −ba, a2 = 0 for all a, b ∈ A1 .

Geometrical vs. functorial viewpoints–Grothendieck
In what follows a super-algebra A = A0 ⊕ A1 is assumed to besuper-commutative, so that A includes A0 as a central subalgebra,
and ab = −ba, a2 = 0 for all a, b ∈ A1 .
A prime (resp., maximal) of A is a super-ideal of the form P ⊕ A1 ,where P is a prime (resp., maximal) of A0 .

Geometrical vs. functorial viewpoints–Grothendieck
In what follows a super-algebra A = A0 ⊕ A1 is assumed to besuper-commutative, so that A includes A0 as a central subalgebra,
and ab = −ba, a2 = 0 for all a, b ∈ A1 .
A prime (resp., maximal) of A is a super-ideal of the form P ⊕ A1 ,where P is a prime (resp., maximal) of A0 .
A is said to be local if A0 is so.

Geometrical vs. functorial viewpoints–Grothendieck
In what follows a super-algebra A = A0 ⊕ A1 is assumed to besuper-commutative, so that A includes A0 as a central subalgebra,
and ab = −ba, a2 = 0 for all a, b ∈ A1 .
A prime (resp., maximal) of A is a super-ideal of the form P ⊕ A1 ,where P is a prime (resp., maximal) of A0 .
A is said to be local if A0 is so.
Geometrical viewpoint:Geometrical vs. functorial viewpoints–Grothendieck
In what follows a super-algebra A = A0 ⊕ A1 is assumed to besuper-commutative, so that A includes A0 as a central subalgebra,
and ab = −ba, a2 = 0 for all a, b ∈ A1 .
A prime (resp., maximal) of A is a super-ideal of the form P ⊕ A1 ,where P is a prime (resp., maximal) of A0 .
A is said to be local if A0 is so.
Geometrical viewpoint: A local-super-ringed space is a
topological space X equipped with a structure sheaf OX of
super-algebras such that the stalk OX ,P at every point P is a local
super-algebra.

Geometrical vs. functorial viewpoints–Grothendieck
In what follows a super-algebra A = A0 ⊕ A1 is assumed to besuper-commutative, so that A includes A0 as a central subalgebra,
and ab = −ba, a2 = 0 for all a, b ∈ A1 .
A prime (resp., maximal) of A is a super-ideal of the form P ⊕ A1 ,where P is a prime (resp., maximal) of A0 .
A is said to be local if A0 is so.
Geometrical viewpoint: A local-super-ringed space is a
topological space X equipped with a structure sheaf OX of
super-algebras such that the stalk OX ,P at every point P is a local
super-algebra.
A super-scheme is a local-super-ringed space locally isomorphic toan aﬃne super-scheme, Spec A.

Geometrical vs. functorial viewpoints–Grothendieck
In what follows a super-algebra A = A0 ⊕ A1 is assumed to besuper-commutative, so that A includes A0 as a central subalgebra,
and ab = −ba, a2 = 0 for all a, b ∈ A1 .
A prime (resp., maximal) of A is a super-ideal of the form P ⊕ A1 ,where P is a prime (resp., maximal) of A0 .
A is said to be local if A0 is so.
Geometrical viewpoint: A local-super-ringed space is a
topological space X equipped with a structure sheaf OX of
super-algebras such that the stalk OX ,P at every point P is a local
super-algebra.
A super-scheme is a local-super-ringed space locally isomorphic toan aﬃne super-scheme, Spec A.
The aﬃne super-scheme Spec A has Spec(A0 ) as the underlying
topological space, and the structure sheaf OSpec A is such that
OSpec A (D(x)) = A ⊗A0 (A0 )x ,OSpec A,P = A ⊗A0 (A0 )P ,where x ∈ A0 , D(x) = {x ∈/ P ∈ Spec(A0 )} and P ∈ Spec(A0 ).

Functorial viewpoint:Functorial viewpoint: A k-functor is a set-valued functor
F : (super-algebra) → (set)deﬁned on the category of super-algebras (over k).

Functorial viewpoint: A k-functor is a set-valued functor
F : (super-algebra) → (set)deﬁned on the category of super-algebras (over k).
A functorial aﬃne super-scheme is a representable k-functor,

Functorial viewpoint: A k-functor is a set-valued functor
F : (super-algebra) → (set)deﬁned on the category of super-algebras (over k).
A functorial aﬃne super-scheme is a representable k-functor, whichis thus of the form
Sp A = SAlg(A, −) : R 7→ SAlg(A, R),
where A is a super-algebra.

Functorial viewpoint: A k-functor is a set-valued functor
F : (super-algebra) → (set)deﬁned on the category of super-algebras (over k).
A functorial aﬃne super-scheme is a representable k-functor, whichis thus of the form
Sp A = SAlg(A, −) : R 7→ SAlg(A, R),
where A is a super-algebra.
A functorial super-scheme is a k-functor which is the union ofsome open sub-functors that are functorial aﬃne super-schemes.

Functorial viewpoint: A k-functor is a set-valued functor
F : (super-algebra) → (set)deﬁned on the category of super-algebras (over k).
A functorial aﬃne super-scheme is a representable k-functor, whichis thus of the form
Sp A = SAlg(A, −) : R 7→ SAlg(A, R),
where A is a super-algebra.
A functorial super-scheme is a k-functor which is the union ofsome open sub-functors that are functorial aﬃne super-schemes.
Comparison Thm (M-Zubkov 2011): Spec A 7→ Sp A extends to acategory equivalence()
functorial(super-scheme) ≈super-scheme
which assigns to X , the k-functor Mor(Spec(−), X ) represented byX.

An advantage of the functorial viewpoint:An advantage of the functorial viewpoint: The k-functors are so
ﬂexible that they include sheaf-like objects which indeed behave
like sheaves with respect to some “topology” on super-algebrasthat you may choose as you like.

An advantage of the functorial viewpoint: The k-functors are so
ﬂexible that they include sheaf-like objects which indeed behave
like sheaves with respect to some “topology” on super-algebrasthat you may choose as you like. We choose the fppf topology,
to get faisceaux.

An advantage of the functorial viewpoint: The k-functors are so
ﬂexible that they include sheaf-like objects which indeed behave
like sheaves with respect to some “topology” on super-algebrasthat you may choose as you like. We choose the fppf topology,
to get faisceaux.
Def (Grothendieck): A faisceau (or k-sheaf) is a k-functor F
which preserve ﬁnite products, and turns every equalizer diagram
R → S ⇒ S ⊗R S
associated with an fppf (= faith. ﬂat and ﬁnitely presented) map
R → S of super-algebras into the equalizer diagram of sets
F (R) → F (S) ⇒ F (S ⊗R S).

Geometrical side()
aﬃnesuper-scheme∩(super-scheme)≈≈( Functorial side )/ functorial aﬃnesuper-schemes∩)(
functorial/
super-schemes∩(faisceaux)Geometrical side()
aﬃnesuper-scheme∩(super-scheme)≈≈( Functorial side )/ functorial aﬃnesuper-schemes∩)(
functorial/
super-schemes∩(faisceaux)Fact: Given a k-functor X there uniquely exists a faisceau X̃
equipped with a morphism ι : X → X̃ , such that the induced map
Mor(ι, Y ) : Mor(X̃ , Y ) → Mor(X , Y )is a bijection for every faisceau Y .

Geometrical side()
aﬃnesuper-scheme∩(super-scheme)≈≈( Functorial side )/ functorial aﬃnesuper-schemes∩)(
functorial/
super-schemes∩(faisceaux)Fact: Given a k-functor X there uniquely exists a faisceau X̃
equipped with a morphism ι : X → X̃ , such that the induced map
Mor(ι, Y ) : Mor(X̃ , Y ) → Mor(X , Y )is a bijection for every faisceau Y .
Appl: Let G /̃H be the faisceau associated with the coset functor
R 7→ G (R)/H(R).

Geometrical side()
aﬃnesuper-scheme∩(super-scheme)≈≈( Functorial side )/ functorial aﬃnesuper-schemes∩)(
functorial/
super-schemes∩(faisceaux)Fact: Given a k-functor X there uniquely exists a faisceau X̃
equipped with a morphism ι : X → X̃ , such that the induced map
Mor(ι, Y ) : Mor(X̃ , Y ) → Mor(X , Y )is a bijection for every faisceau Y .
Appl: Let G /̃H be the faisceau associated with the coset functor
R 7→ G (R)/H(R). If this G /̃H happens to be a super-scheme, thatis the quotient super-scheme G /H.

Super-algebraic groups
Algebraicsuper-group() (schemes) G are the group objects in(functorial) algebraic.
super-schemes

Super-algebraic groups
Algebraicsuper-group() (schemes) G are the group objects in(functorial) algebraic. We omit “schemes”, and assume G to besuper-schemesaﬃne in what follows.

Super-algebraic groups
Algebraicsuper-group() (schemes) G are the group objects in(functorial) algebraic. We omit “schemes”, and assume G to besuper-schemesaﬃne in what follows. Thus
G = Spec A = Sp A,
where A is ﬁnitely generated (super-comm.) super-Hopf algebra.

Super-algebraic groups
Algebraicsuper-group() (schemes) G are the group objects in(functorial) algebraic. We omit “schemes”, and assume G to besuper-schemesaﬃne in what follows. Thus
G = Spec A = Sp A,
where A is ﬁnitely generated (super-comm.) super-Hopf algebra.
Ex G = GL(m|n):Super-algebraic groups
Algebraicsuper-group() (schemes) G are the group objects in(functorial) algebraic. We omit “schemes”, and assume G to besuper-schemesaﬃne in what follows. Thus
G = Spec A = Sp A,
where A is ﬁnitely generated (super-comm.) super-Hopf algebra.
Ex G = GL(m|n): the k-group functor()
xij piℓ
R 7→ invertible, x , y ∈ R0 , piℓ , qkj ∈ R1
qkj ykℓ 1≤i,j≤m ij kℓeven, odd
elements
1≤k,ℓ≤nSuper-algebraic groups
Algebraicsuper-group() (schemes) G are the group objects in(functorial) algebraic. We omit “schemes”, and assume G to besuper-schemesaﬃne in what follows. Thus
G = Spec A = Sp A,
where A is ﬁnitely generated (super-comm.) super-Hopf algebra.
Ex G = GL(m|n): the k-group functor()
xij piℓ
R 7→ invertible, x , y ∈ R0 , piℓ , qkj ∈ R1
qkj ykℓ 1≤i,j≤m ij kℓeven, odd
elements
1≤k,ℓ≤nrepresented by the super-Hopf algebra[]
A = k xij , ykℓ , det(X )−1 , det(Y )−1 ⊗ ∧(piℓ , qkj );
(
) (
)
X P
xij piℓ=, xij , ykℓ even, piℓ , qkj odd variables
Q Y
qkj ykℓ
(
) (
) (
)
(
) (
)
X P
X P
X P
X P
I O
∆
=
⊗
, ε
=
.
Q Y
Q Y
Q Y
Q Y
O I

Alternative description of super-Lie algebras.

Alternative description of super-Lie algebras. A super-Lie algebra
may be understood as a pair (g0 , V ) of a Lie algebra g0 and rightg0 -module V , equipped with a g0 -equivariant symmetric linear
map [ , ] : V ⊗ V → g0 such thatv ◁ [v , v ] = 0, v ∈ V .

(♡)

Alternative description of super-Lie algebras. A super-Lie algebra
may be understood as a pair (g0 , V ) of a Lie algebra g0 and rightg0 -module V , equipped with a g0 -equivariant symmetric linear
map [ , ] : V ⊗ V → g0 such thatv ◁ [v , v ] = 0, v ∈ V .

(♡)

Every super-Lie algebra g arises uniquely arises from such a pair,g = g0 ⋉ V with the deformed relation [(0, v ), (0, w )] = ([v , w ], 0).

Alternative description of super-Lie algebras. A super-Lie algebra
may be understood as a pair (g0 , V ) of a Lie algebra g0 and rightg0 -module V , equipped with a g0 -equivariant symmetric linear
map [ , ] : V ⊗ V → g0 such thatv ◁ [v , v ] = 0, v ∈ V .

(♡)

Every super-Lie algebra g arises uniquely arises from such a pair,g = g0 ⋉ V with the deformed relation [(0, v ), (0, w )] = ([v , w ], 0).
Analogous description of super-algebraic groups (M-Shibata 2017).

Alternative description of super-Lie algebras. A super-Lie algebra
may be understood as a pair (g0 , V ) of a Lie algebra g0 and rightg0 -module V , equipped with a g0 -equivariant symmetric linear
map [ , ] : V ⊗ V → g0 such thatv ◁ [v , v ] = 0, v ∈ V .

(♡)

Every super-Lie algebra g arises uniquely arises from such a pair,g = g0 ⋉ V with the deformed relation [(0, v ), (0, w )] = ([v , w ], 0).
Analogous description of super-algebraic groups (M-Shibata 2017).
A super-algebraic group may be understood a pair (G0 , V ) of analgebraic group G0 and right G0 -module V , equipped with a G0 equivariant symmetric linear map [ , ] : V ⊗ V → Lie(G0 ) s.t. (♡).

Alternative description of super-Lie algebras. A super-Lie algebra
may be understood as a pair (g0 , V ) of a Lie algebra g0 and rightg0 -module V , equipped with a g0 -equivariant symmetric linear
map [ , ] : V ⊗ V → g0 such thatv ◁ [v , v ] = 0, v ∈ V .

(♡)

Every super-Lie algebra g arises uniquely arises from such a pair,g = g0 ⋉ V with the deformed relation [(0, v ), (0, w )] = ([v , w ], 0).
Analogous description of super-algebraic groups (M-Shibata 2017).
A super-algebraic group may be understood a pair (G0 , V ) of analgebraic group G0 and right G0 -module V , equipped with a G0 equivariant symmetric linear map [ , ] : V ⊗ V → Lie(G0 ) s.t. (♡).
Every super-algebraic group arises uniquely from such a pair,Alternative description of super-Lie algebras. A super-Lie algebra
may be understood as a pair (g0 , V ) of a Lie algebra g0 and rightg0 -module V , equipped with a g0 -equivariant symmetric linear
map [ , ] : V ⊗ V → g0 such thatv ◁ [v , v ] = 0, v ∈ V .

(♡)

Every super-Lie algebra g arises uniquely arises from such a pair,g = g0 ⋉ V with the deformed relation [(0, v ), (0, w )] = ([v , w ], 0).
Analogous description of super-algebraic groups (M-Shibata 2017).
A super-algebraic group may be understood a pair (G0 , V ) of analgebraic group G0 and right G0 -module V , equipped with a G0 equivariant symmetric linear map [ , ] : V ⊗ V → Lie(G0 ) s.t. (♡).
Every super-algebraic group arises uniquely from such a pair,deforming the semi-direct productR 7→ G0 (R0 ) ⋉ (V ⊗ R1 )so that (v ⊗ a)(w ⊗ b) = exp([v , w ] ⊗ ab)(w ⊗ b)(v ⊗ a).

Example. Super-Grassmanians
Let r ≤ m, s ≤ n be integers ≥ 0.

Example. Super-Grassmanians
Let r ≤ m, s ≤ n be integers ≥ 0. Let km|n := km ⊕ k[1]n .

Example. Super-Grassmanians
Let r ≤ m, s ≤ n be integers ≥ 0. Let km|n := km ⊕ k[1]n .
Given R ∈ (super-algebra), let R m|n := km|n ⊗ R = R m ⊕ R[1]n .

Example. Super-Grassmanians
Let r ≤ m, s ≤ n be integers ≥ 0. Let km|n := km ⊕ k[1]n .
Given R ∈ (super-algebra), let R m|n := km|n ⊗ R = R m ⊕ R[1]n .
The super-Grassmanian Gr(r |s, m|n) is the k-functor{}
r |sGr(r |s, m|n)(R) = M <⊕ R m|n | MP ≃ RP ∀P ∈ Spec(R0 ) .
This is a super-scheme, covered by aﬃne super-schemes FW ,{}FW (R) = M | M ⊕ (W ⊗ R) = R m|n ,where W ⊂ km|n are sub-super-vector spaces ≃ km−r |n−s .

Example. Super-Grassmanians
Let r ≤ m, s ≤ n be integers ≥ 0. Let km|n := km ⊕ k[1]n .
Given R ∈ (super-algebra), let R m|n := km|n ⊗ R = R m ⊕ R[1]n .
The super-Grassmanian Gr(r |s, m|n) is the k-functor{}
r |sGr(r |s, m|n)(R) = M <⊕ R m|n | MP ≃ RP ∀P ∈ Spec(R0 ) .
This is a super-scheme, covered by aﬃne super-schemes FW ,{}FW (R) = M | M ⊕ (W ⊗ R) = R m|n ,where W ⊂ km|n are sub-super-vector spaces ≃ km−r |n−s .
Fix a sub-super-vector space U ⊂ km|n s.t. U ≃ kr |s .

Example. Super-Grassmanians
Let r ≤ m, s ≤ n be integers ≥ 0. Let km|n := km ⊕ k[1]n .
Given R ∈ (super-algebra), let R m|n := km|n ⊗ R = R m ⊕ R[1]n .
The super-Grassmanian Gr(r |s, m|n) is the k-functor{}
r |sGr(r |s, m|n)(R) = M <⊕ R m|n | MP ≃ RP ∀P ∈ Spec(R0 ) .
This is a super-scheme, covered by aﬃne super-schemes FW ,{}FW (R) = M | M ⊕ (W ⊗ R) = R m|n ,where W ⊂ km|n are sub-super-vector spaces ≃ km−r |n−s .
Fix a sub-super-vector space U ⊂ km|n s.t. U ≃ kr |s . Then
GL(m|n)(R) → Gr(r |s, m|n)(R), g 7→ g (U ⊗ R)
gives rise to≃GL(m|n)/̃StabGL(m|n) (U) −→ Gr(r |s, m|n).

Our construction
Let G = Spec A be a super-algebraic group.

Our construction
Let G = Spec A be a super-algebraic group. There is associatedthe (ordinary) algebraic group
G0 = Spec(A/(A1 )) : (algebra) ∋ R 7→ SAlg(A, R) ∈ (group).

Our construction
Let G = Spec A be a super-algebraic group. There is associatedthe (ordinary) algebraic group
G0 = Spec(A/(A1 )) : (algebra) ∋ R 7→ SAlg(A, R) ∈ (group).
The underlying topological spaces coincide, |G | = |G0 |.

Our construction
Let G = Spec A be a super-algebraic group. There is associatedthe (ordinary) algebraic group
G0 = Spec(A/(A1 )) : (algebra) ∋ R 7→ SAlg(A, R) ∈ (group).
The underlying topological spaces coincide, |G | = |G0 |.
Suppose H = Spec B is a closed sub-super-group of G , so that B
is a quotient super-Hopf algebra of A.

Our construction
Let G = Spec A be a super-algebraic group. There is associatedthe (ordinary) algebraic group
G0 = Spec(A/(A1 )) : (algebra) ∋ R 7→ SAlg(A, R) ∈ (group).
The underlying topological spaces coincide, |G | = |G0 |.
Suppose H = Spec B is a closed sub-super-group of G , so that B
is a quotient super-Hopf algebra of A. The associated algebraic
group H0 = Spec(B/(B1 )) is a closed subgroup of G0 .

Our construction
Let G = Spec A be a super-algebraic group. There is associatedthe (ordinary) algebraic group
G0 = Spec(A/(A1 )) : (algebra) ∋ R 7→ SAlg(A, R) ∈ (group).
The underlying topological spaces coincide, |G | = |G0 |.
Suppose H = Spec B is a closed sub-super-group of G , so that B
is a quotient super-Hopf algebra of A. The associated algebraic
group H0 = Spec(B/(B1 )) is a closed subgroup of G0 .
Classical result (Grothendieck): There exists the quotientscheme G0 /H0 , which is Noetherian and represents the faisceau
G0 /̃H0 . The morphism π : G0 → G0 /H0 is aﬃne and fppf.

Our construction
Let G = Spec A be a super-algebraic group. There is associatedthe (ordinary) algebraic group
G0 = Spec(A/(A1 )) : (algebra) ∋ R 7→ SAlg(A, R) ∈ (group).
The underlying topological spaces coincide, |G | = |G0 |.
Suppose H = Spec B is a closed sub-super-group of G , so that B
is a quotient super-Hopf algebra of A. The associated algebraic
group H0 = Spec(B/(B1 )) is a closed subgroup of G0 .
Classical result (Grothendieck): There exists the quotientscheme G0 /H0 , which is Noetherian and represents the faisceau
G0 /̃H0 . The morphism π : G0 → G0 /H0 is aﬃne and fppf.
Let U ⊂ G0 /H0 be aﬃne open, and regard π −1 (U) (⊂ |G0 | = |G |)as an open sub-super-scheme of G .

Our construction
Let G = Spec A be a super-algebraic group. There is associatedthe (ordinary) algebraic group
G0 = Spec(A/(A1 )) : (algebra) ∋ R 7→ SAlg(A, R) ∈ (group).
The underlying topological spaces coincide, |G | = |G0 |.
Suppose H = Spec B is a closed sub-super-group of G , so that B
is a quotient super-Hopf algebra of A. The associated algebraic
group H0 = Spec(B/(B1 )) is a closed subgroup of G0 .
Classical result (Grothendieck): There exists the quotientscheme G0 /H0 , which is Noetherian and represents the faisceau
G0 /̃H0 . The morphism π : G0 → G0 /H0 is aﬃne and fppf.
Let U ⊂ G0 /H0 be aﬃne open, and regard π −1 (U) (⊂ |G0 | = |G |)as an open sub-super-scheme of G . The key is to construct a nice
H-equivariant aﬃne super-scheme XU together with an H-equiv.
open embedding≃
XU −→ π −1 (U) ⊂ G .

Using Hopf-algebra techniques we construct XU so as()
XU = Spec (OG0 (π −1 (U)) ⊗ ∧(Z )) ×H0 H ,where Z = (Lie(G )1 /Lie(H)1 )∗ .

Using Hopf-algebra techniques we construct XU so as()
XU = Spec (OG0 (π −1 (U)) ⊗ ∧(Z )) ×H0 H ,where Z = (Lie(G )1 /Lie(H)1 )∗ . This is so nice that the faisceau
XU /̃H is aﬃne,()
XU /̃H = Spec ∧OG /H (U) (OG0 (π −1 (U)) ×H0 Z ) .
00

Using Hopf-algebra techniques we construct XU so as()
XU = Spec (OG0 (π −1 (U)) ⊗ ∧(Z )) ×H0 H ,where Z = (Lie(G )1 /Lie(H)1 )∗ . This is so nice that the faisceau
XU /̃H is aﬃne,()
XU /̃H = Spec ∧OG /H (U) (OG0 (π −1 (U)) ×H0 Z ) .
00

Thm (M-Takahashi): The aﬃne super-schemes XU /̃H, where U
ranges over the aﬃne open subschemes of G0 /H0 , are glued into a
Noetherian super-scheme with the underlying topological space|G0 /H0 |, whose struc. sheaf is locally isomorphic to∧OG0 /H0 (π∗ OG0 ×H0 Z ).

Using Hopf-algebra techniques we construct XU so as()
XU = Spec (OG0 (π −1 (U)) ⊗ ∧(Z )) ×H0 H ,where Z = (Lie(G )1 /Lie(H)1 )∗ . This is so nice that the faisceau
XU /̃H is aﬃne,()
XU /̃H = Spec ∧OG /H (U) (OG0 (π −1 (U)) ×H0 Z ) .
00

Thm (M-Takahashi): The aﬃne super-schemes XU /̃H, where U
ranges over the aﬃne open subschemes of G0 /H0 , are glued into a
Noetherian super-scheme with the underlying topological space|G0 /H0 |, whose struc. sheaf is locally isomorphic to∧OG0 /H0 (π∗ OG0 ×H0 Z ).
The resulting super-scheme is the quotient G /H, and represents
the faisceau G /̃H.

Using Hopf-algebra techniques we construct XU so as()
XU = Spec (OG0 (π −1 (U)) ⊗ ∧(Z )) ×H0 H ,where Z = (Lie(G )1 /Lie(H)1 )∗ . This is so nice that the faisceau
XU /̃H is aﬃne,()
XU /̃H = Spec ∧OG /H (U) (OG0 (π −1 (U)) ×H0 Z ) .
00

Thm (M-Takahashi): The aﬃne super-schemes XU /̃H, where U
ranges over the aﬃne open subschemes of G0 /H0 , are glued into a
Noetherian super-scheme with the underlying topological space|G0 /H0 |, whose struc. sheaf is locally isomorphic to∧OG0 /H0 (π∗ OG0 ×H0 Z ).
The resulting super-scheme is the quotient G /H, and represents
the faisceau G /̃H. It has additional desirable properties including:an open subset U ⊂ |G /H| (= |G0 /H0 |) is aﬃne in G /H ⇔ it isaﬃne in G0 /H0 .

Thanks, also to the organizers!