classical integral transforms in semi-commutative algebraic geometry
TRANSCRIPT
![Page 1: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/1.jpg)
Classical Integral Transforms in Semi-commutativeAlgebraic Geometry
Adam Nyman
Western Washington University
August 27, 2009
Adam Nyman
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Conventions and Notation
Adam Nyman
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Conventions and Notation
always work over commutative ring k,
Adam Nyman
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Conventions and Notation
always work over commutative ring k,
X is (comm.) quasi-compact separated k-scheme
Adam Nyman
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Conventions and Notation
always work over commutative ring k,
X is (comm.) quasi-compact separated k-scheme
“scheme”=commutative k-scheme
Adam Nyman
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Conventions and Notation
always work over commutative ring k,
X is (comm.) quasi-compact separated k-scheme
“scheme”=commutative k-scheme
“bimodule”=object in Bimodk(C,D)
Adam Nyman
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Part 1
Integral Transforms and Bimod: Examples
Adam Nyman
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Integral Transforms and Bimod: Examples
Example 1R , S rings, F an R − S-bimodule
−⊗R F : ModR → ModS
Adam Nyman
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Integral Transforms and Bimod: Examples
Example 1R , S rings, F an R − S-bimodule
−⊗R F : ModR → ModS
F =“integral kernel”
Adam Nyman
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Integral Transforms and Bimod: Examples
Example 1R , S rings, F an R − S-bimodule
−⊗R F : ModR → ModS
F =“integral kernel”
Theorem (Eilenberg, Watts 1960)
Every F ∈ Bimodk(ModR ,ModS) is an integral transform.
Adam Nyman
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Integral Transforms and Bimod: Examples
Example 1R , S rings, F an R − S-bimodule
−⊗R F : ModR → ModS
F =“integral kernel”
Theorem (Eilenberg, Watts 1960)
Every F ∈ Bimodk(ModR ,ModS) is an integral transform. Moregenerally, F 7→ −⊗R F induces an equivalence
Mod(Rop ⊗k S) → Bimodk(ModR ,ModS).
Adam Nyman
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Integral Transforms and Bimod: A Generalization ofExample 1
A = cocomplete abelian cat.
Adam Nyman
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Integral Transforms and Bimod: A Generalization ofExample 1
A = cocomplete abelian cat.
RA = cat. of left R-objects in A
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Integral Transforms and Bimod: A Generalization ofExample 1
A = cocomplete abelian cat.
RA = cat. of left R-objects in A
ob RA : (F , ρ) where F ∈ obA, ρ : R → EndA F
Adam Nyman
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Integral Transforms and Bimod: A Generalization ofExample 1
A = cocomplete abelian cat.
RA = cat. of left R-objects in A
ob RA : (F , ρ) where F ∈ obA, ρ : R → EndA F
Remark: AR defined similarly
Adam Nyman
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Integral Transforms and Bimod: A Generalization ofExample 1
A = cocomplete abelian cat.
RA = cat. of left R-objects in A
ob RA : (F , ρ) where F ∈ obA, ρ : R → EndA F
Remark: AR defined similarly
e.g.
if A = ModS then RA ≡ Mod(Rop ⊗k S).
Adam Nyman
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Integral Transforms and Bimod: A Generalization ofExample 1
A = cocomplete abelian cat.
RA = cat. of left R-objects in A
ob RA : (F , ρ) where F ∈ obA, ρ : R → EndA F
Remark: AR defined similarly
e.g.
if A = ModS then RA ≡ Mod(Rop ⊗k S).
For F ∈ RA, define
Adam Nyman
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Integral Transforms and Bimod: A Generalization ofExample 1
A = cocomplete abelian cat.
RA = cat. of left R-objects in A
ob RA : (F , ρ) where F ∈ obA, ρ : R → EndA F
Remark: AR defined similarly
e.g.
if A = ModS then RA ≡ Mod(Rop ⊗k S).
For F ∈ RA, define
−⊗R F : ModR → A
Adam Nyman
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Integral Transforms and Bimod: A Generalization ofExample 1
A = cocomplete abelian cat.
RA = cat. of left R-objects in A
ob RA : (F , ρ) where F ∈ obA, ρ : R → EndA F
Remark: AR defined similarly
e.g.
if A = ModS then RA ≡ Mod(Rop ⊗k S).
For F ∈ RA, define
−⊗R F : ModR → A
as left adjoint to
HomA(F ,−) : A → ModR
Adam Nyman
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Integral Transforms and Bimod: A Generalization ofExample 1
Theorem (N-Smith 2007)
Every F ∈ Bimodk(ModR ,A) is an integral transform.
Adam Nyman
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Integral Transforms and Bimod: A Generalization ofExample 1
Theorem (N-Smith 2007)
Every F ∈ Bimodk(ModR ,A) is an integral transform. Moregenerally, F 7→ −⊗R F induces an equivalence
RA → Bimodk(ModR ,A).
Adam Nyman
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Integral Transforms and Bimod: Examples
Example 2 Y scheme
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Integral Transforms and Bimod: Examples
Example 2 Y scheme X × Y
τ
{{wwww
wwww
wπ
##GGGG
GGGG
G
X Y
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Integral Transforms and Bimod: Examples
Example 2 Y scheme X × Y
τ
{{wwww
wwww
wπ
##GGGG
GGGG
G
X Y
F ∈ Qcoh(X ×Y )
Adam Nyman
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Integral Transforms and Bimod: Examples
Example 2 Y scheme X × Y
τ
{{wwww
wwww
wπ
##GGGG
GGGG
G
X Y
F ∈ Qcoh(X ×Y )
π∗(τ∗(−) ⊗OX×Y
F) : QcohX → QcohY
Adam Nyman
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Integral Transforms and Bimod: Examples
Example 2 Y scheme X × Y
τ
{{wwww
wwww
wπ
##GGGG
GGGG
G
X Y
F ∈ Qcoh(X ×Y )
π∗(τ∗(−) ⊗OX×Y
F) : QcohX → QcohY
e.g.
If f : Y → X is a morphism of schemes thenf ∗ : QcohX → QcohY is an integral transform.
Adam Nyman
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Integral Transforms and Bimod: Examples
Example 2 Y scheme X × Y
τ
{{wwww
wwww
wπ
##GGGG
GGGG
G
X Y
F ∈ Qcoh(X ×Y )
π∗(τ∗(−) ⊗OX×Y
F) : QcohX → QcohY
e.g.
If f : Y → X is a morphism of schemes thenf ∗ : QcohX → QcohY is an integral transform.
e.g.
Let X = P1 and Y = Spec k, F = OX .
Adam Nyman
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Integral Transforms and Bimod: Examples
Example 2 Y scheme X × Y
τ
{{wwww
wwww
wπ
##GGGG
GGGG
G
X Y
F ∈ Qcoh(X ×Y )
π∗(τ∗(−) ⊗OX×Y
F) : QcohX → QcohY
e.g.
If f : Y → X is a morphism of schemes thenf ∗ : QcohX → QcohY is an integral transform.
e.g.
Let X = P1 and Y = Spec k, F = OX .
H1(X ,−) ∈ Bimodk(QcohX ,QcohY ) is not an integraltransform.
Adam Nyman
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Integral Transforms and Bimod: Examples
Example 2 Y scheme X × Y
τ
{{wwww
wwww
wπ
##GGGG
GGGG
G
X Y
F ∈ Qcoh(X ×Y )
π∗(τ∗(−) ⊗OX×Y
F) : QcohX → QcohY
e.g.
If f : Y → X is a morphism of schemes thenf ∗ : QcohX → QcohY is an integral transform.
e.g.
Let X = P1 and Y = Spec k, F = OX .
H1(X ,−) ∈ Bimodk(QcohX ,QcohY ) is not an integraltransform.
Int. trans. not always rt exact
Adam Nyman
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Integral Transforms and Bimod: Examples
Example 2 Y scheme X × Y
τ
{{wwww
wwww
wπ
##GGGG
GGGG
G
X Y
F ∈ Qcoh(X ×Y )
π∗(τ∗(−) ⊗OX×Y
F) : QcohX → QcohY
e.g.
If f : Y → X is a morphism of schemes thenf ∗ : QcohX → QcohY is an integral transform.
e.g.
Let X = P1 and Y = Spec k, F = OX .
H1(X ,−) ∈ Bimodk(QcohX ,QcohY ) is not an integraltransform.
Int. trans. not always rt exact: π∗(τ∗(−) ⊗OX
F) ∼= Γ(P1,−).
Adam Nyman
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Integral Transforms and Bimod: Examples
Example 3
k a field
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Integral Transforms and Bimod: Examples
Example 3
k a field, X , Y finite type over k
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Integral Transforms and Bimod: Examples
Example 3
k a field, X , Y finite type over k
A finite OX -algebra
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Integral Transforms and Bimod: Examples
Example 3
k a field, X , Y finite type over k
A finite OX -algebra, B finite OY -algebra
Adam Nyman
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Integral Transforms and Bimod: Examples
Example 3
k a field, X , Y finite type over k
A finite OX -algebra, B finite OY -algebra
Theorem (Artin-Zhang 1994)
If F : modA → modB is an equivalence
Adam Nyman
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Integral Transforms and Bimod: Examples
Example 3
k a field, X , Y finite type over k
A finite OX -algebra, B finite OY -algebra
Theorem (Artin-Zhang 1994)
If F : modA → modB is an equivalence then
F ∼= π∗(τ∗(−) ⊗τ∗A F)
for some F ∈ mod(Aop ⊗k B).
Adam Nyman
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Integral Transforms and Bimod: Examples
Problems
Adam Nyman
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Integral Transforms and Bimod: Examples
Problems
1 Find notion of integral transform generalizing examples
Adam Nyman
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Integral Transforms and Bimod: Examples
Problems
1 Find notion of integral transform generalizing examples
2 When is an integral transform a bimodule i.e. when is it rt.exact?
Adam Nyman
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Integral Transforms and Bimod: Examples
Problems
1 Find notion of integral transform generalizing examples
2 When is an integral transform a bimodule i.e. when is it rt.exact?
3 When is a bimodule an integral transform?
Adam Nyman
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Integral Transforms and Bimod: Examples
Problems
1 Find notion of integral transform generalizing examples
2 When is an integral transform a bimodule i.e. when is it rt.exact?
3 When is a bimodule an integral transform? If it is not, howclose is it to being an integral transform?
Adam Nyman
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Integral Transforms and Bimod: Examples
Problems
1 Find notion of integral transform generalizing examples
2 When is an integral transform a bimodule i.e. when is it rt.exact?
3 When is a bimodule an integral transform? If it is not, howclose is it to being an integral transform?
“Classical”=not between derived or dg or ∞-categories
Adam Nyman
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Part 2
Non- and Semi-commutative Algebraic Geometry
Adam Nyman
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Non-commutative Algebraic Geometry: NoncommutativeSpaces
Non-commutative Space := Grothendieck Category
Adam Nyman
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Non-commutative Algebraic Geometry: NoncommutativeSpaces
Non-commutative Space := Grothendieck Category =
(k-linear) abelian category with
Adam Nyman
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Non-commutative Algebraic Geometry: NoncommutativeSpaces
Non-commutative Space := Grothendieck Category =
(k-linear) abelian category with
exact direct limits and
Adam Nyman
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Non-commutative Algebraic Geometry: NoncommutativeSpaces
Non-commutative Space := Grothendieck Category =
(k-linear) abelian category with
exact direct limits and
a generator.
Adam Nyman
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Non-commutative Algebraic Geometry: NoncommutativeSpaces
Non-commutative Space := Grothendieck Category =
(k-linear) abelian category with
exact direct limits and
a generator.
Notation: Y geometry
Adam Nyman
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Non-commutative Algebraic Geometry: NoncommutativeSpaces
Non-commutative Space := Grothendieck Category =
(k-linear) abelian category with
exact direct limits and
a generator.
Notation: Y geometry or ModY category theory
Adam Nyman
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Non-commutative Algebraic Geometry: NoncommutativeSpaces
Non-commutative Space := Grothendieck Category =
(k-linear) abelian category with
exact direct limits and
a generator.
Notation: Y geometry or ModY category theory
The following are non-commutative spaces:
Adam Nyman
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Non-commutative Algebraic Geometry: NoncommutativeSpaces
Non-commutative Space := Grothendieck Category =
(k-linear) abelian category with
exact direct limits and
a generator.
Notation: Y geometry or ModY category theory
The following are non-commutative spaces:
Mod R , R a ring
Adam Nyman
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Non-commutative Algebraic Geometry: NoncommutativeSpaces
Non-commutative Space := Grothendieck Category =
(k-linear) abelian category with
exact direct limits and
a generator.
Notation: Y geometry or ModY category theory
The following are non-commutative spaces:
Mod R , R a ring
Qcoh X
Adam Nyman
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Non-commutative Algebraic Geometry: NoncommutativeSpaces
Non-commutative Space := Grothendieck Category =
(k-linear) abelian category with
exact direct limits and
a generator.
Notation: Y geometry or ModY category theory
The following are non-commutative spaces:
Mod R , R a ring
Qcoh X
Proj A := GrA/TorsA where A is Z-graded
Adam Nyman
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Non-commutative Algebraic Geometry: MorphismsBetween Noncommutative Spaces
Y , Z non-commutative spaces
Adam Nyman
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Non-commutative Algebraic Geometry: MorphismsBetween Noncommutative Spaces
Y , Z non-commutative spaces
Yf→ Z denotes adjoint pair (f ∗, f∗) in the diagram
ModYf∗⇋
f ∗ModZ
Adam Nyman
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Non-commutative Algebraic Geometry: MorphismsBetween Noncommutative Spaces
Y , Z non-commutative spaces
Yf→ Z denotes adjoint pair (f ∗, f∗) in the diagram
ModYf∗⇋
f ∗ModZ
Motivation
If f : Y → X is a morphism of commutative schemes, (f ∗, f∗) is anadjoint pair.
Adam Nyman
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Non-commutative Algebraic Geometry: MorphismsBetween Noncommutative Spaces
Y , Z non-commutative spaces
Yf→ Z denotes adjoint pair (f ∗, f∗) in the diagram
ModYf∗⇋
f ∗ModZ
Motivation
If f : Y → X is a morphism of commutative schemes, (f ∗, f∗) is anadjoint pair.
Adjoint functor theorem ⇒
Morphisms f : Y → Z ↔ Bimodk(ModZ ,ModY ).
Adam Nyman
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Non-commutative Algebraic Geometry: MorphismsBetween Noncommutative Spaces
e.g.
Let f : Y → X denote a morphism of schemes such that (f ∗, f∗, f!)
is an adjoint triple (e.g. a closed immersion of varieties).
Adam Nyman
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Non-commutative Algebraic Geometry: MorphismsBetween Noncommutative Spaces
e.g.
Let f : Y → X denote a morphism of schemes such that (f ∗, f∗, f!)
is an adjoint triple (e.g. a closed immersion of varieties).Then
QcohYf∗⇋
f ∗QcohX
and
QcohXf !
⇋
f∗QcohY
are morphisms of noncommutative spaces Y → X and X → Y .
Adam Nyman
![Page 60: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/60.jpg)
Non-commutative Algebraic Geometry: MorphismsBetween Noncommutative Spaces
e.g.
Let f : Y → X denote a morphism of schemes such that (f ∗, f∗, f!)
is an adjoint triple (e.g. a closed immersion of varieties).Then
QcohYf∗⇋
f ∗QcohX
and
QcohXf !
⇋
f∗QcohY
are morphisms of noncommutative spaces Y → X and X → Y .
The latter may not come from a morphism of schemes.
Adam Nyman
![Page 61: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/61.jpg)
Semi-commutative Algebraic Geometry
A = quasi-coherent sheaf of OX -algebras (throughout)
Adam Nyman
![Page 62: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/62.jpg)
Semi-commutative Algebraic Geometry
A = quasi-coherent sheaf of OX -algebras (throughout)
Y = non-commutative space
Adam Nyman
![Page 63: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/63.jpg)
Semi-commutative Algebraic Geometry
A = quasi-coherent sheaf of OX -algebras (throughout)
Y = non-commutative space
Semi-comm. Alg. Geom.
Adam Nyman
![Page 64: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/64.jpg)
Semi-commutative Algebraic Geometry
A = quasi-coherent sheaf of OX -algebras (throughout)
Y = non-commutative space
Semi-comm. Alg. Geom. = Study of mapsY → (X ,A)
Adam Nyman
![Page 65: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/65.jpg)
Semi-commutative Algebraic Geometry
A = quasi-coherent sheaf of OX -algebras (throughout)
Y = non-commutative space
Semi-comm. Alg. Geom. = Study of mapsY → (X ,A)
= Study of Bimodk(ModA,ModY )
Adam Nyman
![Page 66: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/66.jpg)
Semi-commutative Algebraic Geometry
A = quasi-coherent sheaf of OX -algebras (throughout)
Y = non-commutative space
Semi-comm. Alg. Geom. = Study of mapsY → (X ,A)
= Study of Bimodk(ModA,ModY )
Examples in Part 1 are semi-commutative
Adam Nyman
![Page 67: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/67.jpg)
Semi-commutative Algebraic Geometry
A = quasi-coherent sheaf of OX -algebras (throughout)
Y = non-commutative space
Semi-comm. Alg. Geom. = Study of mapsY → (X ,A)
= Study of Bimodk(ModA,ModY )
Examples in Part 1 are semi-commutative
1 Example 1: F : ModR → ModY .
Adam Nyman
![Page 68: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/68.jpg)
Semi-commutative Algebraic Geometry
A = quasi-coherent sheaf of OX -algebras (throughout)
Y = non-commutative space
Semi-comm. Alg. Geom. = Study of mapsY → (X ,A)
= Study of Bimodk(ModA,ModY )
Examples in Part 1 are semi-commutative
1 Example 1: F : ModR → ModY .If X = Spec k, A ↔ R then ModR ≡ ModA.
Adam Nyman
![Page 69: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/69.jpg)
Semi-commutative Algebraic Geometry
A = quasi-coherent sheaf of OX -algebras (throughout)
Y = non-commutative space
Semi-comm. Alg. Geom. = Study of mapsY → (X ,A)
= Study of Bimodk(ModA,ModY )
Examples in Part 1 are semi-commutative
1 Example 1: F : ModR → ModY .If X = Spec k, A ↔ R then ModR ≡ ModA.
2 Example 2: F : QcohX → QcohY .
Adam Nyman
![Page 70: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/70.jpg)
Semi-commutative Algebraic Geometry
A = quasi-coherent sheaf of OX -algebras (throughout)
Y = non-commutative space
Semi-comm. Alg. Geom. = Study of mapsY → (X ,A)
= Study of Bimodk(ModA,ModY )
Examples in Part 1 are semi-commutative
1 Example 1: F : ModR → ModY .If X = Spec k, A ↔ R then ModR ≡ ModA.
2 Example 2: F : QcohX → QcohY .If A = OX QcohX ≡ ModA. Let ModY = QcohY .
Adam Nyman
![Page 71: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/71.jpg)
Semi-commutative Algebraic Geometry
A = quasi-coherent sheaf of OX -algebras (throughout)
Y = non-commutative space
Semi-comm. Alg. Geom. = Study of mapsY → (X ,A)
= Study of Bimodk(ModA,ModY )
Examples in Part 1 are semi-commutative
1 Example 1: F : ModR → ModY .If X = Spec k, A ↔ R then ModR ≡ ModA.
2 Example 2: F : QcohX → QcohY .If A = OX QcohX ≡ ModA. Let ModY = QcohY .
3 Example 3: F : ModA → ModB.
Adam Nyman
![Page 72: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/72.jpg)
Semi-commutative Algebraic Geometry
A = quasi-coherent sheaf of OX -algebras (throughout)
Y = non-commutative space
Semi-comm. Alg. Geom. = Study of mapsY → (X ,A)
= Study of Bimodk(ModA,ModY )
Examples in Part 1 are semi-commutative
1 Example 1: F : ModR → ModY .If X = Spec k, A ↔ R then ModR ≡ ModA.
2 Example 2: F : QcohX → QcohY .If A = OX QcohX ≡ ModA. Let ModY = QcohY .
3 Example 3: F : ModA → ModB.Let ModY = ModB.
Adam Nyman
![Page 73: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/73.jpg)
Refined Problems
Adam Nyman
![Page 74: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/74.jpg)
Refined Problems
1 Find notion of integral transform in the semi-comm. setting,i.e. from ModA to ModY .
Adam Nyman
![Page 75: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/75.jpg)
Refined Problems
1 Find notion of integral transform in the semi-comm. setting,i.e. from ModA to ModY .
2 When is an integral transform in Bimodk(ModA,ModY ) i.e.when is it rt. exact?
Adam Nyman
![Page 76: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/76.jpg)
Refined Problems
1 Find notion of integral transform in the semi-comm. setting,i.e. from ModA to ModY .
2 When is an integral transform in Bimodk(ModA,ModY ) i.e.when is it rt. exact?
3 When is F ∈ Bimodk(ModA,ModY ) an integral transform?
Adam Nyman
![Page 77: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/77.jpg)
Refined Problems
1 Find notion of integral transform in the semi-comm. setting,i.e. from ModA to ModY .
2 When is an integral transform in Bimodk(ModA,ModY ) i.e.when is it rt. exact?
3 When is F ∈ Bimodk(ModA,ModY ) an integral transform?If it isn’t, how close is it to being an integral transform?
Adam Nyman
![Page 78: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/78.jpg)
Part 3
Integral Transforms in Semi-commutative AlgebraicGeometry w/ D. Chan in case A = OX
Adam Nyman
![Page 79: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/79.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry w/ D. Chan in case A = OX
An Integral Transform is a functor of the form
π∗(−⊗A F) : ModA → ModY
where
Adam Nyman
![Page 80: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/80.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry w/ D. Chan in case A = OX
An Integral Transform is a functor of the form
π∗(−⊗A F) : ModA → ModY
where
F is a “left A-object” in ModY
Adam Nyman
![Page 81: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/81.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry w/ D. Chan in case A = OX
An Integral Transform is a functor of the form
π∗(−⊗A F) : ModA → ModY
where
F is a “left A-object” in ModY
−⊗A F denotes tensoring a right A-module with F and
Adam Nyman
![Page 82: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/82.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry w/ D. Chan in case A = OX
An Integral Transform is a functor of the form
π∗(−⊗A F) : ModA → ModY
where
F is a “left A-object” in ModY
−⊗A F denotes tensoring a right A-module with F and
π∗ is semi-comm. analogue of the pushforward ofπ : X × Y → Y
Adam Nyman
![Page 83: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/83.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry w/ D. Chan in case A = OX
An Integral Transform is a functor of the form
π∗(−⊗A F) : ModA → ModY
where
F is a “left A-object” in ModY
−⊗A F denotes tensoring a right A-module with F and
π∗ is semi-comm. analogue of the pushforward ofπ : X × Y → Y
Main Idea (Artin-Zhang 2001)
Constructions from commutative algebraic geometry which arelocal only over X exist in semi-commutative setting.
Adam Nyman
![Page 84: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/84.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry: Left A-objects
A left A-object in Y , denoted M, consists of following data:
Adam Nyman
![Page 85: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/85.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry: Left A-objects
A left A-object in Y , denoted M, consists of following data:
For U ⊂ X affine open, an object M(U) ∈ A(U)ModY
Adam Nyman
![Page 86: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/86.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry: Left A-objects
A left A-object in Y , denoted M, consists of following data:
For U ⊂ X affine open, an object M(U) ∈ A(U)ModY
Given U × Vpr1
//
pr2��
U
��
V // X
an isomorphism
ψU,V : pr∗1 M(U) → pr∗2 M(V ) (sat. cocycle cond.)
Adam Nyman
![Page 87: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/87.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry: Left A-objects
A left A-object in Y , denoted M, consists of following data:
For U ⊂ X affine open, an object M(U) ∈ A(U)ModY
Given U × Vpr1
//
pr2��
U
��
V // X
an isomorphism
ψU,V : pr∗1 M(U) → pr∗2 M(V ) (sat. cocycle cond.)
AModY := cat. of left A-objects in Y
Adam Nyman
![Page 88: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/88.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry: Left A-objects
A left A-object in Y , denoted M, consists of following data:
For U ⊂ X affine open, an object M(U) ∈ A(U)ModY
Given U × Vpr1
//
pr2��
U
��
V // X
an isomorphism
ψU,V : pr∗1 M(U) → pr∗2 M(V ) (sat. cocycle cond.)
AModY := cat. of left A-objects in Y
e.g.
X = Spec k, A ↔ R ⇒ AModY ≡ RModY
Adam Nyman
![Page 89: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/89.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry: Tensor products with Left A-objects
If F ∈ AModY
Adam Nyman
![Page 90: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/90.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry: Tensor products with Left A-objects
If F ∈ AModY define
−⊗A F : ModA → ModYOX
over open affine U ⊂ X by
Adam Nyman
![Page 91: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/91.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry: Tensor products with Left A-objects
If F ∈ AModY define
−⊗A F : ModA → ModYOX
over open affine U ⊂ X by
(N ⊗A F)(U) := N (U) ⊗A(U) F(U)
Adam Nyman
![Page 92: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/92.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry: Tensor products with Left A-objects
If F ∈ AModY define
−⊗A F : ModA → ModYOX
over open affine U ⊂ X by
(N ⊗A F)(U) := N (U) ⊗A(U) F(U)
Define gluing isomorphisms locally as well.
Adam Nyman
![Page 93: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/93.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry: Tensor products with Left A-objects
If F ∈ AModY define
−⊗A F : ModA → ModYOX
over open affine U ⊂ X by
(N ⊗A F)(U) := N (U) ⊗A(U) F(U)
Define gluing isomorphisms locally as well.
F is flat/A ⇔ −⊗A F is exact.
Adam Nyman
![Page 94: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/94.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry: Cech Cohomology
Define π∗ : ModYOX→ ModY via rel. Cech Cohomology:
Adam Nyman
![Page 95: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/95.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry: Cech Cohomology
Define π∗ : ModYOX→ ModY via rel. Cech Cohomology:
X quasi-compact ⇒ X has finite affine open cover U : {Ui}
Adam Nyman
![Page 96: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/96.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry: Cech Cohomology
Define π∗ : ModYOX→ ModY via rel. Cech Cohomology:
X quasi-compact ⇒ X has finite affine open cover U : {Ui}
For N ∈ ModYOX, let
Adam Nyman
![Page 97: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/97.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry: Cech Cohomology
Define π∗ : ModYOX→ ModY via rel. Cech Cohomology:
X quasi-compact ⇒ X has finite affine open cover U : {Ui}
For N ∈ ModYOX, let
Cp(U,N ) :=⊕
i0<i1<···<ip
N (Ui0 ∩ · · · ∩ Uip)
Adam Nyman
![Page 98: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/98.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry: Cech Cohomology
Define π∗ : ModYOX→ ModY via rel. Cech Cohomology:
X quasi-compact ⇒ X has finite affine open cover U : {Ui}
For N ∈ ModYOX, let
Cp(U,N ) :=⊕
i0<i1<···<ip
N (Ui0 ∩ · · · ∩ Uip)
d : Cp(U,N ) → Cp+1(U,N ) defined via restriction as usual
Adam Nyman
![Page 99: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/99.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry: Cech Cohomology
Define π∗ : ModYOX→ ModY via rel. Cech Cohomology:
X quasi-compact ⇒ X has finite affine open cover U : {Ui}
For N ∈ ModYOX, let
Cp(U,N ) :=⊕
i0<i1<···<ip
N (Ui0 ∩ · · · ∩ Uip)
d : Cp(U,N ) → Cp+1(U,N ) defined via restriction as usual
Ri π∗N := Hi(C �(U,N ))
Adam Nyman
![Page 100: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/100.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry
Refined Problems
Adam Nyman
![Page 101: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/101.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry
Refined Problems
1 Find notion of integral transform in the semi-comm. setting,i.e. from ModA to ModY .
Adam Nyman
![Page 102: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/102.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry
Refined Problems
1 Find notion of integral transform in the semi-comm. setting,i.e. from ModA to ModY .
2 When is an integral transform in Bimodk(ModA,ModY )
Adam Nyman
![Page 103: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/103.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry
Refined Problems
1 Find notion of integral transform in the semi-comm. setting,i.e. from ModA to ModY .
2 When is an integral transform in Bimodk(ModA,ModY ) i.e.when does π∗(−⊗A F) induce a morphism ofnoncommutative spaces
Y → (X ,A)?
Adam Nyman
![Page 104: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/104.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry
Refined Problems
1 Find notion of integral transform in the semi-comm. setting,i.e. from ModA to ModY .
2 When is an integral transform in Bimodk(ModA,ModY ) i.e.when does π∗(−⊗A F) induce a morphism ofnoncommutative spaces
Y → (X ,A)?
3 When is F ∈ Bimodk(ModA,ModY ) an integral transform?
Adam Nyman
![Page 105: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/105.jpg)
Integral Transforms in Semi-commutative AlgebraicGeometry
Refined Problems
1 Find notion of integral transform in the semi-comm. setting,i.e. from ModA to ModY .
2 When is an integral transform in Bimodk(ModA,ModY ) i.e.when does π∗(−⊗A F) induce a morphism ofnoncommutative spaces
Y → (X ,A)?
3 When is F ∈ Bimodk(ModA,ModY ) an integral transform?If it isn’t, how close is it to being an integral transform?
Adam Nyman
![Page 106: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/106.jpg)
Part 4
Semi-commutative Algebraic Geometry: Maps fromn.c. spaces to curves (w/D. Chan)
Throughout Part 4, k = k
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Commutative example
X =smooth curvef : Y → X is comm. ruled surface with fiber C .
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Commutative example
X =smooth curvef : Y → X is comm. ruled surface with fiber C .If Γ ⊂ Y × X = graph of f , then
Adam Nyman
![Page 109: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/109.jpg)
Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Commutative example
X =smooth curvef : Y → X is comm. ruled surface with fiber C .If Γ ⊂ Y × X = graph of f , then
1 comp. of Hilb OY corresponding to C is X
Adam Nyman
![Page 110: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/110.jpg)
Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Commutative example
X =smooth curvef : Y → X is comm. ruled surface with fiber C .If Γ ⊂ Y × X = graph of f , then
1 comp. of Hilb OY corresponding to C is X
2 OΓtr = corresponding universal quotient of OY , and
Adam Nyman
![Page 111: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/111.jpg)
Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Commutative example
X =smooth curvef : Y → X is comm. ruled surface with fiber C .If Γ ⊂ Y × X = graph of f , then
1 comp. of Hilb OY corresponding to C is X
2 OΓtr = corresponding universal quotient of OY , and
3 If π : X × Y → Y is projection, then f ∗ ∼= π∗(−⊗OXOΓtr ).
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Commutative example
X =smooth curvef : Y → X is comm. ruled surface with fiber C .If Γ ⊂ Y × X = graph of f , then
1 comp. of Hilb OY corresponding to C is X
2 OΓtr = corresponding universal quotient of OY , and
3 If π : X × Y → Y is projection, then f ∗ ∼= π∗(−⊗OXOΓtr ).
Problem
To what extent do 1-3 hold in the semi-commutative setting?
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
If Y is a non-commutative smooth proper d-fold one can usethe following to study Y :
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
If Y is a non-commutative smooth proper d-fold one can usethe following to study Y :
Sheaf Cohomology (Artin-Zhang 1994),
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
If Y is a non-commutative smooth proper d-fold one can usethe following to study Y :
Sheaf Cohomology (Artin-Zhang 1994),
Intersection theory (Mori-Smith 2001),
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
If Y is a non-commutative smooth proper d-fold one can usethe following to study Y :
Sheaf Cohomology (Artin-Zhang 1994),
Intersection theory (Mori-Smith 2001),
Serre duality (Bondal-Kapranov 1990),
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
If Y is a non-commutative smooth proper d-fold one can usethe following to study Y :
Sheaf Cohomology (Artin-Zhang 1994),
Intersection theory (Mori-Smith 2001),
Serre duality (Bondal-Kapranov 1990),
Dimension theory,
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
If Y is a non-commutative smooth proper d-fold one can usethe following to study Y :
Sheaf Cohomology (Artin-Zhang 1994),
Intersection theory (Mori-Smith 2001),
Serre duality (Bondal-Kapranov 1990),
Dimension theory,
Hilbert schemes (Artin-Zhang 2001)
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
If Y is a non-commutative smooth proper d-fold one can usethe following to study Y :
Sheaf Cohomology (Artin-Zhang 1994),
Intersection theory (Mori-Smith 2001),
Serre duality (Bondal-Kapranov 1990),
Dimension theory,
Hilbert schemes (Artin-Zhang 2001)
Adam Nyman
![Page 120: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/120.jpg)
Semi-commutative Algebraic Geometry: Examples of n.c.smooth proper d -folds
Adam Nyman
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Semi-commutative Algebraic Geometry: Examples of n.c.smooth proper d -folds
e.g.
A = Skl(a, b, c) :=k〈x0, x1, x2〉
(axixi+1 + bxi+1xi + cx2i+2 : i = 1, 2, 3 mod 3)
for generic (a : b : c) ∈ P2.
Adam Nyman
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Semi-commutative Algebraic Geometry: Examples of n.c.smooth proper d -folds
e.g.
A = Skl(a, b, c) :=k〈x0, x1, x2〉
(axixi+1 + bxi+1xi + cx2i+2 : i = 1, 2, 3 mod 3)
for generic (a : b : c) ∈ P2. Then ProjA is a n.c. smooth proper
2-fold.
Adam Nyman
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Semi-commutative Algebraic Geometry: Examples of n.c.smooth proper d -folds
e.g.
A = Skl(a, b, c) :=k〈x0, x1, x2〉
(axixi+1 + bxi+1xi + cx2i+2 : i = 1, 2, 3 mod 3)
for generic (a : b : c) ∈ P2. Then ProjA is a n.c. smooth proper
2-fold.
e.g.
Homogenized U(sl2)
Adam Nyman
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Semi-commutative Algebraic Geometry: Examples of n.c.smooth proper d -folds
e.g.
A = Skl(a, b, c) :=k〈x0, x1, x2〉
(axixi+1 + bxi+1xi + cx2i+2 : i = 1, 2, 3 mod 3)
for generic (a : b : c) ∈ P2. Then ProjA is a n.c. smooth proper
2-fold.
e.g.
Homogenized U(sl2) =
A =k〈e, f , h, z〉
(ef − fe − zh, eh − he − 2ze, fh − hf + 2zf , z central).
Adam Nyman
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Semi-commutative Algebraic Geometry: Examples of n.c.smooth proper d -folds
e.g.
A = Skl(a, b, c) :=k〈x0, x1, x2〉
(axixi+1 + bxi+1xi + cx2i+2 : i = 1, 2, 3 mod 3)
for generic (a : b : c) ∈ P2. Then ProjA is a n.c. smooth proper
2-fold.
e.g.
Homogenized U(sl2) =
A =k〈e, f , h, z〉
(ef − fe − zh, eh − he − 2ze, fh − hf + 2zf , z central).
ProjA is a n.c. smooth proper 3-fold.
Adam Nyman
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Semi-commutative Algebraic Geometry: Examples of n.c.smooth proper d -folds
e.g.
A = Skl(a, b, c) :=k〈x0, x1, x2〉
(axixi+1 + bxi+1xi + cx2i+2 : i = 1, 2, 3 mod 3)
for generic (a : b : c) ∈ P2. Then ProjA is a n.c. smooth proper
2-fold.
e.g.
Homogenized U(sl2) =
A =k〈e, f , h, z〉
(ef − fe − zh, eh − he − 2ze, fh − hf + 2zf , z central).
ProjA is a n.c. smooth proper 3-fold. Given t ∈ Z (A)2, ProjA/(t)is a n.c. smooth proper 2-fold.
Adam Nyman
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Maps from n.c. spaces to curves (w/D. Chan): Geometricmachinery
Adam Nyman
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Maps from n.c. spaces to curves (w/D. Chan): Geometricmachinery
Y = ProjA n.c. smooth proper surface w/ struct. sheaf OY := πA
Adam Nyman
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Maps from n.c. spaces to curves (w/D. Chan): Geometricmachinery
Y = ProjA n.c. smooth proper surface w/ struct. sheaf OY := πA
For M,N ∈ modY let
Adam Nyman
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Maps from n.c. spaces to curves (w/D. Chan): Geometricmachinery
Y = ProjA n.c. smooth proper surface w/ struct. sheaf OY := πA
For M,N ∈ modY let
Hi (M) := ExtiY (OY ,M).
Adam Nyman
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Maps from n.c. spaces to curves (w/D. Chan): Geometricmachinery
Y = ProjA n.c. smooth proper surface w/ struct. sheaf OY := πA
For M,N ∈ modY let
Hi (M) := ExtiY (OY ,M).
Theorem (Artin-Zhang 1994)
H i (M) and more generally ExtiY (M,N )
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Maps from n.c. spaces to curves (w/D. Chan): Geometricmachinery
Y = ProjA n.c. smooth proper surface w/ struct. sheaf OY := πA
For M,N ∈ modY let
Hi (M) := ExtiY (OY ,M).
Theorem (Artin-Zhang 1994)
H i (M) and more generally ExtiY (M,N ) are finite dimensional.
Adam Nyman
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Maps from n.c. spaces to curves (w/D. Chan): Geometricmachinery
Y = ProjA n.c. smooth proper surface w/ struct. sheaf OY := πA
For M,N ∈ modY let
Hi (M) := ExtiY (OY ,M).
Theorem (Artin-Zhang 1994)
H i (M) and more generally ExtiY (M,N ) are finite dimensional.
Thus intersection defined (Mori-Smith 2001) by
Adam Nyman
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Maps from n.c. spaces to curves (w/D. Chan): Geometricmachinery
Y = ProjA n.c. smooth proper surface w/ struct. sheaf OY := πA
For M,N ∈ modY let
Hi (M) := ExtiY (OY ,M).
Theorem (Artin-Zhang 1994)
H i (M) and more generally ExtiY (M,N ) are finite dimensional.
Thus intersection defined (Mori-Smith 2001) by
M.N := −2Σi=0
(−1)i dimExtiY (M,N )
Adam Nyman
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Maps from n.c. spaces to curves (w/D. Chan): Geometricmachinery
Y = ProjA n.c. smooth proper surface w/ struct. sheaf OY := πA
For M,N ∈ modY let
Hi (M) := ExtiY (OY ,M).
Theorem (Artin-Zhang 1994)
H i (M) and more generally ExtiY (M,N ) are finite dimensional.
Thus intersection defined (Mori-Smith 2001) by
M.N := −2Σi=0
(−1)i dimExtiY (M,N )
is well defined.
Adam Nyman
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Maps from n.c. spaces to curves (w/D. Chan): Geometricmachinery
Y = ProjA n.c. smooth proper surface w/ struct. sheaf OY := πA
For M,N ∈ modY let
Hi (M) := ExtiY (OY ,M).
Theorem (Artin-Zhang 1994)
H i (M) and more generally ExtiY (M,N ) are finite dimensional.
Thus intersection defined (Mori-Smith 2001) by
M.N := −2Σi=0
(−1)i dimExtiY (M,N )
is well defined.Remark: This specializes to the intersection product for curves ona comm. surface.
Adam Nyman
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Maps from n.c. spaces to curves (w/D. Chan): Geometricmachinery
Y = n.c. smooth proper d-fold.
Adam Nyman
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Maps from n.c. spaces to curves (w/D. Chan): Geometricmachinery
Y = n.c. smooth proper d-fold.
Theorem (Artin-Zhang)
For P ∈ modY , there exists a Hilbert scheme Hilb P
parameterizing quotients of P .
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Maps from n.c. spaces to curves (w/D. Chan): Geometricmachinery
Y = n.c. smooth proper d-fold.
Theorem (Artin-Zhang)
For P ∈ modY , there exists a Hilbert scheme Hilb P
parameterizing quotients of P . Hilb P is countable union ofprojective schemes which is locally of finite type.
Adam Nyman
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Maps from n.c. spaces to curves (w/D. Chan):K -non-effective rational curves
Adam Nyman
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Maps from n.c. spaces to curves (w/D. Chan):K -non-effective rational curves
Y = n.c. smooth proper surface with structure sheaf OY
Adam Nyman
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Maps from n.c. spaces to curves (w/D. Chan):K -non-effective rational curves
Y = n.c. smooth proper surface with structure sheaf OY
Say F ∈ modY is K -non-effective rational curve withself-intersection 0 if
Adam Nyman
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Maps from n.c. spaces to curves (w/D. Chan):K -non-effective rational curves
Y = n.c. smooth proper surface with structure sheaf OY
Say F ∈ modY is K -non-effective rational curve withself-intersection 0 if
1 F is 1-critical quotient of OY
Adam Nyman
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Maps from n.c. spaces to curves (w/D. Chan):K -non-effective rational curves
Y = n.c. smooth proper surface with structure sheaf OY
Say F ∈ modY is K -non-effective rational curve withself-intersection 0 if
1 F is 1-critical quotient of OY
2 H0(F ) = k, H1(F ) = 0, F 2 = 0
Adam Nyman
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Maps from n.c. spaces to curves (w/D. Chan):K -non-effective rational curves
Y = n.c. smooth proper surface with structure sheaf OY
Say F ∈ modY is K -non-effective rational curve withself-intersection 0 if
1 F is 1-critical quotient of OY
2 H0(F ) = k, H1(F ) = 0, F 2 = 0
3 H0(F ⊗ ωY ) = 0
Adam Nyman
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Maps from n.c. spaces to curves (w/D. Chan):K -non-effective rational curves
Y = n.c. smooth proper surface with structure sheaf OY
Say F ∈ modY is K -non-effective rational curve withself-intersection 0 if
1 F is 1-critical quotient of OY
2 H0(F ) = k, H1(F ) = 0, F 2 = 0
3 H0(F ⊗ ωY ) = 0
Remark: In comm. case, 1-2 ⇒ F is struct. sheaf of K -negative,rational curve with self-intersection 0
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Maps from n.c. spaces to curves (w/D. Chan):K -non-effective rational curves
Y = n.c. smooth proper surface with structure sheaf OY
Say F ∈ modY is K -non-effective rational curve withself-intersection 0 if
1 F is 1-critical quotient of OY
2 H0(F ) = k, H1(F ) = 0, F 2 = 0
3 H0(F ⊗ ωY ) = 0
Remark: In comm. case, 1-2 ⇒ F is struct. sheaf of K -negative,rational curve with self-intersection 0. 3 is substitute forK -negativity, since we don’t know K .C < 0 ⇒ H0(OC ⊗ ω) = 0.
Adam Nyman
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Maps from n.c. spaces to curves (w/D. Chan):K -non-effective rational curves
Y = n.c. smooth proper surface with structure sheaf OY
Say F ∈ modY is K -non-effective rational curve withself-intersection 0 if
1 F is 1-critical quotient of OY
2 H0(F ) = k, H1(F ) = 0, F 2 = 0
3 H0(F ⊗ ωY ) = 0
Remark: In comm. case, 1-2 ⇒ F is struct. sheaf of K -negative,rational curve with self-intersection 0. 3 is substitute forK -negativity, since we don’t know K .C < 0 ⇒ H0(OC ⊗ ω) = 0.
e.g.
If f : Y → X is a comm. or n.c. ruled surface
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Maps from n.c. spaces to curves (w/D. Chan):K -non-effective rational curves
Y = n.c. smooth proper surface with structure sheaf OY
Say F ∈ modY is K -non-effective rational curve withself-intersection 0 if
1 F is 1-critical quotient of OY
2 H0(F ) = k, H1(F ) = 0, F 2 = 0
3 H0(F ⊗ ωY ) = 0
Remark: In comm. case, 1-2 ⇒ F is struct. sheaf of K -negative,rational curve with self-intersection 0. 3 is substitute forK -negativity, since we don’t know K .C < 0 ⇒ H0(OC ⊗ ω) = 0.
e.g.
If f : Y → X is a comm. or n.c. ruled surface then the structuresheaf of fiber is K -non-effective rational curve with self-intersection0.
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Theorem (Chan-N, 2009)
Suppose Y is a n.c. smooth proper surface.
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Theorem (Chan-N, 2009)
Suppose Y is a n.c. smooth proper surface. If
1 F is K -non-effective rational curve with F 2 = 0
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Theorem (Chan-N, 2009)
Suppose Y is a n.c. smooth proper surface. If
1 F is K -non-effective rational curve with F 2 = 0, and
2 for every simple 0-dim quotient P ∈ modY of F we haveF .P = 0
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Theorem (Chan-N, 2009)
Suppose Y is a n.c. smooth proper surface. If
1 F is K -non-effective rational curve with F 2 = 0, and
2 for every simple 0-dim quotient P ∈ modY of F we haveF .P = 0,
then the component of Hilb OY containing F is a smooth curve X
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Theorem (Chan-N, 2009)
Suppose Y is a n.c. smooth proper surface. If
1 F is K -non-effective rational curve with F 2 = 0, and
2 for every simple 0-dim quotient P ∈ modY of F we haveF .P = 0,
then the component of Hilb OY containing F is a smooth curve X ,and the corresponding family F over X is such that the followingis exact and preserves noetherian objects:
π∗(−⊗OXF) : QcohX → ModY
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Theorem (Chan-N, 2009)
Suppose Y is a n.c. smooth proper surface. If
1 F is K -non-effective rational curve with F 2 = 0, and
2 for every simple 0-dim quotient P ∈ modY of F we haveF .P = 0,
then the component of Hilb OY containing F is a smooth curve X ,and the corresponding family F over X is such that the followingis exact and preserves noetherian objects:
π∗(−⊗OXF) : QcohX → ModY
e.g.
If X = smooth curve, f : Y → X = n.c. ruled surface and F =struct. sheaf of fiber
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Theorem (Chan-N, 2009)
Suppose Y is a n.c. smooth proper surface. If
1 F is K -non-effective rational curve with F 2 = 0, and
2 for every simple 0-dim quotient P ∈ modY of F we haveF .P = 0,
then the component of Hilb OY containing F is a smooth curve X ,and the corresponding family F over X is such that the followingis exact and preserves noetherian objects:
π∗(−⊗OXF) : QcohX → ModY
e.g.
If X = smooth curve, f : Y → X = n.c. ruled surface and F =struct. sheaf of fiber then F satisfies 1,2
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Theorem (Chan-N, 2009)
Suppose Y is a n.c. smooth proper surface. If
1 F is K -non-effective rational curve with F 2 = 0, and
2 for every simple 0-dim quotient P ∈ modY of F we haveF .P = 0,
then the component of Hilb OY containing F is a smooth curve X ,and the corresponding family F over X is such that the followingis exact and preserves noetherian objects:
π∗(−⊗OXF) : QcohX → ModY
e.g.
If X = smooth curve, f : Y → X = n.c. ruled surface and F =struct. sheaf of fiber then F satisfies 1,2 and f ∗ ∼= π∗(−⊗OX
F).
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
More generally:
Y =n.c. smooth proper d-fold
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
More generally:
Y =n.c. smooth proper d-fold
X=integral proj. curve
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
More generally:
Y =n.c. smooth proper d-fold
X=integral proj. curve
F ∈ OXModY flat/X .
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
More generally:
Y =n.c. smooth proper d-fold
X=integral proj. curve
F ∈ OXModY flat/X .
Say F is base point free if for any simple P ∈ modY ,HomY (F ,P) = 0 for a generic fibre F ∈ F
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
More generally:
Y =n.c. smooth proper d-fold
X=integral proj. curve
F ∈ OXModY flat/X .
Say F is base point free if for any simple P ∈ modY ,HomY (F ,P) = 0 for a generic fibre F ∈ F
Theorem (Chan-N 2009)
If F is base point free
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
More generally:
Y =n.c. smooth proper d-fold
X=integral proj. curve
F ∈ OXModY flat/X .
Say F is base point free if for any simple P ∈ modY ,HomY (F ,P) = 0 for a generic fibre F ∈ F
Theorem (Chan-N 2009)
If F is base point free then π∗(−⊗OXF) is exact
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
More generally:
Y =n.c. smooth proper d-fold
X=integral proj. curve
F ∈ OXModY flat/X .
Say F is base point free if for any simple P ∈ modY ,HomY (F ,P) = 0 for a generic fibre F ∈ F
Theorem (Chan-N 2009)
If F is base point free then π∗(−⊗OXF) is exact hence in
Bimodk(QcohX ,ModY )
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
More generally:
Y =n.c. smooth proper d-fold
X=integral proj. curve
F ∈ OXModY flat/X .
Say F is base point free if for any simple P ∈ modY ,HomY (F ,P) = 0 for a generic fibre F ∈ F
Theorem (Chan-N 2009)
If F is base point free then π∗(−⊗OXF) is exact hence in
Bimodk(QcohX ,ModY ), and preserves noetherian objects.
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Refined Problems
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Refined Problems
1 Find notion of integral transform in the semi-comm. setting,i.e. from ModA to ModY .
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Refined Problems
1 Find notion of integral transform in the semi-comm. setting,i.e. from ModA to ModY .
2 When is an integral transform in Bimodk(ModA,ModY )
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Refined Problems
1 Find notion of integral transform in the semi-comm. setting,i.e. from ModA to ModY .
2 When is an integral transform in Bimodk(ModA,ModY ) i.e.when does π∗(−⊗A F) induce a morphism ofnoncommutative spaces
Y → (X ,A)?
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Refined Problems
1 Find notion of integral transform in the semi-comm. setting,i.e. from ModA to ModY .
2 When is an integral transform in Bimodk(ModA,ModY ) i.e.when does π∗(−⊗A F) induce a morphism ofnoncommutative spaces
Y → (X ,A)?
3 When is F ∈ Bimodk(ModA,ModY ) an integral transform?
Adam Nyman
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Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Refined Problems
1 Find notion of integral transform in the semi-comm. setting,i.e. from ModA to ModY .
2 When is an integral transform in Bimodk(ModA,ModY ) i.e.when does π∗(−⊗A F) induce a morphism ofnoncommutative spaces
Y → (X ,A)?
3 When is F ∈ Bimodk(ModA,ModY ) an integral transform?If it isn’t, how close is it to being an integral transform?
Adam Nyman
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Part 5
Integral Transforms and Bimod revisited
Adam Nyman
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Integral Transforms and Bimod revisited: The Functor(−)♭.
Adam Nyman
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Integral Transforms and Bimod revisited: The Functor(−)♭.
F ∈ Bimodk(ModA,ModY )
Adam Nyman
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Integral Transforms and Bimod revisited: The Functor(−)♭.
F ∈ Bimodk(ModA,ModY )
u : U → X denotes inclusion of affine open
Adam Nyman
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Integral Transforms and Bimod revisited: The Functor(−)♭.
F ∈ Bimodk(ModA,ModY )
u : U → X denotes inclusion of affine open
Fu∗ ∈ Bimodk(ModA(U),ModY ) ⇒
Fu∗∼= −⊗A(U) FU
for some FU ∈ A(U)ModY .
Adam Nyman
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Integral Transforms and Bimod revisited: The Functor(−)♭.
F ∈ Bimodk(ModA,ModY )
u : U → X denotes inclusion of affine open
Fu∗ ∈ Bimodk(ModA(U),ModY ) ⇒
Fu∗∼= −⊗A(U) FU
for some FU ∈ A(U)ModY .
Theorem (Van den Bergh, Chan-N.)
The collection F ♭(U) = FU induces a functor
(−)♭ : Bimodk(ModA,ModY ) → AModY
Adam Nyman
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Integral Transforms and Bimod revisited: The Functor(−)♭.
F ∈ Bimodk(ModA,ModY )
u : U → X denotes inclusion of affine open
Fu∗ ∈ Bimodk(ModA(U),ModY ) ⇒
Fu∗∼= −⊗A(U) FU
for some FU ∈ A(U)ModY .
Theorem (Van den Bergh, Chan-N.)
The collection F ♭(U) = FU induces a functor
(−)♭ : Bimodk(ModA,ModY ) → AModY
Furthermore, if F = π∗(− ⊗A F) then
Adam Nyman
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Integral Transforms and Bimod revisited: The Functor(−)♭.
F ∈ Bimodk(ModA,ModY )
u : U → X denotes inclusion of affine open
Fu∗ ∈ Bimodk(ModA(U),ModY ) ⇒
Fu∗∼= −⊗A(U) FU
for some FU ∈ A(U)ModY .
Theorem (Van den Bergh, Chan-N.)
The collection F ♭(U) = FU induces a functor
(−)♭ : Bimodk(ModA,ModY ) → AModY
Furthermore, if F = π∗(− ⊗A F) then F ♭ ∼= F .
Adam Nyman
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Integral Transforms and Bimod revisited
Goals
Let F ∈ Bimodk(ModA,ModY ).
Adam Nyman
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Integral Transforms and Bimod revisited
Goals
Let F ∈ Bimodk(ModA,ModY ).
1 Find relationship between F and π∗(−⊗A F ♭)
Adam Nyman
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Integral Transforms and Bimod revisited
Goals
Let F ∈ Bimodk(ModA,ModY ).
1 Find relationship between F and π∗(−⊗A F ♭)
2 Find necessary and sufficient conditions for F ∼= π∗(−⊗A F ♭)
Adam Nyman
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Integral Transforms and Bimod revisited: Totally GlobalFunctors
F ∈ Bimodk(ModA,ModY ) such that F ♭ = 0 are totally global
Adam Nyman
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Integral Transforms and Bimod revisited: Totally GlobalFunctors
F ∈ Bimodk(ModA,ModY ) such that F ♭ = 0 are totally global
e.g.
F = H1(P1,−) ∈ Bimodk(QcohP1,Modk) is totally global.
Adam Nyman
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Integral Transforms and Bimod revisited: Totally GlobalFunctors
F ∈ Bimodk(ModA,ModY ) such that F ♭ = 0 are totally global
e.g.
F = H1(P1,−) ∈ Bimodk(QcohP1,Modk) is totally global.
Theorem (N-Smith, 2008)
k = k, F ∈ Bimodk(QcohP1,Modk). If
Adam Nyman
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Integral Transforms and Bimod revisited: Totally GlobalFunctors
F ∈ Bimodk(ModA,ModY ) such that F ♭ = 0 are totally global
e.g.
F = H1(P1,−) ∈ Bimodk(QcohP1,Modk) is totally global.
Theorem (N-Smith, 2008)
k = k, F ∈ Bimodk(QcohP1,Modk). If
F is totally global and
Adam Nyman
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Integral Transforms and Bimod revisited: Totally GlobalFunctors
F ∈ Bimodk(ModA,ModY ) such that F ♭ = 0 are totally global
e.g.
F = H1(P1,−) ∈ Bimodk(QcohP1,Modk) is totally global.
Theorem (N-Smith, 2008)
k = k, F ∈ Bimodk(QcohP1,Modk). If
F is totally global and
F preserves noetherian objects, then
Adam Nyman
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Integral Transforms and Bimod revisited: Totally GlobalFunctors
F ∈ Bimodk(ModA,ModY ) such that F ♭ = 0 are totally global
e.g.
F = H1(P1,−) ∈ Bimodk(QcohP1,Modk) is totally global.
Theorem (N-Smith, 2008)
k = k, F ∈ Bimodk(QcohP1,Modk). If
F is totally global and
F preserves noetherian objects, then
F ∼=⊕∞
i=m H1(P1, (−)(i))⊕ni
Adam Nyman
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Integral Transforms and Bimod revisited
Throughout F ∈ Bimodk(ModA,ModY ).
Adam Nyman
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Integral Transforms and Bimod revisited
Throughout F ∈ Bimodk(ModA,ModY ).
Claim A
There is a natural transformation
ΓF : F → π∗(−⊗A F♭)
Adam Nyman
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Integral Transforms and Bimod revisited
Throughout F ∈ Bimodk(ModA,ModY ).
Claim A
There is a natural transformation
ΓF : F → π∗(−⊗A F♭)
such that ker ΓF and cok ΓF are totally global.
Adam Nyman
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Integral Transforms and Bimod revisited
Throughout F ∈ Bimodk(ModA,ModY ).
Claim A
There is a natural transformation
ΓF : F → π∗(−⊗A F♭)
such that ker ΓF and cok ΓF are totally global.
It follows that ΓF is an isomorphism if
Adam Nyman
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Integral Transforms and Bimod revisited
Throughout F ∈ Bimodk(ModA,ModY ).
Claim A
There is a natural transformation
ΓF : F → π∗(−⊗A F♭)
such that ker ΓF and cok ΓF are totally global.
It follows that ΓF is an isomorphism if
1 X is affine or
Adam Nyman
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Integral Transforms and Bimod revisited
Throughout F ∈ Bimodk(ModA,ModY ).
Claim A
There is a natural transformation
ΓF : F → π∗(−⊗A F♭)
such that ker ΓF and cok ΓF are totally global.
It follows that ΓF is an isomorphism if
1 X is affine or
2 F is exact
Adam Nyman
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Integral Transforms and Bimod revisited
Remarks:
Adam Nyman
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Integral Transforms and Bimod revisited
Remarks:
1 Claim A confirmed in case A = OX and Y = scheme (N2009)
Adam Nyman
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Integral Transforms and Bimod revisited
Remarks:
1 Claim A confirmed in case A = OX and Y = scheme (N2009)
2 Claim A 1 confirmed (N-Smith 2008)
Adam Nyman
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Integral Transforms and Bimod revisited
Remarks:
1 Claim A confirmed in case A = OX and Y = scheme (N2009)
2 Claim A 1 confirmed (N-Smith 2008)
3 Claim A 2 confirmed in case A = OX (Chan-N 2009)
Adam Nyman
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Integral Transforms and Bimod revisited
Remarks:
1 Claim A confirmed in case A = OX and Y = scheme (N2009)
2 Claim A 1 confirmed (N-Smith 2008)
3 Claim A 2 confirmed in case A = OX (Chan-N 2009)
4 Test Problem: Classify noetherian preservingF ∈ Bimodk(QcohP
1,Modk) in case k = k.
Adam Nyman
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Integral Transforms and Bimod revisited
Proof (that X affine ⇒ ΓF is an isomorphism):
Adam Nyman
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Integral Transforms and Bimod revisited
Proof (that X affine ⇒ ΓF is an isomorphism):
Easy fact: F totally global ⇔ Fu∗ = 0 ∀ open affine u : U → X .
Adam Nyman
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Integral Transforms and Bimod revisited
Proof (that X affine ⇒ ΓF is an isomorphism):
Easy fact: F totally global ⇔ Fu∗ = 0 ∀ open affine u : U → X .
X affine ⇒ idX is inclusion of affine open
Adam Nyman
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Integral Transforms and Bimod revisited
Proof (that X affine ⇒ ΓF is an isomorphism):
Easy fact: F totally global ⇔ Fu∗ = 0 ∀ open affine u : U → X .
X affine ⇒ idX is inclusion of affine open
ker ΓF = ker ΓF ◦ idX∗
Adam Nyman
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Integral Transforms and Bimod revisited
Proof (that X affine ⇒ ΓF is an isomorphism):
Easy fact: F totally global ⇔ Fu∗ = 0 ∀ open affine u : U → X .
X affine ⇒ idX is inclusion of affine open
ker ΓF = ker ΓF ◦ idX∗
Since ker ΓF is totally global,
Adam Nyman
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Integral Transforms and Bimod revisited
Proof (that X affine ⇒ ΓF is an isomorphism):
Easy fact: F totally global ⇔ Fu∗ = 0 ∀ open affine u : U → X .
X affine ⇒ idX is inclusion of affine open
ker ΓF = ker ΓF ◦ idX∗
Since ker ΓF is totally global, ker ΓF ◦ idX∗ = 0.
Adam Nyman
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Integral Transforms and Bimod revisited
Proof (that X affine ⇒ ΓF is an isomorphism):
Easy fact: F totally global ⇔ Fu∗ = 0 ∀ open affine u : U → X .
X affine ⇒ idX is inclusion of affine open
ker ΓF = ker ΓF ◦ idX∗
Since ker ΓF is totally global, ker ΓF ◦ idX∗ = 0.
Similarly, cok ΓF = 0.
Adam Nyman
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Integral Transforms and Bimod revisited
U = finite affine open cover of X
Adam Nyman
![Page 209: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/209.jpg)
Integral Transforms and Bimod revisited
U = finite affine open cover of X
M ∈ ModA
Adam Nyman
![Page 210: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/210.jpg)
Integral Transforms and Bimod revisited
U = finite affine open cover of X
M ∈ ModA
d : C 0(U,M) → C 1(U,M) diff. of sheafified Cech complex
Adam Nyman
![Page 211: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/211.jpg)
Integral Transforms and Bimod revisited
U = finite affine open cover of X
M ∈ ModA
d : C 0(U,M) → C 1(U,M) diff. of sheafified Cech complex
Claim B
ΓF : F → π∗(−⊗A F ♭) is an isomorphism iff
Adam Nyman
![Page 212: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/212.jpg)
Integral Transforms and Bimod revisited
U = finite affine open cover of X
M ∈ ModA
d : C 0(U,M) → C 1(U,M) diff. of sheafified Cech complex
Claim B
ΓF : F → π∗(−⊗A F ♭) is an isomorphism iff
1 the canonical map F (ker d) → ker(Fd) is an isomorphism forall flat M.
Adam Nyman
![Page 213: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/213.jpg)
Integral Transforms and Bimod revisited
U = finite affine open cover of X
M ∈ ModA
d : C 0(U,M) → C 1(U,M) diff. of sheafified Cech complex
Claim B
ΓF : F → π∗(−⊗A F ♭) is an isomorphism iff
1 the canonical map F (ker d) → ker(Fd) is an isomorphism forall flat M.
2 π∗(− ⊗A F ♭) is right exact
Adam Nyman
![Page 214: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/214.jpg)
Integral Transforms and Bimod revisited
U = finite affine open cover of X
M ∈ ModA
d : C 0(U,M) → C 1(U,M) diff. of sheafified Cech complex
Claim B
ΓF : F → π∗(−⊗A F ♭) is an isomorphism iff
1 the canonical map F (ker d) → ker(Fd) is an isomorphism forall flat M.
2 π∗(− ⊗A F ♭) is right exact
Claim B confirmed in case A = OX and Y = scheme (N 2009)
Adam Nyman
![Page 215: Classical Integral Transforms in Semi-commutative Algebraic Geometry](https://reader035.vdocument.in/reader035/viewer/2022071519/613c623cf237e1331c513c30/html5/thumbnails/215.jpg)
Thank you for your attention!
Adam Nyman