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Classical Integral Transforms in Semi-commutativeAlgebraic Geometry
Adam Nyman
Western Washington University
August 27, 2009
Adam Nyman
Conventions and Notation
Adam Nyman
Conventions and Notation
always work over commutative ring k,
Adam Nyman
Conventions and Notation
always work over commutative ring k,
X is (comm.) quasi-compact separated k-scheme
Adam Nyman
Conventions and Notation
always work over commutative ring k,
X is (comm.) quasi-compact separated k-scheme
“scheme”=commutative k-scheme
Adam Nyman
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
Part 1
Integral Transforms and Bimod: Examples
Adam Nyman
Integral Transforms and Bimod: Examples
Example 1R , S rings, F an R − S-bimodule
−⊗R F : ModR → ModS
Adam Nyman
Integral Transforms and Bimod: Examples
Example 1R , S rings, F an R − S-bimodule
−⊗R F : ModR → ModS
F =“integral kernel”
Adam Nyman
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
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
Integral Transforms and Bimod: A Generalization ofExample 1
A = cocomplete abelian cat.
Adam Nyman
Integral Transforms and Bimod: A Generalization ofExample 1
A = cocomplete abelian cat.
RA = cat. of left R-objects in A
Adam Nyman
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
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
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
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
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
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
Integral Transforms and Bimod: A Generalization ofExample 1
Theorem (N-Smith 2007)
Every F ∈ Bimodk(ModR ,A) is an integral transform.
Adam Nyman
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
Integral Transforms and Bimod: Examples
Example 2 Y scheme
Adam Nyman
Integral Transforms and Bimod: Examples
Example 2 Y scheme X × Y
τ
{{wwww
wwww
wπ
##GGGG
GGGG
G
X Y
Adam Nyman
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
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
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
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
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
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
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
Integral Transforms and Bimod: Examples
Example 3
k a field
Adam Nyman
Integral Transforms and Bimod: Examples
Example 3
k a field, X , Y finite type over k
Adam Nyman
Integral Transforms and Bimod: Examples
Example 3
k a field, X , Y finite type over k
A finite OX -algebra
Adam Nyman
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
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
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
Integral Transforms and Bimod: Examples
Problems
Adam Nyman
Integral Transforms and Bimod: Examples
Problems
1 Find notion of integral transform generalizing examples
Adam Nyman
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
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
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
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
Part 2
Non- and Semi-commutative Algebraic Geometry
Adam Nyman
Non-commutative Algebraic Geometry: NoncommutativeSpaces
Non-commutative Space := Grothendieck Category
Adam Nyman
Non-commutative Algebraic Geometry: NoncommutativeSpaces
Non-commutative Space := Grothendieck Category =
(k-linear) abelian category with
Adam Nyman
Non-commutative Algebraic Geometry: NoncommutativeSpaces
Non-commutative Space := Grothendieck Category =
(k-linear) abelian category with
exact direct limits and
Adam Nyman
Non-commutative Algebraic Geometry: NoncommutativeSpaces
Non-commutative Space := Grothendieck Category =
(k-linear) abelian category with
exact direct limits and
a generator.
Adam Nyman
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
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
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
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
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
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
Non-commutative Algebraic Geometry: MorphismsBetween Noncommutative Spaces
Y , Z non-commutative spaces
Adam Nyman
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
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
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
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
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
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
Semi-commutative Algebraic Geometry
A = quasi-coherent sheaf of OX -algebras (throughout)
Adam Nyman
Semi-commutative Algebraic Geometry
A = quasi-coherent sheaf of OX -algebras (throughout)
Y = non-commutative space
Adam Nyman
Semi-commutative Algebraic Geometry
A = quasi-coherent sheaf of OX -algebras (throughout)
Y = non-commutative space
Semi-comm. Alg. Geom.
Adam Nyman
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
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
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
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
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
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
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
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
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
Refined Problems
Adam Nyman
Refined Problems
1 Find notion of integral transform in the semi-comm. setting,i.e. from ModA to ModY .
Adam Nyman
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
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
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
Part 3
Integral Transforms in Semi-commutative AlgebraicGeometry w/ D. Chan in case A = OX
Adam Nyman
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
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
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
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
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
Integral Transforms in Semi-commutative AlgebraicGeometry: Left A-objects
A left A-object in Y , denoted M, consists of following data:
Adam Nyman
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
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
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
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
Integral Transforms in Semi-commutative AlgebraicGeometry: Tensor products with Left A-objects
If F ∈ AModY
Adam Nyman
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
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
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
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
Integral Transforms in Semi-commutative AlgebraicGeometry: Cech Cohomology
Define π∗ : ModYOX→ ModY via rel. Cech Cohomology:
Adam Nyman
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
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
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
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
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
Integral Transforms in Semi-commutative AlgebraicGeometry
Refined Problems
Adam Nyman
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
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
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
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
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
Part 4
Semi-commutative Algebraic Geometry: Maps fromn.c. spaces to curves (w/D. Chan)
Throughout Part 4, k = k
Adam Nyman
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
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
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
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
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
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
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
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
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
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
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
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
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
Semi-commutative Algebraic Geometry: Examples of n.c.smooth proper d -folds
Adam Nyman
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
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
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
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
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
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
Maps from n.c. spaces to curves (w/D. Chan): Geometricmachinery
Adam Nyman
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
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
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
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 )
Adam Nyman
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
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
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
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
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
Maps from n.c. spaces to curves (w/D. Chan): Geometricmachinery
Y = n.c. smooth proper d-fold.
Adam Nyman
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 .
Adam Nyman
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
Maps from n.c. spaces to curves (w/D. Chan):K -non-effective rational curves
Adam Nyman
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
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
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
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
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
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
Adam Nyman
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
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
Adam Nyman
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
Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Adam Nyman
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
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
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
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
Adam Nyman
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
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
Adam Nyman
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
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
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
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
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
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
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
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
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
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
Semi-commutative Algebraic Geometry: Maps from n.c.spaces to curves (w/D. Chan)
Refined Problems
Adam Nyman
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
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
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
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
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
Part 5
Integral Transforms and Bimod revisited
Adam Nyman
Integral Transforms and Bimod revisited: The Functor(−)♭.
Adam Nyman
Integral Transforms and Bimod revisited: The Functor(−)♭.
F ∈ Bimodk(ModA,ModY )
Adam Nyman
Integral Transforms and Bimod revisited: The Functor(−)♭.
F ∈ Bimodk(ModA,ModY )
u : U → X denotes inclusion of affine open
Adam Nyman
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
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
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
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
Integral Transforms and Bimod revisited
Goals
Let F ∈ Bimodk(ModA,ModY ).
Adam Nyman
Integral Transforms and Bimod revisited
Goals
Let F ∈ Bimodk(ModA,ModY ).
1 Find relationship between F and π∗(−⊗A F ♭)
Adam Nyman
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
Integral Transforms and Bimod revisited: Totally GlobalFunctors
F ∈ Bimodk(ModA,ModY ) such that F ♭ = 0 are totally global
Adam Nyman
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
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
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
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
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
Integral Transforms and Bimod revisited
Throughout F ∈ Bimodk(ModA,ModY ).
Adam Nyman
Integral Transforms and Bimod revisited
Throughout F ∈ Bimodk(ModA,ModY ).
Claim A
There is a natural transformation
ΓF : F → π∗(−⊗A F♭)
Adam Nyman
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
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
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
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
Integral Transforms and Bimod revisited
Remarks:
Adam Nyman
Integral Transforms and Bimod revisited
Remarks:
1 Claim A confirmed in case A = OX and Y = scheme (N2009)
Adam Nyman
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
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
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
Integral Transforms and Bimod revisited
Proof (that X affine ⇒ ΓF is an isomorphism):
Adam Nyman
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
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
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
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
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
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
Integral Transforms and Bimod revisited
U = finite affine open cover of X
Adam Nyman
Integral Transforms and Bimod revisited
U = finite affine open cover of X
M ∈ ModA
Adam Nyman
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
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
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
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
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
Thank you for your attention!
Adam Nyman
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