Classical Integral Transforms in Semi-commutative Algebraic Geometry

Classical Integral Transforms in Semi-commutativeAlgebraic Geometry

Adam Nyman

Western Washington University

August 27, 2009

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Conventions and Notation

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always work over commutative ring k,

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X is (comm.) quasi-compact separated k-scheme

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“scheme”=commutative k-scheme

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“scheme”=commutative k-scheme

“bimodule”=object in Bimodk(C,D)

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Part 1

Integral Transforms and Bimod: Examples

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Example 1R , S rings, F an R − S-bimodule

−⊗R F : ModR → ModS

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F =“integral kernel”

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Theorem (Eilenberg, Watts 1960)

Every F ∈ Bimodk(ModR ,ModS) is an integral transform.

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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).

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Integral Transforms and Bimod: A Generalization ofExample 1

A = cocomplete abelian cat.

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RA = cat. of left R-objects in A

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ob RA : (F , ρ) where F ∈ obA, ρ : R → EndA F

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Remark: AR defined similarly

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e.g.

if A = ModS then RA ≡ Mod(Rop ⊗k S).

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e.g.


For F ∈ RA, define

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e.g.



−⊗R F : ModR → A

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e.g.



−⊗R F : ModR → A

as left adjoint to

HomA(F ,−) : A → ModR

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Theorem (N-Smith 2007)

Every F ∈ Bimodk(ModR ,A) is an integral transform.

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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).

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Example 2 Y scheme

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Example 2 Y scheme X × Y

τ

{{wwww

wwww

wπ

##GGGG

GGGG

G

X Y

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τ

{{wwww

wwww

wπ

##GGGG

GGGG

G

X Y

F ∈ Qcoh(X ×Y )

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τ

{{wwww

wwww

wπ

##GGGG

GGGG

G

X Y

F ∈ Qcoh(X ×Y )

π∗(τ∗(−) ⊗OX×Y

F) : QcohX → QcohY

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τ

{{wwww

wwww

wπ

##GGGG

GGGG

G

X Y

F ∈ Qcoh(X ×Y )

π∗(τ∗(−) ⊗OX×Y


e.g.

If f : Y → X is a morphism of schemes thenf ∗ : QcohX → QcohY is an integral transform.

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τ

{{wwww

wwww

wπ

##GGGG

GGGG

G

X Y

F ∈ Qcoh(X ×Y )

π∗(τ∗(−) ⊗OX×Y


e.g.


e.g.

Let X = P1 and Y = Spec k, F = OX .

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τ

{{wwww

wwww

wπ

##GGGG

GGGG

G

X Y

F ∈ Qcoh(X ×Y )

π∗(τ∗(−) ⊗OX×Y


e.g.


e.g.


H1(X ,−) ∈ Bimodk(QcohX ,QcohY ) is not an integraltransform.

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τ

{{wwww

wwww

wπ

##GGGG

GGGG

G

X Y

F ∈ Qcoh(X ×Y )

π∗(τ∗(−) ⊗OX×Y


e.g.


e.g.



Int. trans. not always rt exact

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τ

{{wwww

wwww

wπ

##GGGG

GGGG

G

X Y

F ∈ Qcoh(X ×Y )

π∗(τ∗(−) ⊗OX×Y


e.g.


e.g.



Int. trans. not always rt exact: π∗(τ∗(−) ⊗OX

F) ∼= Γ(P1,−).

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Example 3

k a field

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Example 3

k a field, X , Y finite type over k

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Example 3


A finite OX -algebra

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Example 3


A finite OX -algebra, B finite OY -algebra

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Example 3



Theorem (Artin-Zhang 1994)

If F : modA → modB is an equivalence

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Example 3




If F : modA → modB is an equivalence then

F ∼= π∗(τ∗(−) ⊗τ∗A F)

for some F ∈ mod(Aop ⊗k B).

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Problems

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Problems

1 Find notion of integral transform generalizing examples

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Problems


2 When is an integral transform a bimodule i.e. when is it rt.exact?

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Problems



3 When is a bimodule an integral transform?

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Problems



3 When is a bimodule an integral transform? If it is not, howclose is it to being an integral transform?

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Problems



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

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Part 2

Non- and Semi-commutative Algebraic Geometry

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Non-commutative Algebraic Geometry: NoncommutativeSpaces

Non-commutative Space := Grothendieck Category

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Non-commutative Space := Grothendieck Category =

(k-linear) abelian category with

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exact direct limits and

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a generator.

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a generator.

Notation: Y geometry

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a generator.

Notation: Y geometry or ModY category theory

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a generator.


The following are non-commutative spaces:

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a generator.



Mod R , R a ring

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a generator.



Mod R , R a ring

Qcoh X

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a generator.



Mod R , R a ring

Qcoh X

Proj A := GrA/TorsA where A is Z-graded

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Non-commutative Algebraic Geometry: MorphismsBetween Noncommutative Spaces

Y , Z non-commutative spaces

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Yf→ Z denotes adjoint pair (f ∗, f∗) in the diagram

ModYf∗⇋

f ∗ModZ

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ModYf∗⇋

f ∗ModZ

Motivation

If f : Y → X is a morphism of commutative schemes, (f ∗, f∗) is anadjoint pair.

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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 ).

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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).

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e.g.


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 .

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e.g.


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.

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Semi-commutative Algebraic Geometry

A = quasi-coherent sheaf of OX -algebras (throughout)

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Y = non-commutative space

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Semi-comm. Alg. Geom.

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Semi-comm. Alg. Geom. = Study of mapsY → (X ,A)

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= Study of Bimodk(ModA,ModY )

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Examples in Part 1 are semi-commutative

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1 Example 1: F : ModR → ModY .

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1 Example 1: F : ModR → ModY .If X = Spec k, A ↔ R then ModR ≡ ModA.

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2 Example 2: F : QcohX → QcohY .

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2 Example 2: F : QcohX → QcohY .If A = OX QcohX ≡ ModA. Let ModY = QcohY .

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3 Example 3: F : ModA → ModB.

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3 Example 3: F : ModA → ModB.Let ModY = ModB.

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Refined Problems

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Refined Problems

1 Find notion of integral transform in the semi-comm. setting,i.e. from ModA to ModY .

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Refined Problems


2 When is an integral transform in Bimodk(ModA,ModY ) i.e.when is it rt. exact?

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Refined Problems



3 When is F ∈ Bimodk(ModA,ModY ) an integral transform?

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Refined Problems



3 When is F ∈ Bimodk(ModA,ModY ) an integral transform?If it isn’t, how close is it to being an integral transform?

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Part 3

Integral Transforms in Semi-commutative AlgebraicGeometry w/ D. Chan in case A = OX

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An Integral Transform is a functor of the form

π∗(−⊗A F) : ModA → ModY

where

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where

F is a “left A-object” in ModY

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where


−⊗A F denotes tensoring a right A-module with F and

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where



π∗ is semi-comm. analogue of the pushforward ofπ : X × Y → Y

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where



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.

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Integral Transforms in Semi-commutative AlgebraicGeometry: Left A-objects

A left A-object in Y , denoted M, consists of following data:

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For U ⊂ X affine open, an object M(U) ∈ A(U)ModY

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Given U × Vpr1

//

pr2��

U

��

V // X

an isomorphism

ψU,V : pr∗1 M(U) → pr∗2 M(V ) (sat. cocycle cond.)

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Given U × Vpr1

//

pr2��

U

��

V // X

an isomorphism


AModY := cat. of left A-objects in Y

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Given U × Vpr1

//

pr2��

U

��

V // X

an isomorphism


AModY := cat. of left A-objects in Y

e.g.

X = Spec k, A ↔ R ⇒ AModY ≡ RModY

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Integral Transforms in Semi-commutative AlgebraicGeometry: Tensor products with Left A-objects

If F ∈ AModY

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If F ∈ AModY define

−⊗A F : ModA → ModYOX

over open affine U ⊂ X by

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(N ⊗A F)(U) := N (U) ⊗A(U) F(U)

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(N ⊗A F)(U) := N (U) ⊗A(U) F(U)

Define gluing isomorphisms locally as well.

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(N ⊗A F)(U) := N (U) ⊗A(U) F(U)

Define gluing isomorphisms locally as well.

F is flat/A ⇔ −⊗A F is exact.

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Integral Transforms in Semi-commutative AlgebraicGeometry: Cech Cohomology

Define π∗ : ModYOX→ ModY via rel. Cech Cohomology:

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X quasi-compact ⇒ X has finite affine open cover U : {Ui}

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For N ∈ ModYOX, let

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Cp(U,N ) :=⊕

i0<i1<···<ip

N (Ui0 ∩ · · · ∩ Uip)

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Cp(U,N ) :=⊕

i0<i1<···<ip

N (Ui0 ∩ · · · ∩ Uip)

d : Cp(U,N ) → Cp+1(U,N ) defined via restriction as usual

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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 ))

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Integral Transforms in Semi-commutative AlgebraicGeometry

Refined Problems

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Refined Problems


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Refined Problems


2 When is an integral transform in Bimodk(ModA,ModY )

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Refined Problems


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)?

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Refined Problems



Y → (X ,A)?


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Refined Problems



Y → (X ,A)?


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Part 4

Semi-commutative Algebraic Geometry: Maps fromn.c. spaces to curves (w/D. Chan)

Throughout Part 4, k = k

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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 .

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Commutative example

X =smooth curvef : Y → X is comm. ruled surface with fiber C .If Γ ⊂ Y × X = graph of f , then

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Commutative example


1 comp. of Hilb OY corresponding to C is X

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Commutative example



2 OΓtr = corresponding universal quotient of OY , and

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Commutative example




3 If π : X × Y → Y is projection, then f ∗ ∼= π∗(−⊗OXOΓtr ).

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Commutative example




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?

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If Y is a non-commutative smooth proper d-fold one can usethe following to study Y :

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Sheaf Cohomology (Artin-Zhang 1994),

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Intersection theory (Mori-Smith 2001),

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Serre duality (Bondal-Kapranov 1990),

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Dimension theory,

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Dimension theory,

Hilbert schemes (Artin-Zhang 2001)

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Dimension theory,

Hilbert schemes (Artin-Zhang 2001)

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Semi-commutative Algebraic Geometry: Examples of n.c.smooth proper d -folds

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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.

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e.g.

A = Skl(a, b, c) :=k〈x0, x1, x2〉


for generic (a : b : c) ∈ P2. Then ProjA is a n.c. smooth proper

2-fold.

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e.g.

A = Skl(a, b, c) :=k〈x0, x1, x2〉



2-fold.

e.g.

Homogenized U(sl2)

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e.g.

A = Skl(a, b, c) :=k〈x0, x1, x2〉



2-fold.

e.g.

Homogenized U(sl2) =

A =k〈e, f , h, z〉

(ef − fe − zh, eh − he − 2ze, fh − hf + 2zf , z central).

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e.g.

A = Skl(a, b, c) :=k〈x0, x1, x2〉



2-fold.

e.g.


A =k〈e, f , h, z〉


ProjA is a n.c. smooth proper 3-fold.

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e.g.

A = Skl(a, b, c) :=k〈x0, x1, x2〉



2-fold.

e.g.


A =k〈e, f , h, z〉


ProjA is a n.c. smooth proper 3-fold. Given t ∈ Z (A)2, ProjA/(t)is a n.c. smooth proper 2-fold.

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Maps from n.c. spaces to curves (w/D. Chan): Geometricmachinery

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Y = ProjA n.c. smooth proper surface w/ struct. sheaf OY := πA

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For M,N ∈ modY let

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Hi (M) := ExtiY (OY ,M).

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H i (M) and more generally ExtiY (M,N )

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H i (M) and more generally ExtiY (M,N ) are finite dimensional.

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Thus intersection defined (Mori-Smith 2001) by

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M.N := −2Σi=0

(−1)i dimExtiY (M,N )

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M.N := −2Σi=0


is well defined.

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M.N := −2Σi=0


is well defined.Remark: This specializes to the intersection product for curves ona comm. surface.

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Y = n.c. smooth proper d-fold.

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Theorem (Artin-Zhang)

For P ∈ modY , there exists a Hilbert scheme Hilb P

parameterizing quotients of P .

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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.

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Maps from n.c. spaces to curves (w/D. Chan):K -non-effective rational curves

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Y = n.c. smooth proper surface with structure sheaf OY

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Say F ∈ modY is K -non-effective rational curve withself-intersection 0 if

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1 F is 1-critical quotient of OY

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2 H0(F ) = k, H1(F ) = 0, F 2 = 0

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2 H0(F ) = k, H1(F ) = 0, F 2 = 0

3 H0(F ⊗ ωY ) = 0

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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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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.

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2 H0(F ) = k, H1(F ) = 0, F 2 = 0

3 H0(F ⊗ ωY ) = 0


e.g.

If f : Y → X is a comm. or n.c. ruled surface

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2 H0(F ) = k, H1(F ) = 0, F 2 = 0

3 H0(F ⊗ ωY ) = 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.

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Theorem (Chan-N, 2009)

Suppose Y is a n.c. smooth proper surface.

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Suppose Y is a n.c. smooth proper surface. If

1 F is K -non-effective rational curve with F 2 = 0

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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

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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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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

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e.g.

If X = smooth curve, f : Y → X = n.c. ruled surface and F =struct. sheaf of fiber

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e.g.

If X = smooth curve, f : Y → X = n.c. ruled surface and F =struct. sheaf of fiber then F satisfies 1,2

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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).

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More generally:

Y =n.c. smooth proper d-fold

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More generally:


X=integral proj. curve

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More generally:



F ∈ OXModY flat/X .

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More generally:




Say F is base point free if for any simple P ∈ modY ,HomY (F ,P) = 0 for a generic fibre F ∈ F

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More generally:





Theorem (Chan-N 2009)

If F is base point free

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More generally:






If F is base point free then π∗(−⊗OXF) is exact

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More generally:






If F is base point free then π∗(−⊗OXF) is exact hence in

Bimodk(QcohX ,ModY )

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More generally:






If F is base point free then π∗(−⊗OXF) is exact hence in

Bimodk(QcohX ,ModY ), and preserves noetherian objects.

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Refined Problems

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Refined Problems


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Refined Problems


2 When is an integral transform in Bimodk(ModA,ModY )

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Refined Problems



Y → (X ,A)?

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Refined Problems



Y → (X ,A)?


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Refined Problems



Y → (X ,A)?


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Part 5

Integral Transforms and Bimod revisited

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Integral Transforms and Bimod revisited: The Functor(−)♭.

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F ∈ Bimodk(ModA,ModY )

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u : U → X denotes inclusion of affine open

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Fu∗ ∈ Bimodk(ModA(U),ModY ) ⇒

Fu∗∼= −⊗A(U) FU

for some FU ∈ A(U)ModY .

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Theorem (Van den Bergh, Chan-N.)

The collection F ♭(U) = FU induces a functor

(−)♭ : Bimodk(ModA,ModY ) → AModY

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Furthermore, if F = π∗(− ⊗A F) then

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Furthermore, if F = π∗(− ⊗A F) then F ♭ ∼= F .

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Goals

Let F ∈ Bimodk(ModA,ModY ).

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Goals


1 Find relationship between F and π∗(−⊗A F ♭)

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Goals


1 Find relationship between F and π∗(−⊗A F ♭)

2 Find necessary and sufficient conditions for F ∼= π∗(−⊗A F ♭)

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Integral Transforms and Bimod revisited: Totally GlobalFunctors

F ∈ Bimodk(ModA,ModY ) such that F ♭ = 0 are totally global

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e.g.

F = H1(P1,−) ∈ Bimodk(QcohP1,Modk) is totally global.

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e.g.


Theorem (N-Smith, 2008)

k = k, F ∈ Bimodk(QcohP1,Modk). If

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e.g.




F is totally global and

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e.g.





F preserves noetherian objects, then

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e.g.





F preserves noetherian objects, then

F ∼=⊕∞

i=m H1(P1, (−)(i))⊕ni

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Throughout F ∈ Bimodk(ModA,ModY ).

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Claim A

There is a natural transformation

ΓF : F → π∗(−⊗A F♭)

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Claim A


ΓF : F → π∗(−⊗A F♭)

such that ker ΓF and cok ΓF are totally global.

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Claim A


ΓF : F → π∗(−⊗A F♭)


It follows that ΓF is an isomorphism if

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Claim A


ΓF : F → π∗(−⊗A F♭)



1 X is affine or

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Claim A


ΓF : F → π∗(−⊗A F♭)



1 X is affine or

2 F is exact

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Remarks:

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Remarks:

1 Claim A confirmed in case A = OX and Y = scheme (N2009)

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Remarks:


2 Claim A 1 confirmed (N-Smith 2008)

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Remarks:



3 Claim A 2 confirmed in case A = OX (Chan-N 2009)

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Remarks:



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.

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Proof (that X affine ⇒ ΓF is an isomorphism):

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Easy fact: F totally global ⇔ Fu∗ = 0 ∀ open affine u : U → X .

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X affine ⇒ idX is inclusion of affine open

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ker ΓF = ker ΓF ◦ idX∗

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Since ker ΓF is totally global,

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Since ker ΓF is totally global, ker ΓF ◦ idX∗ = 0.

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Since ker ΓF is totally global, ker ΓF ◦ idX∗ = 0.

Similarly, cok ΓF = 0.

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U = finite affine open cover of X

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M ∈ ModA

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M ∈ ModA

d : C 0(U,M) → C 1(U,M) diff. of sheafified Cech complex

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M ∈ ModA


Claim B

ΓF : F → π∗(−⊗A F ♭) is an isomorphism iff

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M ∈ ModA


Claim B


1 the canonical map F (ker d) → ker(Fd) is an isomorphism forall flat M.

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M ∈ ModA


Claim B



2 π∗(− ⊗A F ♭) is right exact

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M ∈ ModA


Claim B



2 π∗(− ⊗A F ♭) is right exact

Claim B confirmed in case A = OX and Y = scheme (N 2009)

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Thank you for your attention!

Adam Nyman

Documents

Classical Integral Transforms in Semi-commutative Algebraic Geometry