![Page 1: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/1.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Local Cohomology Sheaves on Algebraic Stacks
Tobias Sitte
PhD Defence
15th July 2014
![Page 2: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/2.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Table of contents
1 Schemes
2 Local cohomology
3 Algebraic stacks
4 Local cohomology sheaves on algebraic stacks
![Page 3: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/3.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Glueing Rn
An (n-dimensional) manifold M is a topological space that locally looks like Rn,
i.e. M can be covered by opens {Ui}i s.t.
every Ui is homeomorphic to an open subset of Rn via ϕi : Ui'→ Vi ⊂ Rn,
compatible on intersections.
U1 U2
ϕ1
ϕ2
ϕ2 ◦ ϕ−11
In other words: A manifold is obtained by glueing copies of Rn.
![Page 4: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/4.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Glueing Rn
An (n-dimensional) manifold M is a topological space that locally looks like Rn,i.e. M can be covered by opens {Ui}i s.t.
every Ui is homeomorphic to an open subset of Rn via ϕi : Ui'→ Vi ⊂ Rn,
compatible on intersections.
U1 U2
ϕ1
ϕ2
ϕ2 ◦ ϕ−11
In other words: A manifold is obtained by glueing copies of Rn.
![Page 5: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/5.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Glueing Rn
An (n-dimensional) manifold M is a topological space that locally looks like Rn,i.e. M can be covered by opens {Ui}i s.t.
every Ui is homeomorphic to an open subset of Rn via ϕi : Ui'→ Vi ⊂ Rn,
compatible on intersections.
U1 U2
ϕ1
ϕ2
ϕ2 ◦ ϕ−11
In other words: A manifold is obtained by glueing copies of Rn.
![Page 6: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/6.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Glueing Rn
An (n-dimensional) manifold M is a topological space that locally looks like Rn,i.e. M can be covered by opens {Ui}i s.t.
every Ui is homeomorphic to an open subset of Rn via ϕi : Ui'→ Vi ⊂ Rn,
compatible on intersections.
U1 U2
ϕ1
ϕ2
ϕ2 ◦ ϕ−11
In other words: A manifold is obtained by glueing copies of Rn.
![Page 7: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/7.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Glueing Rn
An (n-dimensional) manifold M is a topological space that locally looks like Rn,i.e. M can be covered by opens {Ui}i s.t.
every Ui is homeomorphic to an open subset of Rn via ϕi : Ui'→ Vi ⊂ Rn,
compatible on intersections.
U1 U2
ϕ1
ϕ2
ϕ2 ◦ ϕ−11
In other words: A manifold is obtained by glueing copies of Rn.
![Page 8: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/8.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: affine schemes
A
Spec(A)
commutative ring
set of prime ideals p ⊂ A with Zariski topology
Zariski topology: Every closed set is of the form
V (I ) ={p ∈ Spec(A)
∣∣ I ⊂ p}
I C A ideal .
Example (of affine schemes)
Spec(k) = {(0)} for a field k.
Spec(Z) ={
(0), (2), (3), (5), (7), . . .}
0 2 3 5 7 11 13 373,587,883
Spec(Z) ={
(0), (2), (3), (5), (7), . . .}
(closed points)
0 2 3 5 7 11 13 373,587,883
A1A := Spec
(A[X ]
)for a ring A.
![Page 9: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/9.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: affine schemes
A Spec(A)commutative ring
set of prime ideals p ⊂ A with Zariski topology
Zariski topology: Every closed set is of the form
V (I ) ={p ∈ Spec(A)
∣∣ I ⊂ p}
I C A ideal .
Example (of affine schemes)
Spec(k) = {(0)} for a field k.
Spec(Z) ={
(0), (2), (3), (5), (7), . . .}
0 2 3 5 7 11 13 373,587,883
Spec(Z) ={
(0), (2), (3), (5), (7), . . .}
(closed points)
0 2 3 5 7 11 13 373,587,883
A1A := Spec
(A[X ]
)for a ring A.
![Page 10: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/10.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: affine schemes
A Spec(A)commutative ring set of prime ideals p ⊂ A with Zariski topology
Zariski topology: Every closed set is of the form
V (I ) ={p ∈ Spec(A)
∣∣ I ⊂ p}
I C A ideal .
Example (of affine schemes)
Spec(k) = {(0)} for a field k.
Spec(Z) ={
(0), (2), (3), (5), (7), . . .}
0 2 3 5 7 11 13 373,587,883
Spec(Z) ={
(0), (2), (3), (5), (7), . . .}
(closed points)
0 2 3 5 7 11 13 373,587,883
A1A := Spec
(A[X ]
)for a ring A.
![Page 11: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/11.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: affine schemes
A Spec(A)commutative ring set of prime ideals p ⊂ A with Zariski topology
Zariski topology: Every closed set is of the form
V (I ) ={p ∈ Spec(A)
∣∣ I ⊂ p}
I C A ideal .
Example (of affine schemes)
Spec(k) = {(0)} for a field k.
Spec(Z) ={
(0), (2), (3), (5), (7), . . .}
0 2 3 5 7 11 13 373,587,883
Spec(Z) ={
(0), (2), (3), (5), (7), . . .}
(closed points)
0 2 3 5 7 11 13 373,587,883
A1A := Spec
(A[X ]
)for a ring A.
![Page 12: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/12.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: affine schemes
A Spec(A)commutative ring set of prime ideals p ⊂ A with Zariski topology
Zariski topology: Every closed set is of the form
V (I ) ={p ∈ Spec(A)
∣∣ I ⊂ p}
I C A ideal .
Example (of affine schemes)
Spec(k) = {(0)} for a field k.
Spec(Z) ={
(0), (2), (3), (5), (7), . . .}
0 2 3 5 7 11 13 373,587,883
Spec(Z) ={
(0), (2), (3), (5), (7), . . .}
(closed points)
0 2 3 5 7 11 13 373,587,883
A1A := Spec
(A[X ]
)for a ring A.
![Page 13: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/13.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: affine schemes
A Spec(A)commutative ring set of prime ideals p ⊂ A with Zariski topology
Zariski topology: Every closed set is of the form
V (I ) ={p ∈ Spec(A)
∣∣ I ⊂ p}
I C A ideal .
Example (of affine schemes)
Spec(k) = {(0)} for a field k.
Spec(Z) ={
(0), (2), (3), (5), (7), . . .}
0 2 3 5 7 11 13 373,587,883
Spec(Z) ={
(0), (2), (3), (5), (7), . . .}
(closed points)
0 2 3 5 7 11 13 373,587,883
A1A := Spec
(A[X ]
)for a ring A.
![Page 14: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/14.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: affine schemes
A Spec(A)commutative ring set of prime ideals p ⊂ A with Zariski topology
Zariski topology: Every closed set is of the form
V (I ) ={p ∈ Spec(A)
∣∣ I ⊂ p}
I C A ideal .
Example (of affine schemes)
Spec(k) = {(0)} for a field k.
Spec(Z) ={
(0), (2), (3), (5), (7), . . .}
0 2 3 5 7 11 13 373,587,883
Spec(Z) ={
(0), (2), (3), (5), (7), . . .}
(closed points)
0 2 3 5 7 11 13 373,587,883
A1A := Spec
(A[X ]
)for a ring A.
![Page 15: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/15.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: affine schemes
A Spec(A)commutative ring set of prime ideals p ⊂ A with Zariski topology
Zariski topology: Every closed set is of the form
V (I ) ={p ∈ Spec(A)
∣∣ I ⊂ p}
I C A ideal .
Example (of affine schemes)
Spec(k) = {(0)} for a field k.
Spec(Z) ={
(0), (2), (3), (5), (7), . . .}
0 2 3 5 7 11 13 373,587,883
Spec(Z) ={
(0), (2), (3), (5), (7), . . .}
(closed points)
0 2 3 5 7 11 13 373,587,883
A1A := Spec
(A[X ]
)for a ring A.
![Page 16: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/16.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: affine schemes
A Spec(A)commutative ring set of prime ideals p ⊂ A with Zariski topology
Zariski topology: Every closed set is of the form
V (I ) ={p ∈ Spec(A)
∣∣ I ⊂ p}
I C A ideal .
Example (of affine schemes)
Spec(k) = {(0)} for a field k.
Spec(Z) ={
(0), (2), (3), (5), (7), . . .}
0 2 3 5 7 11 13 373,587,883
Spec(Z) ={
(0), (2), (3), (5), (7), . . .}
(closed points)
0 2 3 5 7 11 13 373,587,883
A1A := Spec
(A[X ]
)for a ring A.
![Page 17: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/17.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: glueing
Example (glueing)
A1C = Spec(C[X ])
={
(X − a)∣∣ a ∈ C
}∪{
(0)}
Glue two copies of A1C along A1
C \ {(X )} = Spec(C[X ,X−1]
).
C[X ,X−1]∼=→ C[Y ,Y−1],
X 7→ Y
C[X ,X−1]∼=→ C[Y ,Y−1], X 7→ Y−1
P1C
Slogan
A scheme is a (ringed) topological space that locally looks like Spec(A) forsome ring A.
![Page 18: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/18.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: glueing
Example (glueing)
A1C = Spec(C[X ]) =
{(X − a)
∣∣ a ∈ C}∪{
(0)}
Glue two copies of A1C along A1
C \ {(X )} = Spec(C[X ,X−1]
).
C[X ,X−1]∼=→ C[Y ,Y−1],
X 7→ Y
C[X ,X−1]∼=→ C[Y ,Y−1], X 7→ Y−1
P1C
Slogan
A scheme is a (ringed) topological space that locally looks like Spec(A) forsome ring A.
![Page 19: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/19.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: glueing
Example (glueing)
A1C = Spec(C[X ]) =
{(X − a)
∣∣ a ∈ C}∪{
(0)}
0 X − 1 X X + 1
Glue two copies of A1C along A1
C \ {(X )} = Spec(C[X ,X−1]
).
C[X ,X−1]∼=→ C[Y ,Y−1],
X 7→ Y
C[X ,X−1]∼=→ C[Y ,Y−1], X 7→ Y−1
P1C
Slogan
A scheme is a (ringed) topological space that locally looks like Spec(A) forsome ring A.
![Page 20: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/20.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: glueing
Example (glueing)
A1C = Spec(C[X ]) =
{(X − a)
∣∣ a ∈ C}∪{
(0)}
X
Glue two copies of A1C along A1
C \ {(X )} = Spec(C[X ,X−1]
).
C[X ,X−1]∼=→ C[Y ,Y−1],
X 7→ Y
C[X ,X−1]∼=→ C[Y ,Y−1], X 7→ Y−1
P1C
Slogan
A scheme is a (ringed) topological space that locally looks like Spec(A) forsome ring A.
![Page 21: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/21.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: glueing
Example (glueing)
A1C = Spec(C[X ]) =
{(X − a)
∣∣ a ∈ C}∪{
(0)}
X
Glue two copies of A1C along A1
C \ {(X )}
Glue two copies of A1C along
A1C \ {(X )} = Spec
(C[X ,X−1]
).
C[X ,X−1]∼=→ C[Y ,Y−1],
X 7→ Y
C[X ,X−1]∼=→ C[Y ,Y−1], X 7→ Y−1
P1C
Slogan
A scheme is a (ringed) topological space that locally looks like Spec(A) forsome ring A.
![Page 22: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/22.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: glueing
Example (glueing)
A1C = Spec(C[X ]) =
{(X − a)
∣∣ a ∈ C}∪{
(0)}
X
Glue two copies of A1C along A1
C \ {(X )}
= Spec(C[X ,X−1]
).
C[X ,X−1]∼=→ C[Y ,Y−1],
X 7→ Y
C[X ,X−1]∼=→ C[Y ,Y−1], X 7→ Y−1
P1C
Slogan
A scheme is a (ringed) topological space that locally looks like Spec(A) forsome ring A.
![Page 23: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/23.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: glueing
Example (glueing)
A1C = Spec(C[X ]) =
{(X − a)
∣∣ a ∈ C}∪{
(0)}
X
Glue two copies of A1C along A1
C \ {(X )} = Spec(C[X ,X−1]
).
C[X ,X−1]∼=→ C[Y ,Y−1],
X 7→ Y
C[X ,X−1]∼=→ C[Y ,Y−1], X 7→ Y−1
P1C
Slogan
A scheme is a (ringed) topological space that locally looks like Spec(A) forsome ring A.
![Page 24: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/24.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: glueing
Example (glueing)
A1C = Spec(C[X ]) =
{(X − a)
∣∣ a ∈ C}∪{
(0)}
X
Glue two copies of A1C along A1
C \ {(X )} = Spec(C[X ,X−1]
).
C[X ,X−1]∼=→ C[Y ,Y−1],
X 7→ Y
C[X ,X−1]∼=→ C[Y ,Y−1], X 7→ Y−1
P1C
Slogan
A scheme is a (ringed) topological space that locally looks like Spec(A) forsome ring A.
![Page 25: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/25.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: glueing
Example (glueing)
A1C = Spec(C[X ]) =
{(X − a)
∣∣ a ∈ C}∪{
(0)}
X
Glue two copies of A1C along A1
C \ {(X )} = Spec(C[X ,X−1]
).
C[X ,X−1]∼=→ C[Y ,Y−1], X 7→ Y
C[X ,X−1]∼=→ C[Y ,Y−1], X 7→ Y−1
P1C
Slogan
A scheme is a (ringed) topological space that locally looks like Spec(A) forsome ring A.
![Page 26: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/26.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: glueing
Example (glueing)
A1C = Spec(C[X ]) =
{(X − a)
∣∣ a ∈ C}∪{
(0)}
X
Glue two copies of A1C along A1
C \ {(X )} = Spec(C[X ,X−1]
).
C[X ,X−1]∼=→ C[Y ,Y−1], X 7→ Y
C[X ,X−1]∼=→ C[Y ,Y−1], X 7→ Y−1
P1C
Slogan
A scheme is a (ringed) topological space that locally looks like Spec(A) forsome ring A.
![Page 27: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/27.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: glueing
Example (glueing)
A1C = Spec(C[X ]) =
{(X − a)
∣∣ a ∈ C}∪{
(0)}
X
Glue two copies of A1C along A1
C \ {(X )} = Spec(C[X ,X−1]
).
C[X ,X−1]∼=→ C[Y ,Y−1], X 7→ Y
C[X ,X−1]∼=→ C[Y ,Y−1], X 7→ Y−1
P1C
Slogan
A scheme is a (ringed) topological space that locally looks like Spec(A) forsome ring A.
![Page 28: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/28.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: glueing
Example (glueing)
A1C = Spec(C[X ]) =
{(X − a)
∣∣ a ∈ C}∪{
(0)}
X
Glue two copies of A1C along A1
C \ {(X )} = Spec(C[X ,X−1]
).
C[X ,X−1]∼=→ C[Y ,Y−1], X 7→ Y
C[X ,X−1]∼=→ C[Y ,Y−1], X 7→ Y−1
P1C
Slogan
A scheme is a (ringed) topological space that locally looks like Spec(A) forsome ring A.
![Page 29: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/29.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: glueing
Example (glueing)
A1C = Spec(C[X ]) =
{(X − a)
∣∣ a ∈ C}∪{
(0)}
X
Glue two copies of A1C along A1
C \ {(X )} = Spec(C[X ,X−1]
).
C[X ,X−1]∼=→ C[Y ,Y−1], X 7→ Y
C[X ,X−1]∼=→ C[Y ,Y−1], X 7→ Y−1 P1
C
Slogan
A scheme is a (ringed) topological space that locally looks like Spec(A) forsome ring A.
![Page 30: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/30.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
From algebra to geometry: glueing
Example (glueing)
A1C = Spec(C[X ]) =
{(X − a)
∣∣ a ∈ C}∪{
(0)}
X
Glue two copies of A1C along A1
C \ {(X )} = Spec(C[X ,X−1]
).
C[X ,X−1]∼=→ C[Y ,Y−1], X 7→ Y
C[X ,X−1]∼=→ C[Y ,Y−1], X 7→ Y−1 P1
C
Slogan
A scheme is a (ringed) topological space that locally looks like Spec(A) forsome ring A.
![Page 31: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/31.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
What is local cohomology?
X
Z
U := X \ Z
X schemeZ closed subschemeU open complementF sheaf of OX -modules
Slogan
Local cohomology is sheaf cohomology relativeto a closed subscheme.
{Hk
Z (F) ∈ Ab
HkZ (F) ∈ Mod(OX )
k = 0, 1, 2, . . .
![Page 32: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/32.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
What is local cohomology?
X
Z
U := X \ Z
X scheme
Z closed subschemeU open complementF sheaf of OX -modules
Slogan
Local cohomology is sheaf cohomology relativeto a closed subscheme.
{Hk
Z (F) ∈ Ab
HkZ (F) ∈ Mod(OX )
k = 0, 1, 2, . . .
![Page 33: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/33.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
What is local cohomology?
X
Z
U := X \ Z
X schemeZ closed subscheme
U open complementF sheaf of OX -modules
Slogan
Local cohomology is sheaf cohomology relativeto a closed subscheme.
{Hk
Z (F) ∈ Ab
HkZ (F) ∈ Mod(OX )
k = 0, 1, 2, . . .
![Page 34: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/34.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
What is local cohomology?
X
Z
U := X \ Z
X schemeZ closed subschemeU open complement
F sheaf of OX -modules
Slogan
Local cohomology is sheaf cohomology relativeto a closed subscheme.
{Hk
Z (F) ∈ Ab
HkZ (F) ∈ Mod(OX )
k = 0, 1, 2, . . .
![Page 35: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/35.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
What is local cohomology?
X
Z
U := X \ Z
X schemeZ closed subschemeU open complementF sheaf of OX -modules
Slogan
Local cohomology is sheaf cohomology relativeto a closed subscheme.
{Hk
Z (F) ∈ Ab
HkZ (F) ∈ Mod(OX )
k = 0, 1, 2, . . .
![Page 36: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/36.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
What is local cohomology?
X
Z
U := X \ Z
X schemeZ closed subschemeU open complementF sheaf of OX -modules
Slogan
Local cohomology is sheaf cohomology relativeto a closed subscheme.
{Hk
Z (F) ∈ Ab
HkZ (F) ∈ Mod(OX )
k = 0, 1, 2, . . .
![Page 37: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/37.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
What is local cohomology?
X
Z
U := X \ Z
X schemeZ closed subschemeU open complementF sheaf of OX -modules
Slogan
Local cohomology is sheaf cohomology relativeto a closed subscheme.
{Hk
Z (F) ∈ Ab
HkZ (F) ∈ Mod(OX )
k = 0, 1, 2, . . .
![Page 38: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/38.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Example – Affine schemes
geometry algebraaffine scheme X = Spec(A) commutative ring A
Spec(A)→ Spec(B) B → Aclosed subscheme Z = V (I ) ⊂ X ideal I C A
Slogan
The theory of affine schemes and quasi-coherent sheaves is the theory of ringsand modules.
Can translate: H0Z (X ; F) ”=”
{m ∈ M
∣∣ I nm = 0 for some n� 0}.
Example (H0Z (X ;−) for k = 0)
X = Spec(Z), Z = V((p))
for some prime p
H0Z
(Spec(Z);Z
)= 0
H0Z
(Spec(Z);Z/qZ
)=
{
Z/pZ if p = q
0 if (p, q) = 1
![Page 39: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/39.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Example – Affine schemes
geometry algebra
affine scheme X = Spec(A) commutative ring ASpec(A)→ Spec(B) B → A
closed subscheme Z = V (I ) ⊂ X ideal I C A
Slogan
The theory of affine schemes and quasi-coherent sheaves is the theory of ringsand modules.
Can translate: H0Z (X ; F) ”=”
{m ∈ M
∣∣ I nm = 0 for some n� 0}.
Example (H0Z (X ;−) for k = 0)
X = Spec(Z), Z = V((p))
for some prime p
H0Z
(Spec(Z);Z
)= 0
H0Z
(Spec(Z);Z/qZ
)=
{
Z/pZ if p = q
0 if (p, q) = 1
![Page 40: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/40.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Example – Affine schemes
geometry algebraaffine scheme X = Spec(A) commutative ring A
Spec(A)→ Spec(B) B → Aclosed subscheme Z = V (I ) ⊂ X ideal I C A
Slogan
The theory of affine schemes and quasi-coherent sheaves is the theory of ringsand modules.
Can translate: H0Z (X ; F) ”=”
{m ∈ M
∣∣ I nm = 0 for some n� 0}.
Example (H0Z (X ;−) for k = 0)
X = Spec(Z), Z = V((p))
for some prime p
H0Z
(Spec(Z);Z
)= 0
H0Z
(Spec(Z);Z/qZ
)=
{
Z/pZ if p = q
0 if (p, q) = 1
![Page 41: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/41.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Example – Affine schemes
geometry algebraaffine scheme X = Spec(A) commutative ring A
Spec(A)→ Spec(B) B → A
closed subscheme Z = V (I ) ⊂ X ideal I C A
Slogan
The theory of affine schemes and quasi-coherent sheaves is the theory of ringsand modules.
Can translate: H0Z (X ; F) ”=”
{m ∈ M
∣∣ I nm = 0 for some n� 0}.
Example (H0Z (X ;−) for k = 0)
X = Spec(Z), Z = V((p))
for some prime p
H0Z
(Spec(Z);Z
)= 0
H0Z
(Spec(Z);Z/qZ
)=
{
Z/pZ if p = q
0 if (p, q) = 1
![Page 42: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/42.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Example – Affine schemes
geometry algebraaffine scheme X = Spec(A) commutative ring A
Spec(A)→ Spec(B) B → Aclosed subscheme Z = V (I ) ⊂ X ideal I C A
Slogan
The theory of affine schemes and quasi-coherent sheaves is the theory of ringsand modules.
Can translate: H0Z (X ; F) ”=”
{m ∈ M
∣∣ I nm = 0 for some n� 0}.
Example (H0Z (X ;−) for k = 0)
X = Spec(Z), Z = V((p))
for some prime p
H0Z
(Spec(Z);Z
)= 0
H0Z
(Spec(Z);Z/qZ
)=
{
Z/pZ if p = q
0 if (p, q) = 1
![Page 43: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/43.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Example – Affine schemes
geometry algebraaffine scheme X = Spec(A) commutative ring A
Spec(A)→ Spec(B) B → Aclosed subscheme Z = V (I ) ⊂ X ideal I C A
nice OSpec(A)-module sheaf F A-module M
Slogan
The theory of affine schemes and quasi-coherent sheaves is the theory of ringsand modules.
Can translate: H0Z (X ; F) ”=”
{m ∈ M
∣∣ I nm = 0 for some n� 0}.
Example (H0Z (X ;−) for k = 0)
X = Spec(Z), Z = V((p))
for some prime p
H0Z
(Spec(Z);Z
)= 0
H0Z
(Spec(Z);Z/qZ
)=
{
Z/pZ if p = q
0 if (p, q) = 1
![Page 44: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/44.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Example – Affine schemes
geometry algebraaffine scheme X = Spec(A) commutative ring A
Spec(A)→ Spec(B) B → Aclosed subscheme Z = V (I ) ⊂ X ideal I C A
quasi-coherent OSpec(A)-module sheaf F A-module M
Slogan
The theory of affine schemes and quasi-coherent sheaves is the theory of ringsand modules.
Can translate: H0Z (X ; F) ”=”
{m ∈ M
∣∣ I nm = 0 for some n� 0}.
Example (H0Z (X ;−) for k = 0)
X = Spec(Z), Z = V((p))
for some prime p
H0Z
(Spec(Z);Z
)= 0
H0Z
(Spec(Z);Z/qZ
)=
{
Z/pZ if p = q
0 if (p, q) = 1
![Page 45: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/45.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Example – Affine schemes
geometry algebraaffine scheme X = Spec(A) commutative ring A
Spec(A)→ Spec(B) B → Aclosed subscheme Z = V (I ) ⊂ X ideal I C A
quasi-coherent OSpec(A)-module sheaf F A-module M
Slogan
The theory of affine schemes and quasi-coherent sheaves is the theory of ringsand modules.
Can translate: H0Z (X ; F) ”=”
{m ∈ M
∣∣ I nm = 0 for some n� 0}.
Example (H0Z (X ;−) for k = 0)
X = Spec(Z), Z = V((p))
for some prime p
H0Z
(Spec(Z);Z
)= 0
H0Z
(Spec(Z);Z/qZ
)=
{
Z/pZ if p = q
0 if (p, q) = 1
![Page 46: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/46.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Example – Affine schemes
geometry algebraaffine scheme X = Spec(A) commutative ring A
Spec(A)→ Spec(B) B → Aclosed subscheme Z = V (I ) ⊂ X ideal I C A
quasi-coherent OSpec(A)-module sheaf F A-module M
Slogan
The theory of affine schemes and quasi-coherent sheaves is the theory of ringsand modules.
Can translate: H0Z (X ; F) ”=”
{m ∈ M
∣∣ I nm = 0 for some n� 0}.
Example (H0Z (X ;−) for k = 0)
X = Spec(Z), Z = V((p))
for some prime p
H0Z
(Spec(Z);Z
)= 0
H0Z
(Spec(Z);Z/qZ
)=
{
Z/pZ if p = q
0 if (p, q) = 1
![Page 47: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/47.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Example – Affine schemes
geometry algebraaffine scheme X = Spec(A) commutative ring A
Spec(A)→ Spec(B) B → Aclosed subscheme Z = V (I ) ⊂ X ideal I C A
quasi-coherent OSpec(A)-module sheaf F A-module M
Slogan
The theory of affine schemes and quasi-coherent sheaves is the theory of ringsand modules.
Can translate: H0Z (X ; F) ”=”
{m ∈ M
∣∣ I nm = 0 for some n� 0}.
Example (H0Z (X ;−) for k = 0)
X = Spec(Z), Z = V((p))
for some prime p
H0Z
(Spec(Z);Z
)= 0
H0Z
(Spec(Z);Z/qZ
)=
{
Z/pZ if p = q
0 if (p, q) = 1
![Page 48: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/48.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Example – Affine schemes
geometry algebraaffine scheme X = Spec(A) commutative ring A
Spec(A)→ Spec(B) B → Aclosed subscheme Z = V (I ) ⊂ X ideal I C A
quasi-coherent OSpec(A)-module sheaf F A-module M
Slogan
The theory of affine schemes and quasi-coherent sheaves is the theory of ringsand modules.
Can translate: H0Z (X ; F) ”=”
{m ∈ M
∣∣ I nm = 0 for some n� 0}.
Example (H0Z (X ;−) for k = 0)
X = Spec(Z), Z = V((p))
for some prime p
H0Z
(Spec(Z);Z
)= 0
H0Z
(Spec(Z);Z/qZ
)=
{
Z/pZ if p = q
0 if (p, q) = 1
![Page 49: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/49.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Example – Affine schemes
geometry algebraaffine scheme X = Spec(A) commutative ring A
Spec(A)→ Spec(B) B → Aclosed subscheme Z = V (I ) ⊂ X ideal I C A
quasi-coherent OSpec(A)-module sheaf F A-module M
Slogan
The theory of affine schemes and quasi-coherent sheaves is the theory of ringsand modules.
Can translate: H0Z (X ; F) ”=”
{m ∈ M
∣∣ I nm = 0 for some n� 0}.
Example (H0Z (X ;−) for k = 0)
X = Spec(Z), Z = V((p))
for some prime p
H0Z
(Spec(Z);Z
)= 0
H0Z
(Spec(Z);Z/qZ
)=
{
Z/pZ if p = q
0 if (p, q) = 1
![Page 50: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/50.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Example – Affine schemes
geometry algebraaffine scheme X = Spec(A) commutative ring A
Spec(A)→ Spec(B) B → Aclosed subscheme Z = V (I ) ⊂ X ideal I C A
quasi-coherent OSpec(A)-module sheaf F A-module M
Slogan
The theory of affine schemes and quasi-coherent sheaves is the theory of ringsand modules.
Can translate: H0Z (X ; F) ”=”
{m ∈ M
∣∣ I nm = 0 for some n� 0}.
Example (H0Z (X ;−) for k = 0)
X = Spec(Z), Z = V((p))
for some prime p
H0Z
(Spec(Z);Z
)= 0
H0Z
(Spec(Z);Z/qZ
)=
{Z/pZ if p = q
0 if (p, q) = 1
![Page 51: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/51.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Example – Affine schemes
geometry algebraaffine scheme X = Spec(A) commutative ring A
Spec(A)→ Spec(B) B → Aclosed subscheme Z = V (I ) ⊂ X ideal I C A
quasi-coherent OSpec(A)-module sheaf F A-module M
Slogan
The theory of affine schemes and quasi-coherent sheaves is the theory of ringsand modules.
Can translate: H0Z (X ; F) ”=”
{m ∈ M
∣∣ I nm = 0 for some n� 0}.
Example (H0Z (X ;−) for k = 0)
X = Spec(Z), Z = V((p))
for some prime p
H0Z
(Spec(Z);Z
)= 0
H0Z
(Spec(Z);Z/qZ
)=
{Z/pZ if p = q
0 if (p, q) = 1
![Page 52: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/52.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
What is local cohomology good for?
Long exact sequence
. . . Hk−1(X ; F) Hk−1(U; F∣∣U
)
HkZ (X ; F) Hk(X ; F) Hk(U; F
∣∣U
) . . .
Let A = k[X ,Y ,U,V ] for some field k and consider
I = (XU,XV ,YU,YV )C A .
I is radical, i.e. I =√I =
{a ∈ A
∣∣ an ∈ I for some n}
.Up to radical, I can be generated by the three elements XU,YV andYV + YU since
(XV )2 = XV (XV + YU)− (XU)(YV ) .
Question: Are there two elements generating I up to radical?Use: max
{n∣∣Hn
V (I )(Spec(A);A) 6= 0
}≤ min
{n∣∣√I is generated by n elements
}
![Page 53: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/53.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
What is local cohomology good for?
Long exact sequence
. . . Hk−1(X ; F) Hk−1(U; F∣∣U
)
HkZ (X ; F) Hk(X ; F) Hk(U; F
∣∣U
) . . .
Let A = k[X ,Y ,U,V ] for some field k and consider
I = (XU,XV ,YU,YV )C A .
I is radical, i.e. I =√I =
{a ∈ A
∣∣ an ∈ I for some n}
.Up to radical, I can be generated by the three elements XU,YV andYV + YU since
(XV )2 = XV (XV + YU)− (XU)(YV ) .
Question: Are there two elements generating I up to radical?Use: max
{n∣∣Hn
V (I )(Spec(A);A) 6= 0
}≤ min
{n∣∣√I is generated by n elements
}
![Page 54: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/54.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
What is local cohomology good for?
Long exact sequence
. . . Hk−1(X ; F) Hk−1(U; F∣∣U
)
HkZ (X ; F) Hk(X ; F) Hk(U; F
∣∣U
) . . .
Let A = k[X ,Y ,U,V ] for some field k and consider
I = (XU,XV ,YU,YV )C A .
I is radical, i.e. I =√I =
{a ∈ A
∣∣ an ∈ I for some n}
.Up to radical, I can be generated by the three elements XU,YV andYV + YU since
(XV )2 = XV (XV + YU)− (XU)(YV ) .
Question: Are there two elements generating I up to radical?Use: max
{n∣∣Hn
V (I )(Spec(A);A) 6= 0
}≤ min
{n∣∣√I is generated by n elements
}
![Page 55: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/55.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
What is local cohomology good for?
Long exact sequence
. . . Hk−1(X ; F) Hk−1(U; F∣∣U
)
HkZ (X ; F) Hk(X ; F) Hk(U; F
∣∣U
) . . .
Let A = k[X ,Y ,U,V ] for some field k and consider
I = (XU,XV ,YU,YV )C A .
I is radical, i.e. I =√I =
{a ∈ A
∣∣ an ∈ I for some n}
.
Up to radical, I can be generated by the three elements XU,YV andYV + YU since
(XV )2 = XV (XV + YU)− (XU)(YV ) .
Question: Are there two elements generating I up to radical?Use: max
{n∣∣Hn
V (I )(Spec(A);A) 6= 0
}≤ min
{n∣∣√I is generated by n elements
}
![Page 56: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/56.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
What is local cohomology good for?
Long exact sequence
. . . Hk−1(X ; F) Hk−1(U; F∣∣U
)
HkZ (X ; F) Hk(X ; F) Hk(U; F
∣∣U
) . . .
Let A = k[X ,Y ,U,V ] for some field k and consider
I = (XU,XV ,YU,YV )C A .
I is radical, i.e. I =√I =
{a ∈ A
∣∣ an ∈ I for some n}
.Up to radical, I can be generated by the three elements XU,YV andYV + YU since
(XV )2 = XV (XV + YU)− (XU)(YV ) .
Question: Are there two elements generating I up to radical?Use: max
{n∣∣Hn
V (I )(Spec(A);A) 6= 0
}≤ min
{n∣∣√I is generated by n elements
}
![Page 57: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/57.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
What is local cohomology good for?
Long exact sequence
. . . Hk−1(X ; F) Hk−1(U; F∣∣U
)
HkZ (X ; F) Hk(X ; F) Hk(U; F
∣∣U
) . . .
Let A = k[X ,Y ,U,V ] for some field k and consider
I = (XU,XV ,YU,YV )C A .
I is radical, i.e. I =√I =
{a ∈ A
∣∣ an ∈ I for some n}
.Up to radical, I can be generated by the three elements XU,YV andYV + YU since
(XV )2 = XV (XV + YU)− (XU)(YV ) .
Question: Are there two elements generating I up to radical?
Use: max{n∣∣Hn
V (I )(Spec(A);A) 6= 0
}≤ min
{n∣∣√I is generated by n elements
}
![Page 58: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/58.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
What is local cohomology good for?
Long exact sequence
. . . Hk−1(X ; F) Hk−1(U; F∣∣U
)
HkZ (X ; F) Hk(X ; F) Hk(U; F
∣∣U
) . . .
Let A = k[X ,Y ,U,V ] for some field k and consider
I = (XU,XV ,YU,YV )C A .
I is radical, i.e. I =√I =
{a ∈ A
∣∣ an ∈ I for some n}
.Up to radical, I can be generated by the three elements XU,YV andYV + YU since
(XV )2 = XV (XV + YU)− (XU)(YV ) .
Question: Are there two elements generating I up to radical?Use: max
{n∣∣Hn
V (I )(Spec(A);A) 6= 0
}≤ min
{n∣∣√I is generated by n elements
}
![Page 59: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/59.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology?
Slogan
Non-noetherian rings are bad.
The inclusion QCoh(Spec(A)
)↪→ Mod(OSpec(A)) does not preserve injective
objects (resp. acyclic objects for H0Z (X ;−)) in general.
Theorem
Let X be a scheme and F ∈ QCoh(X ). Have morphism of module sheaves
colimnExtkOX
(OX/In,F)→ Hk
Z (X ; F) ,
where I ⊂ OX corresponds to Z . It is an isomorphism if
(Grothendieck) X is locally noetherian (glued by noetherian rings), or
(S.) Z ↪→ X is a regular closed immersion, or
(S.) X is locally coherent and Z ↪→ X is a weakly proregular closedimmersion.
![Page 60: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/60.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology?
Slogan
Non-noetherian rings are bad.
The inclusion QCoh(Spec(A)
)↪→ Mod(OSpec(A)) does not preserve injective
objects (resp. acyclic objects for H0Z (X ;−)) in general.
Theorem
Let X be a scheme and F ∈ QCoh(X ). Have morphism of module sheaves
colimnExtkOX
(OX/In,F)→ Hk
Z (X ; F) ,
where I ⊂ OX corresponds to Z . It is an isomorphism if
(Grothendieck) X is locally noetherian (glued by noetherian rings), or
(S.) Z ↪→ X is a regular closed immersion, or
(S.) X is locally coherent and Z ↪→ X is a weakly proregular closedimmersion.
![Page 61: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/61.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology?
Slogan
Non-noetherian rings are bad.
The inclusion QCoh(Spec(A)
)↪→ Mod(OSpec(A)) does not preserve injective
objects (resp. acyclic objects for H0Z (X ;−)) in general.
Theorem
Let X be a scheme and F ∈ QCoh(X ). Have morphism of module sheaves
colimnExtkOX
(OX/In,F)→ Hk
Z (X ; F) ,
where I ⊂ OX corresponds to Z . It is an isomorphism if
(Grothendieck) X is locally noetherian (glued by noetherian rings), or
(S.) Z ↪→ X is a regular closed immersion, or
(S.) X is locally coherent and Z ↪→ X is a weakly proregular closedimmersion.
![Page 62: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/62.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology?
Slogan
Non-noetherian rings are bad.
The inclusion QCoh(Spec(A)
)↪→ Mod(OSpec(A)) does not preserve injective
objects (resp. acyclic objects for H0Z (X ;−)) in general.
Theorem
Let X be a scheme and F ∈ QCoh(X ).
Have morphism of module sheaves
colimnExtkOX
(OX/In,F)→ Hk
Z (X ; F) ,
where I ⊂ OX corresponds to Z . It is an isomorphism if
(Grothendieck) X is locally noetherian (glued by noetherian rings), or
(S.) Z ↪→ X is a regular closed immersion, or
(S.) X is locally coherent and Z ↪→ X is a weakly proregular closedimmersion.
![Page 63: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/63.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology?
Slogan
Non-noetherian rings are bad.
The inclusion QCoh(Spec(A)
)↪→ Mod(OSpec(A)) does not preserve injective
objects (resp. acyclic objects for H0Z (X ;−)) in general.
Theorem
Let X be a scheme and F ∈ QCoh(X ). Have morphism of module sheaves
colimnExtkOX
(OX/In,F)→ Hk
Z (X ; F) ,
where I ⊂ OX corresponds to Z .
It is an isomorphism if
(Grothendieck) X is locally noetherian (glued by noetherian rings), or
(S.) Z ↪→ X is a regular closed immersion, or
(S.) X is locally coherent and Z ↪→ X is a weakly proregular closedimmersion.
![Page 64: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/64.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology?
Slogan
Non-noetherian rings are bad.
The inclusion QCoh(Spec(A)
)↪→ Mod(OSpec(A)) does not preserve injective
objects (resp. acyclic objects for H0Z (X ;−)) in general.
Theorem
Let X be a scheme and F ∈ QCoh(X ). Have morphism of module sheaves
colimnExtkOX
(OX/In,F)→ Hk
Z (X ; F) ,
where I ⊂ OX corresponds to Z . It is an isomorphism if
(Grothendieck) X is locally noetherian (glued by noetherian rings), or
(S.) Z ↪→ X is a regular closed immersion, or
(S.) X is locally coherent and Z ↪→ X is a weakly proregular closedimmersion.
![Page 65: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/65.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology?
Slogan
Non-noetherian rings are bad.
The inclusion QCoh(Spec(A)
)↪→ Mod(OSpec(A)) does not preserve injective
objects (resp. acyclic objects for H0Z (X ;−)) in general.
Theorem
Let X be a scheme and F ∈ QCoh(X ). Have morphism of module sheaves
colimnExtkOX
(OX/In,F)→ Hk
Z (X ; F) ,
where I ⊂ OX corresponds to Z . It is an isomorphism if
(Grothendieck) X is locally noetherian (glued by noetherian rings), or
(S.) Z ↪→ X is a regular closed immersion, or
(S.) X is locally coherent and Z ↪→ X is a weakly proregular closedimmersion.
![Page 66: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/66.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology?
Slogan
Non-noetherian rings are bad.
The inclusion QCoh(Spec(A)
)↪→ Mod(OSpec(A)) does not preserve injective
objects (resp. acyclic objects for H0Z (X ;−)) in general.
Theorem
Let X be a scheme and F ∈ QCoh(X ). Have morphism of module sheaves
colimnExtkOX
(OX/In,F)→ Hk
Z (X ; F) ,
where I ⊂ OX corresponds to Z . It is an isomorphism if
(Grothendieck) X is locally noetherian (glued by noetherian rings), or
(S.) Z ↪→ X is a regular closed immersion, or
(S.) X is locally coherent and Z ↪→ X is a weakly proregular closedimmersion.
![Page 67: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/67.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology?
Slogan
Non-noetherian rings are bad.
The inclusion QCoh(Spec(A)
)↪→ Mod(OSpec(A)) does not preserve injective
objects (resp. acyclic objects for H0Z (X ;−)) in general.
Theorem
Let X be a scheme and F ∈ QCoh(X ). Have morphism of module sheaves
colimnExtkOX
(OX/In,F)→ Hk
Z (X ; F) ,
where I ⊂ OX corresponds to Z . It is an isomorphism if
(Grothendieck) X is locally noetherian (glued by noetherian rings), or
(S.) Z ↪→ X is a regular closed immersion, or
(S.) X is locally coherent and Z ↪→ X is a weakly proregular closedimmersion.
![Page 68: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/68.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Why algebraic stacks?
schemes
stacks
alg. stacksqc +
semi-sep
quotients?moduli spaces?
quotientsmoduli stacks
Spec(A)
Pn−1A =
(An \ {0}
)/Gm [An/Gm]
Mfg
![Page 69: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/69.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Why algebraic stacks?
schemes
stacks
alg. stacksqc +
semi-sep
quotients?moduli spaces?
quotientsmoduli stacks
Spec(A)
Pn−1A =
(An \ {0}
)/Gm [An/Gm]
Mfg
![Page 70: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/70.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Why algebraic stacks?
schemes
stacks
alg. stacksqc +
semi-sep
quotients?moduli spaces?
quotientsmoduli stacks
Spec(A)
Pn−1A =
(An \ {0}
)/Gm [An/Gm]
Mfg
![Page 71: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/71.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Why algebraic stacks?
schemes
stacks
alg. stacksqc +
semi-sep
quotients?moduli spaces?
quotientsmoduli stacks
Spec(A)
Pn−1A =
(An \ {0}
)/Gm [An/Gm]
Mfg
![Page 72: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/72.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Why algebraic stacks?
schemes
stacks
alg. stacks
qc +semi-sep
quotients?moduli spaces?
quotientsmoduli stacks
Spec(A)
Pn−1A =
(An \ {0}
)/Gm [An/Gm]
Mfg
![Page 73: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/73.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Why algebraic stacks?
schemes
stacks
alg. stacksqc +
semi-sep
quotients?moduli spaces?
quotientsmoduli stacks
Spec(A)
Pn−1A =
(An \ {0}
)/Gm [An/Gm]
Mfg
![Page 74: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/74.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Why algebraic stacks?
schemes
stacks
alg. stacksqc +
semi-sep
quotients?moduli spaces?
quotientsmoduli stacks
Spec(A)
Pn−1A =
(An \ {0}
)/Gm
[An/Gm]
Mfg
![Page 75: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/75.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Why algebraic stacks?
schemes
stacks
alg. stacksqc +
semi-sep
quotients?moduli spaces?
quotientsmoduli stacks
Spec(A)
Pn−1A =
(An \ {0}
)/Gm [An/Gm]
Mfg
![Page 76: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/76.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
What is an algebraic stack?
The category of algebraic stacks is (2-)equivalent to the category of Hopfalgebroids. A Hopf algebroid is given by a pair of rings (A, Γ) together withfive ring morphisms
Γ A Γ Γ⊗A Γε s
t
∆
c
plus lots of commutative diagrams.Have presentation P : Spec(A)� X (covering).
geometry algebraalgebraic stack X Hopf algebroid (A, Γ)
closed substack Z ⊂ X invariant ideal I C Aquasi-coherent OX -module sheaf F Γ-comodule (M, ψM)
Slogan
The theory of algebraic stacks and quasi-coherent sheaves is the theory of Hopfalgebroids and comodules.
![Page 77: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/77.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
What is an algebraic stack?
The category of algebraic stacks is (2-)equivalent to the category of Hopfalgebroids.
A Hopf algebroid is given by a pair of rings (A, Γ) together withfive ring morphisms
Γ A Γ Γ⊗A Γε s
t
∆
c
plus lots of commutative diagrams.Have presentation P : Spec(A)� X (covering).
geometry algebraalgebraic stack X Hopf algebroid (A, Γ)
closed substack Z ⊂ X invariant ideal I C Aquasi-coherent OX -module sheaf F Γ-comodule (M, ψM)
Slogan
The theory of algebraic stacks and quasi-coherent sheaves is the theory of Hopfalgebroids and comodules.
![Page 78: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/78.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
What is an algebraic stack?
The category of algebraic stacks is (2-)equivalent to the category of Hopfalgebroids. A Hopf algebroid is given by a pair of rings (A, Γ) together withfive ring morphisms
Γ A Γ Γ⊗A Γε s
t
∆
c
plus lots of commutative diagrams.
Have presentation P : Spec(A)� X (covering).
geometry algebraalgebraic stack X Hopf algebroid (A, Γ)
closed substack Z ⊂ X invariant ideal I C Aquasi-coherent OX -module sheaf F Γ-comodule (M, ψM)
Slogan
The theory of algebraic stacks and quasi-coherent sheaves is the theory of Hopfalgebroids and comodules.
![Page 79: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/79.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
What is an algebraic stack?
The category of algebraic stacks is (2-)equivalent to the category of Hopfalgebroids. A Hopf algebroid is given by a pair of rings (A, Γ) together withfive ring morphisms
Γ A Γ Γ⊗A Γε s
t
∆
c
plus lots of commutative diagrams.Have presentation P : Spec(A)� X (covering).
geometry algebraalgebraic stack X Hopf algebroid (A, Γ)
closed substack Z ⊂ X invariant ideal I C Aquasi-coherent OX -module sheaf F Γ-comodule (M, ψM)
Slogan
The theory of algebraic stacks and quasi-coherent sheaves is the theory of Hopfalgebroids and comodules.
![Page 80: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/80.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
What is an algebraic stack?
The category of algebraic stacks is (2-)equivalent to the category of Hopfalgebroids. A Hopf algebroid is given by a pair of rings (A, Γ) together withfive ring morphisms
Γ A Γ Γ⊗A Γε s
t
∆
c
plus lots of commutative diagrams.Have presentation P : Spec(A)� X (covering).
geometry algebraaffine scheme X = Spec(A) commutative ring A
closed subscheme Z ⊂ X ideal I C Aquasi-coherent OX -module sheaf F A-module M
geometry algebraalgebraic stack X Hopf algebroid (A, Γ)
closed substack Z ⊂ X invariant ideal I C Aquasi-coherent OX -module sheaf F Γ-comodule (M, ψM)
Slogan
The theory of algebraic stacks and quasi-coherent sheaves is the theory of Hopfalgebroids and comodules.
![Page 81: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/81.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
What is an algebraic stack?
The category of algebraic stacks is (2-)equivalent to the category of Hopfalgebroids. A Hopf algebroid is given by a pair of rings (A, Γ) together withfive ring morphisms
Γ A Γ Γ⊗A Γε s
t
∆
c
plus lots of commutative diagrams.Have presentation P : Spec(A)� X (covering).
geometry algebraalgebraic stack X Hopf algebroid (A, Γ)
closed substack Z ⊂ X invariant ideal I C Aquasi-coherent OX -module sheaf F Γ-comodule (M, ψM)
Slogan
The theory of algebraic stacks and quasi-coherent sheaves is the theory of Hopfalgebroids and comodules.
![Page 82: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/82.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
What is an algebraic stack?
The category of algebraic stacks is (2-)equivalent to the category of Hopfalgebroids. A Hopf algebroid is given by a pair of rings (A, Γ) together withfive ring morphisms
Γ A Γ Γ⊗A Γε s
t
∆
c
plus lots of commutative diagrams.Have presentation P : Spec(A)� X (covering).
geometry algebraalgebraic stack X Hopf algebroid (A, Γ)
closed substack Z ⊂ X invariant ideal I C Aquasi-coherent OX -module sheaf F Γ-comodule (M, ψM)
Slogan
The theory of algebraic stacks and quasi-coherent sheaves is the theory of Hopfalgebroids and comodules.
![Page 83: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/83.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
History
schemes
stacks
alg. stacks
Grothendieck, 1961
Hovey-Strickland, 2005Mfg
Franke, 1996
X algebraic stack
Spec(A)P→ X presentation
Z ↪→ X closed substack
Uj↪→ X inclusion of the open complement (quasi-compact)F quasi-coherent sheaf
![Page 84: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/84.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
History
schemes
stacks
alg. stacks
Mfg
Grothendieck, 1961
Hovey-Strickland, 2005Mfg
Franke, 1996
X algebraic stack
Spec(A)P→ X presentation
Z ↪→ X closed substack
Uj↪→ X inclusion of the open complement (quasi-compact)F quasi-coherent sheaf
![Page 85: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/85.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
History
schemes
stacks
alg. stacks
Mfg
Grothendieck, 1961
Hovey-Strickland, 2005Mfg
Franke, 1996
X algebraic stack
Spec(A)P→ X presentation
Z ↪→ X closed substack
Uj↪→ X inclusion of the open complement (quasi-compact)F quasi-coherent sheaf
![Page 86: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/86.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
History
schemes
stacks
alg. stacks
Grothendieck, 1961
Hovey-Strickland, 2005Mfg
Franke, 1996
X algebraic stack
Spec(A)P→ X presentation
Z ↪→ X closed substack
Uj↪→ X inclusion of the open complement (quasi-compact)F quasi-coherent sheaf
![Page 87: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/87.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
History
schemes
stacks
alg. stacks
Grothendieck, 1961
Hovey-Strickland, 2005Mfg
Franke, 1996
X algebraic stack
Spec(A)P→ X presentation
Z ↪→ X closed substack
Uj↪→ X inclusion of the open complement (quasi-compact)F quasi-coherent sheaf
![Page 88: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/88.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
History
schemes
stacks
alg. stacks
Grothendieck, 1961
Hovey-Strickland, 2005Mfg
Franke, 1996
X algebraic stack
Spec(A)P→ X presentation
Z ↪→ X closed substack
Uj↪→ X inclusion of the open complement (quasi-compact)F quasi-coherent sheaf
![Page 89: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/89.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Local cohomology sheaves on algebraic stacks
Definition (local cohomology sheaves)
ΓZ(−) : QCoh(X )→ QCoh(X ) , F 7→ (ι ◦ t)(F)
Remark
Can identify ΓZ(F) = ker(F
η→ j∗j∗F).
![Page 90: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/90.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Local cohomology sheaves on algebraic stacks
QCohZ(X )︸ ︷︷ ︸={
F∣∣ j∗F=0
} QCoh(X ) QCoh(U)
QCoh(X )/QCohZ(X )
ι j∗
q
t j∗
s
Definition (local cohomology sheaves)
ΓZ(−) : QCoh(X )→ QCoh(X ) , F 7→ (ι ◦ t)(F)
Remark
Can identify ΓZ(F) = ker(F
η→ j∗j∗F).
![Page 91: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/91.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Local cohomology sheaves on algebraic stacks
QCohZ(X )︸ ︷︷ ︸={
F∣∣ j∗F=0
} QCoh(X ) QCoh(U)
QCoh(X )/QCohZ(X )
ι j∗
q
t j∗
s
Definition (local cohomology sheaves)
ΓZ(−) : QCoh(X )→ QCoh(X ) , F 7→ (ι ◦ t)(F)
Remark
Can identify ΓZ(F) = ker(F
η→ j∗j∗F).
![Page 92: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/92.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Local cohomology sheaves on algebraic stacks
QCohZ(X )︸ ︷︷ ︸={
F∣∣ j∗F=0
} QCoh(X ) QCoh(U)
QCoh(X )/QCohZ(X )
ι j∗
q
t j∗
s
Definition (local cohomology sheaves)
ΓZ(−) : QCoh(X )→ QCoh(X ) , F 7→ (ι ◦ t)(F)
Remark
Can identify ΓZ(F) = ker(F
η→ j∗j∗F).
![Page 93: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/93.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Local cohomology sheaves on algebraic stacks
QCohZ(X )︸ ︷︷ ︸={
F∣∣ j∗F=0
} QCoh(X ) QCoh(U)
QCoh(X )/QCohZ(X )
ι j∗
q
t j∗
s
Definition (local cohomology sheaves)
ΓZ(−) : QCoh(X )→ QCoh(X ) , F 7→ (ι ◦ t)(F)
Remark
Can identify ΓZ(F) = ker(F
η→ j∗j∗F).
![Page 94: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/94.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Local cohomology sheaves on algebraic stacks
QCohZ(X )︸ ︷︷ ︸={
F∣∣ j∗F=0
} QCoh(X ) QCoh(U)
QCoh(X )/QCohZ(X )
ι j∗
q
t j∗
∼s
Definition (local cohomology sheaves)
ΓZ(−) : QCoh(X )→ QCoh(X ) , F 7→ (ι ◦ t)(F)
Remark
Can identify ΓZ(F) = ker(F
η→ j∗j∗F).
![Page 95: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/95.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Local cohomology sheaves on algebraic stacks
QCohZ(X )︸ ︷︷ ︸={
F∣∣ j∗F=0
} QCoh(X ) QCoh(U)
QCoh(X )/QCohZ(X )
ι j∗
q
t j∗
∼s
Definition (local cohomology sheaves)
ΓZ(−) : QCoh(X )→ QCoh(X ) , F 7→ (ι ◦ t)(F)
Remark
Can identify ΓZ(F) = ker(F
η→ j∗j∗F).
![Page 96: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/96.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Local cohomology sheaves on algebraic stacks
QCohZ(X )︸ ︷︷ ︸={
F∣∣ j∗F=0
} QCoh(X ) QCoh(U)
QCoh(X )/QCohZ(X )
ι j∗
q
t j∗
∼s
Definition (local cohomology sheaves)
ΓZ(−) : QCoh(X )→ QCoh(X ) , F 7→ (ι ◦ t)(F)
Remark
Can identify ΓZ(F) = ker(F
η→ j∗j∗F).
![Page 97: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/97.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Local cohomology sheaves on algebraic stacks
QCohZ(X )︸ ︷︷ ︸={
F∣∣ j∗F=0
} QCoh(X ) QCoh(U)
QCoh(X )/QCohZ(X )
ι j∗
q
t j∗
∼s
Definition (local cohomology sheaves)
ΓZ(−) : QCoh(X )→ QCoh(X ) , F 7→ (ι ◦ t)(F)
H•Z(−) := R•QCoh(X )ΓZ(−)
Remark
Can identify ΓZ(F) = ker(F
η→ j∗j∗F).
![Page 98: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/98.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Local cohomology sheaves on algebraic stacks
QCohZ(X )︸ ︷︷ ︸={
F∣∣ j∗F=0
} QCoh(X ) QCoh(U)
QCoh(X )/QCohZ(X )
ι j∗
q
t j∗
∼s
Definition (local cohomology sheaves)
ΓZ(−) : QCoh(X )→ QCoh(X ) , F 7→ (ι ◦ t)(F)
H•Z(−) := R•QCoh(X )ΓZ(−)
Remark
Can identify ΓZ(F) = ker(F
η→ j∗j∗F).
![Page 99: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/99.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Local cohomology sheaves on algebraic stacks
QCohZ(X )︸ ︷︷ ︸={
F∣∣ j∗F=0
} QCoh(X ) QCoh(U)
QCoh(X )/QCohZ(X )
ι j∗
q
t j∗
∼s
Definition (local cohomology sheaves)
ΓZ(−) : QCoh(X )→ QCoh(X ) , F 7→ (ι ◦ t)(F)
H•Z(−) := R•QCoh(X )ΓZ(−)
Remark
Can identify ΓZ(F) = ker(F
η→ j∗j∗F).
![Page 100: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/100.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology sheaves? Comparison result
The presentation Spec(A)P→ X induces P∗ : QCoh(X )→ QCoh
(Spec(A)
).
Theorem
Let F ∈ QCoh(X ). We have a natural morphism
in QCoh(X ). It is an isomorphism if
1 Spec(A) can be chosen to be noetherian, or
2 Z ↪→ X is a regular closed immersion, or
3 Z ↪→ X is a weakly proregular closed immersion.
Slogan
In nice situations, we can compare local cohomology for stacks and schemes.
![Page 101: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/101.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology sheaves? Comparison result
The presentation Spec(A)P→ X induces P∗ : QCoh(X )→ QCoh
(Spec(A)
).
Theorem
Let F ∈ QCoh(X ). We have a natural morphism
in QCoh(X ). It is an isomorphism if
1 Spec(A) can be chosen to be noetherian, or
2 Z ↪→ X is a regular closed immersion, or
3 Z ↪→ X is a weakly proregular closed immersion.
Slogan
In nice situations, we can compare local cohomology for stacks and schemes.
![Page 102: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/102.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology sheaves? Comparison result
The presentation Spec(A)P→ X induces P∗ : QCoh(X )→ QCoh
(Spec(A)
).
Theorem
Let F ∈ QCoh(X ). We have a natural morphism
in QCoh(X ). It is an isomorphism if
1 Spec(A) can be chosen to be noetherian, or
2 Z ↪→ X is a regular closed immersion, or
3 Z ↪→ X is a weakly proregular closed immersion.
Slogan
In nice situations, we can compare local cohomology for stacks and schemes.
![Page 103: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/103.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology sheaves? Comparison result
The presentation Spec(A)P→ X induces P∗ : QCoh(X )→ QCoh
(Spec(A)
).
Theorem
Let F ∈ QCoh(X ). We have a natural morphism
HkZ(F) Hk
V (I )(P∗F)
in QCoh(X ). It is an isomorphism if
1 Spec(A) can be chosen to be noetherian, or
2 Z ↪→ X is a regular closed immersion, or
3 Z ↪→ X is a weakly proregular closed immersion.
Slogan
In nice situations, we can compare local cohomology for stacks and schemes.
![Page 104: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/104.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology sheaves? Comparison result
The presentation Spec(A)P→ X induces P∗ : QCoh(X )→ QCoh
(Spec(A)
).
Theorem
Let F ∈ QCoh(X ). We have a natural morphism
HkZ(F)︸ ︷︷ ︸
∈QCoh(X )
HkV (I )(P
∗F)︸ ︷︷ ︸∈Mod(OSpec(A))
in QCoh(X ). It is an isomorphism if
1 Spec(A) can be chosen to be noetherian, or
2 Z ↪→ X is a regular closed immersion, or
3 Z ↪→ X is a weakly proregular closed immersion.
Slogan
In nice situations, we can compare local cohomology for stacks and schemes.
![Page 105: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/105.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology sheaves? Comparison result
The presentation Spec(A)P→ X induces P∗ : QCoh(X )→ QCoh
(Spec(A)
).
Theorem
Let F ∈ QCoh(X ). We have a natural morphism
HkZ(F)︸ ︷︷ ︸
∈QCoh(X )
HkV (I )(P
∗F)︸ ︷︷ ︸∈QCoh(X )
in QCoh(X ). It is an isomorphism if
1 Spec(A) can be chosen to be noetherian, or
2 Z ↪→ X is a regular closed immersion, or
3 Z ↪→ X is a weakly proregular closed immersion.
Slogan
In nice situations, we can compare local cohomology for stacks and schemes.
![Page 106: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/106.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology sheaves? Comparison result
The presentation Spec(A)P→ X induces P∗ : QCoh(X )→ QCoh
(Spec(A)
).
Theorem
Let F ∈ QCoh(X ). We have a natural morphism
HkZ(F)︸ ︷︷ ︸
∈QCoh(X )
HkV (I )(P
∗F)︸ ︷︷ ︸∈QCoh(X )
in QCoh(X ). It is an isomorphism if
1 Spec(A) can be chosen to be noetherian, or
2 Z ↪→ X is a regular closed immersion, or
3 Z ↪→ X is a weakly proregular closed immersion.
Slogan
In nice situations, we can compare local cohomology for stacks and schemes.
![Page 107: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/107.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology sheaves? Comparison result
The presentation Spec(A)P→ X induces P∗ : QCoh(X )→ QCoh
(Spec(A)
).
Theorem
Let F ∈ QCoh(X ). We have a natural morphism
HkZ(F)︸ ︷︷ ︸
∈QCoh(X )
HkV (I )(P
∗F)︸ ︷︷ ︸∈QCoh(X )
in QCoh(X ).
It is an isomorphism if
1 Spec(A) can be chosen to be noetherian, or
2 Z ↪→ X is a regular closed immersion, or
3 Z ↪→ X is a weakly proregular closed immersion.
Slogan
In nice situations, we can compare local cohomology for stacks and schemes.
![Page 108: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/108.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology sheaves? Comparison result
The presentation Spec(A)P→ X induces P∗ : QCoh(X )→ QCoh
(Spec(A)
).
Theorem
Let F ∈ QCoh(X ). We have a natural morphism
HkZ(F)︸ ︷︷ ︸
∈QCoh(X )
HkV (I )(P
∗F)︸ ︷︷ ︸∈QCoh(X )
in QCoh(X ). It is an isomorphism if
1 Spec(A) can be chosen to be noetherian, or
2 Z ↪→ X is a regular closed immersion, or
3 Z ↪→ X is a weakly proregular closed immersion.
Slogan
In nice situations, we can compare local cohomology for stacks and schemes.
![Page 109: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/109.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology sheaves? Comparison result
The presentation Spec(A)P→ X induces P∗ : QCoh(X )→ QCoh
(Spec(A)
).
Theorem
Let F ∈ QCoh(X ). We have a natural morphism
HkZ(F)︸ ︷︷ ︸
∈QCoh(X )
HkV (I )(P
∗F)︸ ︷︷ ︸∈QCoh(X )
in QCoh(X ). It is an isomorphism if
1 Spec(A) can be chosen to be noetherian, or
2 Z ↪→ X is a regular closed immersion, or
3 Z ↪→ X is a weakly proregular closed immersion.
Slogan
In nice situations, we can compare local cohomology for stacks and schemes.
![Page 110: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/110.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology sheaves? Comparison result
The presentation Spec(A)P→ X induces P∗ : QCoh(X )→ QCoh
(Spec(A)
).
Theorem
Let F ∈ QCoh(X ). We have a natural morphism
HkZ(F)︸ ︷︷ ︸
∈QCoh(X )
HkV (I )(P
∗F)︸ ︷︷ ︸∈QCoh(X )
in QCoh(X ). It is an isomorphism if
1 Spec(A) can be chosen to be noetherian, or
2 Z ↪→ X is a regular closed immersion, or
3 Z ↪→ X is a weakly proregular closed immersion.
Slogan
In nice situations, we can compare local cohomology for stacks and schemes.
![Page 111: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/111.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology sheaves? Comparison result
The presentation Spec(A)P→ X induces P∗ : QCoh(X )→ QCoh
(Spec(A)
).
Theorem
Let F ∈ QCoh(X ). We have a natural morphism
HkZ(F)︸ ︷︷ ︸
∈QCoh(X )
HkV (I )(P
∗F)︸ ︷︷ ︸∈QCoh(X )
in QCoh(X ). It is an isomorphism if
1 Spec(A) can be chosen to be noetherian, or
2 Z ↪→ X is a regular closed immersion, or
3 Z ↪→ X is a weakly proregular closed immersion.
Slogan
In nice situations, we can compare local cohomology for stacks and schemes.
![Page 112: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/112.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology sheaves? Comparison result
The presentation Spec(A)P→ X induces P∗ : QCoh(X )→ QCoh
(Spec(A)
).
Theorem
Let F ∈ QCoh(X ). We have a natural morphism
HkZ(F)︸ ︷︷ ︸
∈QCoh(X )
HkV (I )(P
∗F)︸ ︷︷ ︸∈QCoh(X )
in QCoh(X ). It is an isomorphism if
1 Spec(A) can be chosen to be noetherian, or
2 Z ↪→ X is a regular closed immersion, or
3 Z ↪→ X is a weakly proregular closed immersion.
Slogan
In nice situations, we can compare local cohomology for stacks and schemes.
![Page 113: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/113.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology sheaves? Cech complex
Corollary
Let Z ↪→ X be a weakly proregular closed immersion.
If F ∈ QCoh(X ), then
HkZ(F) ”=”Hk(C•I (M)
),
where C•I (M) is the Cech complex of M w.r.t. the ideal I = (u1, . . . , un)C Acorresponding to Z:
0 M⊕i
M[u−1i ]
⊕i<j
M[u−1i , u−1
j ] . . . M[u−11 , . . . , u−1
n ] 0 .
Slogan
The Cech complex is a computational tool for local cohomology sheaves.
![Page 114: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/114.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology sheaves? Cech complex
Corollary
Let Z ↪→ X be a weakly proregular closed immersion.
If F ∈ QCoh(X ), then
HkZ(F) ”=”Hk(C•I (M)
),
where C•I (M) is the Cech complex of M w.r.t. the ideal I = (u1, . . . , un)C Acorresponding to Z:
0 M⊕i
M[u−1i ]
⊕i<j
M[u−1i , u−1
j ] . . . M[u−11 , . . . , u−1
n ] 0 .
Slogan
The Cech complex is a computational tool for local cohomology sheaves.
![Page 115: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/115.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology sheaves? Cech complex
Corollary
Let Z ↪→ X be a weakly proregular closed immersion. If F ∈ QCoh(X ), then
HkZ(F) ”=”Hk(C•I (M)
),
where C•I (M) is the Cech complex of M w.r.t. the ideal I = (u1, . . . , un)C Acorresponding to Z:
0 M⊕i
M[u−1i ]
⊕i<j
M[u−1i , u−1
j ] . . . M[u−11 , . . . , u−1
n ] 0 .
Slogan
The Cech complex is a computational tool for local cohomology sheaves.
![Page 116: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/116.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology sheaves? Cech complex
Corollary
Let Z ↪→ X be a weakly proregular closed immersion. If F ∈ QCoh(X ), then
HkZ(F) ”=”Hk(C•I (M)
),
where C•I (M) is the Cech complex of M w.r.t. the ideal I = (u1, . . . , un)C Acorresponding to Z:
0 M⊕i
M[u−1i ]
⊕i<j
M[u−1i , u−1
j ] . . . M[u−11 , . . . , u−1
n ] 0 .
Slogan
The Cech complex is a computational tool for local cohomology sheaves.
![Page 117: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/117.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology sheaves? Cech complex
Corollary
Let Z ↪→ X be a weakly proregular closed immersion. If F ∈ QCoh(X ), then
HkZ(F) ”=”Hk(C•I (M)
),
where C•I (M) is the Cech complex of M w.r.t. the ideal I = (u1, . . . , un)C Acorresponding to Z:
0 M⊕i
M[u−1i ]
⊕i<j
M[u−1i , u−1
j ] . . . M[u−11 , . . . , u−1
n ] 0 .
Slogan
The Cech complex is a computational tool for local cohomology sheaves.
![Page 118: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/118.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
How to calculate local cohomology sheaves? Cech complex
Corollary
Let Z ↪→ X be a weakly proregular closed immersion. If F ∈ QCoh(X ), then
HkZ(F) ”=”Hk(C•I (M)
),
where C•I (M) is the Cech complex of M w.r.t. the ideal I = (u1, . . . , un)C Acorresponding to Z:
0 M⊕i
M[u−1i ]
⊕i<j
M[u−1i , u−1
j ] . . . M[u−11 , . . . , u−1
n ] 0 .
Slogan
The Cech complex is a computational tool for local cohomology sheaves.
![Page 119: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/119.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Application: Chromatic convergence
Mfg is the (p-local) moduli stack of formal groups. It has a height filtration
Mfg = Z0 ) Z1 ) Z2 ) . . .
given by closed substacks.Let jn : Un ↪→Mfg denote the inclusion of the open complement of Zn+1.
⋃n≥0
Un (Mfg
Theorem (Chromatic convergence)
Have a canonical morphism
F holimn
Rjn∗ j∗n (F)
in D(QCoh(Mfg)
). If F ∈ QCoh(Mfg) is finitely presentable, this morphism
is an isomorphism.
One can give a proof using the comparison result given before.
![Page 120: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/120.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Application: Chromatic convergence
Mfg is the (p-local) moduli stack of formal groups.
It has a height filtration
Mfg = Z0 ) Z1 ) Z2 ) . . .
given by closed substacks.Let jn : Un ↪→Mfg denote the inclusion of the open complement of Zn+1.
⋃n≥0
Un (Mfg
Theorem (Chromatic convergence)
Have a canonical morphism
F holimn
Rjn∗ j∗n (F)
in D(QCoh(Mfg)
). If F ∈ QCoh(Mfg) is finitely presentable, this morphism
is an isomorphism.
One can give a proof using the comparison result given before.
![Page 121: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/121.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Application: Chromatic convergence
Mfg is the (p-local) moduli stack of formal groups. It has a height filtration
Mfg = Z0 ) Z1 ) Z2 ) . . .
given by closed substacks.
Let jn : Un ↪→Mfg denote the inclusion of the open complement of Zn+1.
⋃n≥0
Un (Mfg
Theorem (Chromatic convergence)
Have a canonical morphism
F holimn
Rjn∗ j∗n (F)
in D(QCoh(Mfg)
). If F ∈ QCoh(Mfg) is finitely presentable, this morphism
is an isomorphism.
One can give a proof using the comparison result given before.
![Page 122: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/122.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Application: Chromatic convergence
Mfg is the (p-local) moduli stack of formal groups. It has a height filtration
Mfg = Z0 ) Z1 ) Z2 ) . . .
given by closed substacks.Let jn : Un ↪→Mfg denote the inclusion of the open complement of Zn+1.
⋃n≥0
Un (Mfg
Theorem (Chromatic convergence)
Have a canonical morphism
F holimn
Rjn∗ j∗n (F)
in D(QCoh(Mfg)
). If F ∈ QCoh(Mfg) is finitely presentable, this morphism
is an isomorphism.
One can give a proof using the comparison result given before.
![Page 123: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/123.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Application: Chromatic convergence
Mfg is the (p-local) moduli stack of formal groups. It has a height filtration
Mfg = Z0 ) Z1 ) Z2 ) . . .
given by closed substacks.Let jn : Un ↪→Mfg denote the inclusion of the open complement of Zn+1.
⋃n≥0
Un (Mfg
Theorem (Chromatic convergence)
Have a canonical morphism
F holimn
Rjn∗ j∗n (F)
in D(QCoh(Mfg)
). If F ∈ QCoh(Mfg) is finitely presentable, this morphism
is an isomorphism.
One can give a proof using the comparison result given before.
![Page 124: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/124.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Application: Chromatic convergence
Mfg is the (p-local) moduli stack of formal groups. It has a height filtration
Mfg = Z0 ) Z1 ) Z2 ) . . .
given by closed substacks.Let jn : Un ↪→Mfg denote the inclusion of the open complement of Zn+1.
⋃n≥0
Un (Mfg
Theorem (Chromatic convergence)
Have a canonical morphism
F holimn
Rjn∗ j∗n (F)
in D(QCoh(Mfg)
).
If F ∈ QCoh(Mfg) is finitely presentable, this morphismis an isomorphism.
One can give a proof using the comparison result given before.
![Page 125: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/125.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Application: Chromatic convergence
Mfg is the (p-local) moduli stack of formal groups. It has a height filtration
Mfg = Z0 ) Z1 ) Z2 ) . . .
given by closed substacks.Let jn : Un ↪→Mfg denote the inclusion of the open complement of Zn+1.
⋃n≥0
Un (Mfg
Theorem (Chromatic convergence)
Have a canonical morphism
F holimn
Rjn∗ j∗n (F)
in D(QCoh(Mfg)
). If F ∈ QCoh(Mfg) is finitely presentable, this morphism
is an isomorphism.
One can give a proof using the comparison result given before.
![Page 126: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/126.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Application: Chromatic convergence
Mfg is the (p-local) moduli stack of formal groups. It has a height filtration
Mfg = Z0 ) Z1 ) Z2 ) . . .
given by closed substacks.Let jn : Un ↪→Mfg denote the inclusion of the open complement of Zn+1.
⋃n≥0
Un (Mfg
Theorem (Chromatic convergence)
Have a canonical morphism
F holimn
Rjn∗ j∗n (F)
in D(QCoh(Mfg)
). If F ∈ QCoh(Mfg) is finitely presentable, this morphism
is an isomorphism.
One can give a proof using the comparison result given before.
![Page 127: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/127.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Conclusion
Slogan
A scheme is a topological space that locally looks like Spec(A) for somering A.
Local cohomology is sheaf cohomology relative to a closed subscheme(resp. substack).
The theory of affine schemes and quasi-coherent sheaves is the theory ofrings and modules.
Non-noetherian rings are bad (but interesting).
The theory of algebraic stacks and quasi-coherent sheaves is the theory ofHopf algebroids and comodules.
In nice situations, we can compare local cohomology for stacks andschemes.
The Cech complex is a computational tool for local cohomology sheaves.
![Page 128: Local Cohomology Sheaves on Algebraic Stackstsitte/defence.pdf · Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacksConclusion Glueing](https://reader034.vdocuments.us/reader034/viewer/2022050308/5f6ff94b940a9d7fac60abe3/html5/thumbnails/128.jpg)
Introduction Schemes Local cohomology Algebraic stacks Local cohomology sheaves on algebraic stacks Conclusion
Thank you for your attention.