Local cohomology and G-modules
Emily E. Witt
University of Minnesota
AMS Special Session on Local Commutative Algebra
October 15, 2011
Local cohomology
Local cohomology
I⊆R
Local cohomology
I⊆R
HIi (R)
Local cohomology
I⊆R
HIi (R)
Local cohomology modules capture many properties of R and I,
e.g.,
Local cohomology
I⊆R
HIi (R)
Local cohomology modules capture many properties of R and I,
e.g.,
• the depth of I
Local cohomology
I⊆R
HIi (R)
Local cohomology modules capture many properties of R and I,
e.g.,
• the depth of I
• the dimension of R
Local cohomology
I⊆R
HIi (R)
Local cohomology modules capture many properties of R and I,
e.g.,
• the depth of I
• the dimension of R
• the Cohen-Macaulay and Gorenstein properties
Local cohomology
I⊆R
HIi (R)
Local cohomology modules capture many properties of R and I,
e.g.,
• the depth of I
• the dimension of R
• the Cohen-Macaulay and Gorenstein properties
However, they are often very large (not finitely generated)
Local cohomology
I⊆R
HIi (R)
Local cohomology modules capture many properties of R and I,
e.g.,
• the depth of I
• the dimension of R
• the Cohen-Macaulay and Gorenstein properties
However, they are often very large (not finitely generated) and
can be difficult to describe.
Notation.
Notation.

x11 · · ·
 ..
..
• X= .
.
xr1 · · ·

x1s
..  is an r × s matrix of indeterminants,
. 
xrs
Notation.

x11 · · ·
 ..
..
• X= .
.
xr1 · · ·
r≤s

x1s
..  is an r × s matrix of indeterminants,
. 
xrs
Notation.

x11 · · ·
 ..
..
• X= .
.
xr1 · · ·
r≤s

x1s
..  is an r × s matrix of indeterminants,
. 
xrs
• C[X] is the polynomial ring in the entries of X,
Notation.

x11 · · ·
 ..
..
• X= .
.
xr1 · · ·
r≤s

x1s
..  is an r × s matrix of indeterminants,
. 
xrs
• C[X] is the polynomial ring in the entries of X,
• ∆ = (∆1 , . . . , ∆n ) the ideal generated by the maximal
minors of X
Goal.
i (C[X]).
Describe H∆
i
Local cohomology of H∆
(C[X]).
i
Local cohomology of H∆
(C[X]).
∆ = (∆1 , . . . , ∆n ) ⊆ C[X].
i
Local cohomology of H∆
(C[X]).
∆ = (∆1 , . . . , ∆n ) ⊆ C[X].
i
H∆
(C[X])
i
Local cohomology of H∆
(C[X]).
∆ = (∆1 , . . . , ∆n ) ⊆ C[X].
i
H∆
(C[X]) may be computed as the ith
cohomology of the complex
i
Local cohomology of H∆
(C[X]).
∆ = (∆1 , . . . , ∆n ) ⊆ C[X].
i
H∆
(C[X]) may be computed as the ith
cohomology of the complex
0 → C[X]
i
Local cohomology of H∆
(C[X]).
∆ = (∆1 , . . . , ∆n ) ⊆ C[X].
i
H∆
(C[X]) may be computed as the ith
cohomology of the complex
0 → C[X] →
M
j
C[X]∆j
i
Local cohomology of H∆
(C[X]).
∆ = (∆1 , . . . , ∆n ) ⊆ C[X].
i
H∆
(C[X]) may be computed as the ith
cohomology of the complex
0 → C[X] →
M
j
C[X]∆j →
M
j<k
C[X]∆j ∆k
i
Local cohomology of H∆
(C[X]).
∆ = (∆1 , . . . , ∆n ) ⊆ C[X].
i
H∆
(C[X]) may be computed as the ith
cohomology of the complex
0 → C[X] →
M
j
C[X]∆j →
M
j<k
C[X]∆j ∆k → . . . → C[X]∆1 ...∆n → 0.
Previously known.
Previously known.
N
min{i | H∆
(C[X]) 6= 0}
Previously known.
N
min{i | H∆
(C[X]) 6= 0} = s − r + 1
Previously known.
N
min{i | H∆
(C[X]) 6= 0} = s − r + 1
• In positive characteristic, this is the only nonzero LC
module.
[Peskine-Szpiro, Hochster-Eagon]
Previously known.
N
min{i | H∆
(C[X]) 6= 0} = s − r + 1
• In positive characteristic, this is the only nonzero LC
module.
[Peskine-Szpiro, Hochster-Eagon]
N
max{i | H∆
(C[X]) 6= 0}
Previously known.
N
min{i | H∆
(C[X]) 6= 0} = s − r + 1
• In positive characteristic, this is the only nonzero LC
module.
[Peskine-Szpiro, Hochster-Eagon]
N
max{i | H∆
(C[X]) 6= 0} = r(s − r) + 1 := N
[Hochster, Huneke, Lyubeznik]
Previously known.
N
min{i | H∆
(C[X]) 6= 0} = s − r + 1
• In positive characteristic, this is the only nonzero LC
module.
[Peskine-Szpiro, Hochster-Eagon]
N
max{i | H∆
(C[X]) 6= 0} = r(s − r) + 1 := N
[Hochster, Huneke, Lyubeznik]
•
Note that N = dim C[∆1 , . . . , ∆n ].
Walther’s example.
Walther’s example.
•
r = 2, s = 3
Walther’s example.
•
•
r = 2, s = 3
x11 x12 x13
X=
x21 x22 x23
Walther’s example.
•
•
•
r = 2, s = 3
x11 x12 x13
X=
x21 x22 x23
N = r(s − r) + 1 = 3
Walther’s example.
•
•
•
r = 2, s = 3
x11 x12 x13
X=
x21 x22 x23
N = r(s − r) + 1 = 3
H∆3 (C[X]) ∼
= EC[X](C),
Walther’s example.
•
•
•
r = 2, s = 3
x11 x12 x13
X=
x21 x22 x23
N = r(s − r) + 1 = 3
H∆3 (C[X]) ∼
= EC[X](C),
the injective hull of C as a C[X]-module.
Main Theorem on Minors (N case).
Main Theorem on Minors (N case).
Theorem.
Main Theorem on Minors (N case).
Theorem. For any r ≤ s,
Main Theorem on Minors (N case).
Theorem. For any r ≤ s,
H∆N (C[X]) ∼
= EC[X](C)
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
Supp H∆N (C[X]) = {m},
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
Supp H∆N (C[X]) = {m},
⇓
(Lyubeznik’s D-modules)
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
Supp H∆N (C[X]) = {m},
⇓
(Lyubeznik’s D-modules)
⊕α
H∆N (C[X]) ∼
= EC[X](C)
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
Supp H∆N (C[X]) = {m},
⇓
(Lyubeznik’s D-modules)
⊕α
H∆N (C[X]) ∼
for some α ∈ N.
= EC[X](C)
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
Supp H∆N (C[X]) = {m},
⇓
(Lyubeznik’s D-modules)
⊕α
H∆N (C[X]) ∼
for some α ∈ N.
= EC[X](C)
Goal:
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
Supp H∆N (C[X]) = {m},
⇓
(Lyubeznik’s D-modules)
⊕α
H∆N (C[X]) ∼
for some α ∈ N.
= EC[X](C)
Goal: Show α = 1.
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
Let G = SLr (C).
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
Let G = SLr (C).
G acts on:
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
Let G = SLr (C).
G acts on:
• C[X]:
Given A ∈ G, A 7→ A−1 · X
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
Let G = SLr (C).
G acts on:
• C[X]:
•
Given A ∈ G, A 7→ A−1 · X
[Weyl] =⇒ C[X]G = C[∆1 , . . . , ∆n ]
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
Let G = SLr (C).
G acts on:
• C[X]:
•
Given A ∈ G, A 7→ A−1 · X
[Weyl] =⇒ C[X]G = C[∆1 , . . . , ∆n ]
• ∆ = (∆1 , . . . , ∆n )
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
Let G = SLr (C).
G acts on:
• C[X]:
•
Given A ∈ G, A 7→ A−1 · X
[Weyl] =⇒ C[X]G = C[∆1 , . . . , ∆n ]
• ∆ = (∆1 , . . . , ∆n )
• The complex:
0 → C[X]
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
Let G = SLr (C).
G acts on:
• C[X]:
•
Given A ∈ G, A 7→ A−1 · X
[Weyl] =⇒ C[X]G = C[∆1 , . . . , ∆n ]
• ∆ = (∆1 , . . . , ∆n )
• The complex:
0 → C[X] →
M
j
C[X]∆j
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
Let G = SLr (C).
G acts on:
• C[X]:
•
Given A ∈ G, A 7→ A−1 · X
[Weyl] =⇒ C[X]G = C[∆1 , . . . , ∆n ]
• ∆ = (∆1 , . . . , ∆n )
• The complex:
0 → C[X] →
M
j
C[X]∆j →
M
j<k
C[X]∆j ∆k
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
Let G = SLr (C).
G acts on:
• C[X]:
•
Given A ∈ G, A 7→ A−1 · X
[Weyl] =⇒ C[X]G = C[∆1 , . . . , ∆n ]
• ∆ = (∆1 , . . . , ∆n )
• The complex:
0 → C[X] →
M
j
C[X]∆j →
M
j<k
C[X]∆j ∆k → . . . → C[X]∆1 ...∆n → 0
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
Let G = SLr (C).
G acts on:
• C[X]:
•
Given A ∈ G, A 7→ A−1 · X
[Weyl] =⇒ C[X]G = C[∆1 , . . . , ∆n ]
• ∆ = (∆1 , . . . , ∆n )
• The complex:
0 → C[X] →
M
C[X]∆j →
j
i (C[X])
• Each H∆
M
j<k
C[X]∆j ∆k → . . . → C[X]∆1 ...∆n → 0
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
i
Each H∆
(C[X]) is a rational R[G]-module:
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
i
Each H∆
(C[X]) is a rational R[G]-module:
•
i
For any g ∈ G, r ∈ C[X], and u ∈ H∆
(C[X]),
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
i
Each H∆
(C[X]) is a rational R[G]-module:
•
i
For any g ∈ G, r ∈ C[X], and u ∈ H∆
(C[X]),
g(ru) = (gr)(gu).
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
i
Each H∆
(C[X]) is a rational R[G]-module:
•
i
For any g ∈ G, r ∈ C[X], and u ∈ H∆
(C[X]),
g(ru) = (gr)(gu).
•
Directed union of V s.t.
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
i
Each H∆
(C[X]) is a rational R[G]-module:
•
i
For any g ∈ G, r ∈ C[X], and u ∈ H∆
(C[X]),
g(ru) = (gr)(gu).
•
Directed union of V s.t.
• dimC V < ∞
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
i
Each H∆
(C[X]) is a rational R[G]-module:
•
i
For any g ∈ G, r ∈ C[X], and u ∈ H∆
(C[X]),
g(ru) = (gr)(gu).
•
Directed union of V s.t.
• dimC V < ∞
• G × V → V is regular map
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
i
Each H∆
(C[X]) is a rational R[G]-module:
•
i
For any g ∈ G, r ∈ C[X], and u ∈ H∆
(C[X]),
g(ru) = (gr)(gu).
•
Directed union of V s.t.
• dimC V < ∞
• G × V → V is regular map
Critical:
N
Ingredients of proof that H∆
(C[X]) ∼
= EC[X] (C).
i
Each H∆
(C[X]) is a rational R[G]-module:
•
i
For any g ∈ G, r ∈ C[X], and u ∈ H∆
(C[X]),
g(ru) = (gr)(gu).
•
Directed union of V s.t.
• dimC V < ∞
• G × V → V is regular map
Critical: G is linearly reductive.
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
General Idea:
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
General Idea: Reduce from the study of
N
H∆
(C[X])
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
General Idea: Reduce from the study of
N
N
(C[∆1 , . . . , ∆n ]).
H∆
(C[X]) to that of H∆
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
General Idea: Reduce from the study of
N
N
(C[∆1 , . . . , ∆n ]).
H∆
(C[X]) to that of H∆
N
H∆
(C[X])G
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
General Idea: Reduce from the study of
N
N
(C[∆1 , . . . , ∆n ]).
H∆
(C[X]) to that of H∆
N
N
C[X]G
H∆
(C[X])G = H∆
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
General Idea: Reduce from the study of
N
N
(C[∆1 , . . . , ∆n ]).
H∆
(C[X]) to that of H∆
N
N
C[X]G
H∆
(C[X])G = H∆
N
= H∆
(C[∆1 , . . . , ∆n ])
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
General Idea: Reduce from the study of
N
N
(C[∆1 , . . . , ∆n ]).
H∆
(C[X]) to that of H∆
N
N
C[X]G
H∆
(C[X])G = H∆
N
= H∆
(C[∆1 , . . . , ∆n ])
∼
= EC[∆1 ,...,∆n ] (C)
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
General Idea: Reduce from the study of
N
N
(C[∆1 , . . . , ∆n ]).
H∆
(C[X]) to that of H∆
N
H∆
(C[X])G =
=
∼
=
=
N
H∆
C[X]G
N
H∆
(C[∆1 , . . . , ∆n ])
EC[∆1 ,...,∆n ] (C)
EC[X]G (C) .
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
N
H∆
(C[X])G indecomposable C[X]G -module
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
N
H∆
(C[X])G indecomposable C[X]G -module
⇓
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
N
H∆
(C[X])G indecomposable C[X]G -module
⇓
N
(C[X])
Soc H∆
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
N
H∆
(C[X])G indecomposable C[X]G -module
⇓
N
(C[X]) := AnnH∆N (C[X]) (m)
Soc H∆
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
N
H∆
(C[X])G indecomposable C[X]G -module
⇓
N
(C[X]) := AnnH∆N (C[X]) (m) simple
Soc H∆
G-module
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
Lemma.
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
Lemma. As rational R[G]-modules,
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
Lemma. As rational R[G]-modules,
N
H∆
(C[X]) ∼
= EC[X] (C)
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
Lemma. As rational R[G]-modules,
N
N
H∆
(C[X]) ∼
(C[X]) .
= EC[X] (C) ⊗C Soc H∆
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
Lemma. As rational R[G]-modules,
N
N
H∆
(C[X]) ∼
(C[X]) .
= EC[X] (C) ⊗C Soc H∆
Using invariant theory
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
Lemma. As rational R[G]-modules,
N
N
H∆
(C[X]) ∼
(C[X]) .
= EC[X] (C) ⊗C Soc H∆
Using invariant theory as well as the representation theory of
G = SLr (C)
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
Lemma. As rational R[G]-modules,
N
N
H∆
(C[X]) ∼
(C[X]) .
= EC[X] (C) ⊗C Soc H∆
Using invariant theory as well as the representation theory of
G = SLr (C) we are able to conclude that
N
Sketch of proof that H∆
(C[X]) ∼
= EC[X] (C).
Lemma. As rational R[G]-modules,
N
N
H∆
(C[X]) ∼
(C[X]) .
= EC[X] (C) ⊗C Soc H∆
Using invariant theory as well as the representation theory of
G = SLr (C) we are able to conclude that
N
H∆
(C[X]) ∼
= EC[X] (C).
Main Theorem on Minors.
Main Theorem on Minors.
r(s−r)+1
(a) H∆
(C[X]) ∼
= EC[X] (C).
Main Theorem on Minors.
r(s−r)+1
(a) H∆
(C[X]) ∼
= EC[X] (C).
it
(b) H∆
(C[X]) 6= 0
Main Theorem on Minors.
r(s−r)+1
(a) H∆
(C[X]) ∼
= EC[X] (C).
it
(b) H∆
(C[X]) 6= 0 ⇐⇒ i = it for some
0 ≤ t < r,
Main Theorem on Minors.
r(s−r)+1
(a) H∆
(C[X]) ∼
= EC[X] (C).
it
(b) H∆
(C[X]) 6= 0 ⇐⇒ i = it for some
0 ≤ t < r,
Main Theorem on Minors.
r(s−r)+1
(a) H∆
(C[X]) ∼
= EC[X] (C).
it
(b) H∆
(C[X]) 6= 0 ⇐⇒ i = it for some
0 ≤ t < r, where
it = (r − t)(s − r) + 1.
Main Theorem on Minors.
r(s−r)+1
(a) H∆
(C[X]) ∼
= EC[X] (C).
it
(b) H∆
(C[X]) 6= 0 ⇐⇒ i = it for some
0 ≤ t < r, where
it = (r − t)(s − r) + 1.
Furthermore,
Main Theorem on Minors.
r(s−r)+1
(a) H∆
(C[X]) ∼
= EC[X] (C).
it
(b) H∆
(C[X]) 6= 0 ⇐⇒ i = it for some
0 ≤ t < r, where
it = (r − t)(s − r) + 1.
Furthermore,
Main Theorem on Minors.
r(s−r)+1
(a) H∆
(C[X]) ∼
= EC[X] (C).
it
(b) H∆
(C[X]) 6= 0 ⇐⇒ i = it for some
0 ≤ t < r, where
it = (r − t)(s − r) + 1.
Furthermore,
it
it
H∆
(C[X]) ,→ H∆
(C[X])It+1
Main Theorem on Minors.
r(s−r)+1
(a) H∆
(C[X]) ∼
= EC[X] (C).
it
(b) H∆
(C[X]) 6= 0 ⇐⇒ i = it for some
0 ≤ t < r, where
it = (r − t)(s − r) + 1.
Furthermore,
it
it
H∆
(C[X]) ,→ H∆
(C[X])It+1 ∼
= EC[X] (C[X]/It+1 )
Main Theorem.
Main Theorem.
Let
Main Theorem.
Let
• Let R be a polynomial ring / k,
Main Theorem.
Let
• Let R be a polynomial ring / k, char(k) = 0,
Main Theorem.
Let
• Let R be a polynomial ring / k, char(k) = 0,
• G linearly reductive group / k
Main Theorem.
Let
• Let R be a polynomial ring / k, char(k) = 0,
• G linearly reductive group / k acting “nicely” on R,
Main Theorem.
Let
• Let R be a polynomial ring / k, char(k) = 0,
• G linearly reductive group / k acting “nicely” on R,
• I = mRG R,
Main Theorem.
Let
• Let R be a polynomial ring / k, char(k) = 0,
• G linearly reductive group / k acting “nicely” on R,
• I = mRG R, and N = dim RG .
Main Theorem.
Let
• Let R be a polynomial ring / k, char(k) = 0,
• G linearly reductive group / k acting “nicely” on R,
• I = mRG R, and N = dim RG .
Then HIN (R) 6= 0
Main Theorem.
Let
• Let R be a polynomial ring / k, char(k) = 0,
• G linearly reductive group / k acting “nicely” on R,
• I = mRG R, and N = dim RG .
Then HIN (R) 6= 0 and HIi (R) = 0 for i > d.
Main Theorem.
Let
• Let R be a polynomial ring / k, char(k) = 0,
• G linearly reductive group / k acting “nicely” on R,
• I = mRG R, and N = dim RG .
Then HIN (R) 6= 0 and HIi (R) = 0 for i > d.
(a) If i < N ,
Main Theorem.
Let
• Let R be a polynomial ring / k, char(k) = 0,
• G linearly reductive group / k acting “nicely” on R,
• I = mRG R, and N = dim RG .
Then HIN (R) 6= 0 and HIi (R) = 0 for i > d.
(a) If i < N , then m ∈
/ AssR HIi (R).
Main Theorem.
Let
• Let R be a polynomial ring / k, char(k) = 0,
• G linearly reductive group / k acting “nicely” on R,
• I = mRG R, and N = dim RG .
Then HIN (R) 6= 0 and HIi (R) = 0 for i > d.
(a) If i < N , then m ∈
/ AssR HIi (R).
If Supp HIN (R) = {m}, then
Main Theorem.
Let
• Let R be a polynomial ring / k, char(k) = 0,
• G linearly reductive group / k acting “nicely” on R,
• I = mRG R, and N = dim RG .
Then HIN (R) 6= 0 and HIi (R) = 0 for i > d.
(a) If i < N , then m ∈
/ AssR HIi (R).
If Supp HIN (R) = {m}, then
Main Theorem.
Let
• Let R be a polynomial ring / k, char(k) = 0,
• G linearly reductive group / k acting “nicely” on R,
• I = mRG R, and N = dim RG .
Then HIN (R) 6= 0 and HIi (R) = 0 for i > d.
(a) If i < N , then m ∈
/ AssR HIi (R).
If Supp HIN (R) = {m}, then
(b) V := Soc HIi (R)
Main Theorem.
Let
• Let R be a polynomial ring / k, char(k) = 0,
• G linearly reductive group / k acting “nicely” on R,
• I = mRG R, and N = dim RG .
Then HIN (R) 6= 0 and HIi (R) = 0 for i > d.
(a) If i < N , then m ∈
/ AssR HIi (R).
If Supp HIN (R) = {m}, then
(b) V := Soc HIi (R) is a simple G-module, and
Main Theorem.
Let
• Let R be a polynomial ring / k, char(k) = 0,
• G linearly reductive group / k acting “nicely” on R,
• I = mRG R, and N = dim RG .
Then HIN (R) 6= 0 and HIi (R) = 0 for i > d.
(a) If i < N , then m ∈
/ AssR HIi (R).
If Supp HIN (R) = {m}, then
(b) V := Soc HIi (R) is a simple G-module, and
(c) As rational R[G]-modules,
Main Theorem.
Let
• Let R be a polynomial ring / k, char(k) = 0,
• G linearly reductive group / k acting “nicely” on R,
• I = mRG R, and N = dim RG .
Then HIN (R) 6= 0 and HIi (R) = 0 for i > d.
(a) If i < N , then m ∈
/ AssR HIi (R).
If Supp HIN (R) = {m}, then
(b) V := Soc HIi (R) is a simple G-module, and
(c) As rational R[G]-modules, HIN (R) ∼
= ER (k) ⊗k V.
Example of further application.
Example of further application.
•
char(k) = 0.
Example of further application.
•
•
char(k) = 0.
X an r × t matrix, Y a t × s matrix.
Example of further application.
•
•
•
char(k) = 0.
X an r × t matrix, Y a t × s matrix.
R = k[X, Y ].
Example of further application.
•
•
•
•
char(k) = 0.
X an r × t matrix, Y a t × s matrix.
R = k[X, Y ].
For A ∈ SLt (k),
Example of further application.
•
•
•
•
char(k) = 0.
X an r × t matrix, Y a t × s matrix.
R = k[X, Y ].
For A ∈ SLt (k),
• X 7→ X · A
Example of further application.
•
•
•
•
char(k) = 0.
X an r × t matrix, Y a t × s matrix.
R = k[X, Y ].
For A ∈ SLt (k),
• X 7→ X · A
• Y →
7 A−1 · Y
Example of further application.
•
•
•
•
char(k) = 0.
X an r × t matrix, Y a t × s matrix.
R = k[X, Y ].
For A ∈ SLt (k),
• X 7→ X · A
• Y →
7 A−1 · Y
•
RG = k[X · Y ],
Example of further application.
•
•
•
•
char(k) = 0.
X an r × t matrix, Y a t × s matrix.
R = k[X, Y ].
For A ∈ SLt (k),
• X 7→ X · A
• Y →
7 A−1 · Y
•
RG = k[X · Y ], so I is generated by the
entries of X · Y .
Example of further application.
•
•
•
•
char(k) = 0.
X an r × t matrix, Y a t × s matrix.
R = k[X, Y ].
For A ∈ SLt (k),
• X 7→ X · A
• Y →
7 A−1 · Y
•
•
RG = k[X · Y ], so I is generated by the
entries of X · Y .
We may study HIi (R).