Local cohomology at determinantal ideals
Uli Walther
Purdue University
With Gennady Lyubeznik, Anurag Singh.
AMS Eastern, Halifax, October 2014
Uli Walther
Local cohomology at determinantal ideals
Notation
R = Z[X ], X = m × n matrix of indeterminates.
Rp = R/pR;
R0 = Q ⊗Z R.
It =ideal of t-minors of X .
Object of desire:
HI•t (R), local cohomology modules.
Uli Walther
Local cohomology at determinantal ideals
Notation
R = Z[X ], X = m × n matrix of indeterminates.
Rp = R/pR;
R0 = Q ⊗Z R.
It =ideal of t-minors of X .
Object of desire:
HI•t (R), local cohomology modules.
Uli Walther
Local cohomology at determinantal ideals
Local cohomology
A = Noetherian ring, √
a = A-ideal,
√
f1 , . . . , fk ∈ a with a = f1 , . . . , fk .
L
Cfs =
A[1/(fi1 · · · fis )].
Cfs −→ Cfs+1 sum of signed localization maps
Gives complex Cf• such that
cohomology is supported inside a,
different choices of f give homotopic complexes,
even if they generate different ideals.
Notation: H s (Cf• ) =: Has (A) for any choice f .
Uli Walther
Local cohomology at determinantal ideals
Local cohomology
A = Noetherian ring, √
a = A-ideal,
√
f1 , . . . , fk ∈ a with a = f1 , . . . , fk .
L
Cfs =
A[1/(fi1 · · · fis )].
Cfs −→ Cfs+1 sum of signed localization maps
Gives complex Cf• such that
cohomology is supported inside a,
different choices of f give homotopic complexes,
even if they generate different ideals.
Notation: H s (Cf• ) =: Has (A) for any choice f .
Uli Walther
Local cohomology at determinantal ideals
Local cohomology
A = Noetherian ring, √
a = A-ideal,
√
f1 , . . . , fk ∈ a with a = f1 , . . . , fk .
L
Cfs =
A[1/(fi1 · · · fis )].
Cfs −→ Cfs+1 sum of signed localization maps
Gives complex Cf• such that
cohomology is supported inside a,
different choices of f give homotopic complexes,
even if they generate different ideals.
Notation: H s (Cf• ) =: Has (A) for any choice f .
Uli Walther
Local cohomology at determinantal ideals
Local cohomology, II
Some properties:
Local cohomology vanishes beyond arithmetic rank.
Alternatively: Has (−) = Rs (Γa (−)) where
Γa (M) = {m ∈ M | supp(A · m) ⊆ Var(a)}.
It is the algebraic geometer’s version of relative cohomology:
Y ⊆ X closed; F sheaf on X , then get exact natural
HYs (X , F) −→ H s (X , F) −→ H s (X r Y , F) −→ HYs+1 (X , F)
Has (A) = lim
H s (A, {f1r , . . . , fkr }).
−→
r
related to:
topology if Q ⊆ A;
tight closure if Z/pZ ⊆ A.
Uli Walther
Local cohomology at determinantal ideals
Some methods
Has (A) is usually not Noetherian.
Frivolous example: A = k[x], a = (x).
Then Has (A) = 0 for s 6= 1 while Ha1 (A) is
khx,∂x i• x1
A
S
e (A/F
if Q ⊆ A,
e
if Z/pZ ⊆ A,
(a))
L
xr
and equal to r <0 k · in either case.
Lessons from the example:
Has (A) is module over A[F ]
Has (A) is module over D(A, Q)
Has (A) is module over D(A, Z)
if Z/pZ ⊆ A,
if Q ⊆ A,
in any case
These rings are bigger, not commutative, not always
Noetherian.
Uli Walther
Local cohomology at determinantal ideals
Some methods
Has (A) is usually not Noetherian.
Frivolous example: A = k[x], a = (x).
Then Has (A) = 0 for s 6= 1 while Ha1 (A) is
khx,∂x i• x1
A
S
e (A/F
if Q ⊆ A,
e
if Z/pZ ⊆ A,
(a))
L
xr
and equal to r <0 k · in either case.
Lessons from the example:
Has (A) is module over A[F ]
Has (A) is module over D(A, Q)
Has (A) is module over D(A, Z)
if Z/pZ ⊆ A,
if Q ⊆ A,
in any case
These rings are bigger, not commutative, not always
Noetherian.
Uli Walther
Local cohomology at determinantal ideals
Some history in the determinantal case
In any characteristic: (but over a domain)
It is prime
(de Concini–Eisenbud–Procesi)
Cohen–Macaulay
(Hochster–Eagon)
of height ht := (mt + 1)(nt + 1)
and arithmetic rank mn − t 2 + 1.
(Bruns–Schwänzl)
In positive characteristic,
Frobenius powers of It are also perfect.
t
So HI>h
(Rp ) = 0.
t
In characteristic zero,
Powers of It or (f1r , . . . , fkr ) not CM, even for
m = 2 = n − 1 = t.
t
HI>h
(R) not necessarily zero.
t
Uli Walther
Local cohomology at determinantal ideals
Motivation:
If m = 2 = n − 1 = t, (“the 2 × 3 case”) then h2 = 2,
ara(I2 ) = 3.
HI32 (R0 ) 6= 0
(Hochster, -).
Huneke, Katz, Marley: if φ : R0 −→ A ⊃ Q with
dim(φ(R0 )) < 6 then HI32 (A) = 0
Does not follow from general vanishing theorems á la
Grothendieck–Faltings–Huneke–Lyubeznik.
Back to R:
there is an exact sequence
ι
0 −→ Z-torsion ,→ HI32 (R) −→ HI32 (R0 ) −→ Z-torsion −→ 0
Singh proved: HI32 (R) is Q-space, so ι iso!
Uli Walther
Local cohomology at determinantal ideals
Motivation:
If m = 2 = n − 1 = t, (“the 2 × 3 case”) then h2 = 2,
ara(I2 ) = 3.
HI32 (R0 ) 6= 0
(Hochster, -).
Huneke, Katz, Marley: if φ : R0 −→ A ⊃ Q with
dim(φ(R0 )) < 6 then HI32 (A) = 0
Does not follow from general vanishing theorems á la
Grothendieck–Faltings–Huneke–Lyubeznik.
Back to R:
there is an exact sequence
ι
0 −→ Z-torsion ,→ HI32 (R) −→ HI32 (R0 ) −→ Z-torsion −→ 0
Singh proved: HI32 (R) is Q-space, so ι iso!
Uli Walther
Local cohomology at determinantal ideals
Main results:
Theorem
R = Z[X ], It as above.
1
HIkt (R) is Z-torsion-free for all t, k.
2
If k 6= ht , then HIkt (R) is a Q-space, iso to HIkt (R0 ).
3
Consider the N-grading on R with [R]0 = Z and deg xi = 1. If
2 ≤ t ≤ min(m, n) and t < max(m, n) then
HImn−t
t
2 +1
(Z[X ]) ∼
= ER0 (Q)(mn)
Uli Walther
(this is a shift!)
Local cohomology at determinantal ideals
Main results, II:
As a consequence, one obtains:
Theorem
Let a = t-minors of M ∈ Am×n , A Noetherian, 1 ≤ t ≤ min(m, n),
t < max(m, n).
2
If dim A < mn then Hamn−t +1 (A) = 0.
Uli Walther
Local cohomology at determinantal ideals
Frobenius techniques
Frobenius functor F : (−) −→ A0 ⊗A (−)
A0 left A-module as expected; the right action a0 a = ap a0 .
F -module is a direct limit
M −→ F (M) −→ F (F (M)) −→ · · · −→ M
for some A-module M and some A-morphism M −→ F (M).
F -finite if M is Noetherian.
A{f }-module is a module over A{f } = Ahf i/{r p f − fr }.
F -finite modules are finite length as F -modules.
Hxr (k[x]) ∼
= Ek[x] (k) not injective as F -module
Uli Walther
(Lyubeznik)
(Ma)
Local cohomology at determinantal ideals
Main technical results in char p > 0
Theorem
Rp = (Z/pZ ⊆ k)[x]; m = (x), I graded.
∀k ≥ 0, TFAE:
1
2
3
4
The D(Rp , k)-module HIk (Rp ) has a composition factor with
support {m}.
The graded F -finite module HIk (Rp ) has a composition factor
with support {m}.
HIk (Rp ) has a graded F -module quotient with support {m}.
dim(A)−k
The natural Frobenius action on [Hm
nilpotent.
(Rp /I )]0 is not
Theorem
Hxr (k[x]) injective as graded F -module.
Uli Walther
Local cohomology at determinantal ideals