The Frobenius Functor and Injective Modules
Tom Marley
University of Nebraska-Lincoln
November 3, 2013
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
The Frobenius Functor
Let R be a commutative Noetherian ring of prime characteristic p
and f : R → R given by f (r ) = r p be the Frobenius endomorphism.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
The Frobenius Functor
Let R be a commutative Noetherian ring of prime characteristic p
and f : R → R given by f (r ) = r p be the Frobenius endomorphism.
Let R f denote the ring R with the R − R bimodule structure:
r · s := rs
s · r := sr
(left R-action)
p
(right R-action)
for r ∈ R and s ∈ R f .
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
The Frobenius Functor
Let R be a commutative Noetherian ring of prime characteristic p
and f : R → R given by f (r ) = r p be the Frobenius endomorphism.
Let R f denote the ring R with the R − R bimodule structure:
r · s := rs
s · r := sr
(left R-action)
p
(right R-action)
for r ∈ R and s ∈ R f .
Then FR (−) := R f ⊗R − is a functor from the category of left
R-modules to itself.
FR is called the Frobenius functor of R.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Properties of FR
Being a tensor product, FR is an additive covariant right exact
functor.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Properties of FR
Being a tensor product, FR is an additive covariant right exact
functor.
A
Suppose φ : R n −
→ R m where A = [aij ]m×n .
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Properties of FR
Being a tensor product, FR is an additive covariant right exact
functor.
A
Suppose φ : R n −
→ R m where A = [aij ]m×n .
Since
∼
FR (R) = R f ⊗R R −
→R
s ⊗ r → sr p
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Properties of FR
Being a tensor product, FR is an additive covariant right exact
functor.
A
Suppose φ : R n −
→ R m where A = [aij ]m×n .
Since
∼
FR (R) = R f ⊗R R −
→R
s ⊗ r → sr p
A[p]
we have FR (φ) : R n −−→ R m where A[p] = [aijp ].
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Properties of FR (cont)
Let I = (a1 , . . . , an ) be an ideal of R.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Properties of FR (cont)
Let I = (a1 , . . . , an ) be an ideal of R.
We can apply FR to the exact sequence
[a1 ...an ]
R n −−−−→ R → R/I → 0
to obtain the exactness of
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Properties of FR (cont)
Let I = (a1 , . . . , an ) be an ideal of R.
We can apply FR to the exact sequence
[a1 ...an ]
R n −−−−→ R → R/I → 0
to obtain the exactness of
[ap ...anp ]
R n −−1−−→ R → FR (R/I ) → 0
p
p
to conclude that FR (R/I ) ∼
= R/I [p] where I [p] = (a1 , . . . , an ).
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Usefulness of FR
Of particular interest is how properties of FR relate to classical
properties of R.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Usefulness of FR
Of particular interest is how properties of FR relate to classical
properties of R.
Theorem (E. Kunz, 1969)
FR is exact if and only if R is regular.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Usefulness of FR
Of particular interest is how properties of FR relate to classical
properties of R.
Theorem (E. Kunz, 1969)
FR is exact if and only if R is regular.
The Frobenius functor has been used with great success to deepen
our understanding of local rings of characteristic p:
The homological conjectures
Tight closure
Uniform bounds
Symbolic powers
Local cohomology
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Preservation properties
If G is free then FR (G ) is free.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Preservation properties
If G is free then FR (G ) is free.
If P is projective then FR (P) is projective.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Preservation properties
If G is free then FR (G ) is free.
If P is projective then FR (P) is projective.
If T is flat then FR (T ) is flat.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Preservation properties
If G is free then FR (G ) is free.
If P is projective then FR (P) is projective.
If T is flat then FR (T ) is flat.
We say that Frobenius preserves free, projective, and flat modules.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Preservation properties
If G is free then FR (G ) is free.
If P is projective then FR (P) is projective.
If T is flat then FR (T ) is flat.
We say that Frobenius preserves free, projective, and flat modules.
Question
When does Frobenius preserve injectives?
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Preservation properties
If G is free then FR (G ) is free.
If P is projective then FR (P) is projective.
If T is flat then FR (T ) is flat.
We say that Frobenius preserves free, projective, and flat modules.
Question
When does Frobenius preserve injectives?
Definition
A ring R is said to be FPI if FR (I ) is injective for every injective
R-module I . We say R is weakly FPI if FR (I ) is injective for every
Artinian injective R-module I .
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Matlis decomposition
By Matlis’s decomposition theorem for injective modules, if I is
injective then
M
I ∼
ER (R/P)αP .
=
P∈Spec R
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Matlis decomposition
By Matlis’s decomposition theorem for injective modules, if I is
injective then
M
I ∼
ER (R/P)αP .
=
P∈Spec R
Hence
R is FPI if and only if FR (ER (R/P)) is injective for every
P ∈ Spec R.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Matlis decomposition
By Matlis’s decomposition theorem for injective modules, if I is
injective then
M
I ∼
ER (R/P)αP .
=
P∈Spec R
Hence
R is FPI if and only if FR (ER (R/P)) is injective for every
P ∈ Spec R.
R is weakly FPI if and only if FR (ER (R/m)) is injective for
every maximal ideal m of R.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Matlis decomposition
By Matlis’s decomposition theorem for injective modules, if I is
injective then
M
I ∼
ER (R/P)αP .
=
P∈Spec R
Hence
R is FPI if and only if FR (ER (R/P)) is injective for every
P ∈ Spec R.
R is weakly FPI if and only if FR (ER (R/m)) is injective for
every maximal ideal m of R.
R is FPI if and only if RP is weakly FPI for every P ∈ Spec R.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Gorenstein rings are FPI
Remark
Let M be an R-module of dimension s and I an ideal of R. Then
FR (HIs (M)) ∼
= HIs (FR (M)).
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Gorenstein rings are FPI
Remark
Let M be an R-module of dimension s and I an ideal of R. Then
FR (HIs (M)) ∼
= HIs (FR (M)).
Proof: Since HIi (M) = 0 for any i > s, one has
R f ⊗R HIs (M) ∼
= HIs[p] (R f ⊗R M) ∼
= HIs (FR (M)).
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Gorenstein rings are FPI
Remark
Let M be an R-module of dimension s and I an ideal of R. Then
FR (HIs (M)) ∼
= HIs (FR (M)).
Proof: Since HIi (M) = 0 for any i > s, one has
R f ⊗R HIs (M) ∼
= HIs[p] (R f ⊗R M) ∼
= HIs (FR (M)).
Theorem (Huneke-Sharp, 1993)
If R is Gorenstein then R is FPI.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Gorenstein rings are FPI
Remark
Let M be an R-module of dimension s and I an ideal of R. Then
FR (HIs (M)) ∼
= HIs (FR (M)).
Proof: Since HIi (M) = 0 for any i > s, one has
R f ⊗R HIs (M) ∼
= HIs[p] (R f ⊗R M) ∼
= HIs (FR (M)).
Theorem (Huneke-Sharp, 1993)
If R is Gorenstein then R is FPI.
Proof: Since Gorenstein localizes, it suffices to show R is weakly
FPI when (R, m) is a local Gorenstein ring. In this case,
d (R), d = dim R. Apply the remark with M = R.
ER (R/m) ∼
= Hm
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Zero-dimensional FPI rings are Gorenstein
Question: Is every FPI ring Gorenstein?
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Zero-dimensional FPI rings are Gorenstein
Question: Is every FPI ring Gorenstein?
Answer: No. In fact, every quasi-Gorenstein ring is FPI, and there
exist quasi-Gorenstein rings which are not Cohen-Macaulay. (A
d (R) ∼ E (R/m).)
local ring R is quasi-Gorenstein if Hm
= R
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Zero-dimensional FPI rings are Gorenstein
Question: Is every FPI ring Gorenstein?
Answer: No. In fact, every quasi-Gorenstein ring is FPI, and there
exist quasi-Gorenstein rings which are not Cohen-Macaulay. (A
d (R) ∼ E (R/m).)
local ring R is quasi-Gorenstein if Hm
= R
Proposition
Every zero-dimensional FPI ring is Gorenstein.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Zero-dimensional FPI rings are Gorenstein
Question: Is every FPI ring Gorenstein?
Answer: No. In fact, every quasi-Gorenstein ring is FPI, and there
exist quasi-Gorenstein rings which are not Cohen-Macaulay. (A
d (R) ∼ E (R/m).)
local ring R is quasi-Gorenstein if Hm
= R
Proposition
Every zero-dimensional FPI ring is Gorenstein.
Proof: We may assume (R, m) is local. Let E = ER (R/m) and
A
consider a presentation R n −
→ R ` → E → 0. Repeated applications
A[q]
of FR gives the exact sequence: R n −−→ R ` → FRe (E ) → 0, where
q = p e . Since R is zero-dimensional, A[q] = 0 for some q. Hence,
R` ∼
= FRe (E ), which is injective as R is FPI. Hence R is injective.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Properties of FPI rings
Theorem A
If I and FR (I ) are injective then FR (I ) ∼
= I.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Properties of FPI rings
Theorem A
If I and FR (I ) are injective then FR (I ) ∼
= I.
Theorem B
If R is weakly FPI then R satisfies Serre’s condition S1 .
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Properties of FPI rings
Theorem A
If I and FR (I ) are injective then FR (I ) ∼
= I.
Theorem B
If R is weakly FPI then R satisfies Serre’s condition S1 .
Theorem C
Let (R, m) be a local ring and E := ER (R/m). TFAE:
1
f
R is weakly FPI and TorR
i (R , E ) = 0 for 1 ≤ i ≤ depth R;
2
R is Gorenstein.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Properties of FPI rings
Theorem D
Suppose R is a quotient of a Gorenstein ring. Then R is FPI if and
only if R is weakly FPI.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Properties of FPI rings
Theorem D
Suppose R is a quotient of a Gorenstein ring. Then R is FPI if and
only if R is weakly FPI.
Remark
If R is local then R is weakly FPI if and only if R̂ is weakly FPI.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Properties of FPI rings
Theorem D
Suppose R is a quotient of a Gorenstein ring. Then R is FPI if and
only if R is weakly FPI.
Remark
If R is local then R is weakly FPI if and only if R̂ is weakly FPI.
Corollary
If R is local and a quotient of a Gorenstein ring then R is FPI if
and only if R̂ is FPI.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
One-dimensional FPI rings
Theorem E
Let R be a one-dimensional local ring. TFAE:
1
R is weakly FPI;
2
R is FPI;
3
R is Cohen-Macaulay and has a canonical ideal ωR such that
[p]
ωR ∼
= ωR .
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
One-dimensional FPI rings
Theorem E
Let R be a one-dimensional local ring. TFAE:
1
R is weakly FPI;
2
R is FPI;
3
R is Cohen-Macaulay and has a canonical ideal ωR such that
[p]
ωR ∼
= ωR .
Example
Let R = Fp [x, y , z]/(xy , xz, yz). Then R is a one-dimensional FPI
ring which is not Gorenstein.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
One-dimensional FPI rings
Theorem E
Let R be a one-dimensional local ring. TFAE:
1
R is weakly FPI;
2
R is FPI;
3
R is Cohen-Macaulay and has a canonical ideal ωR such that
[p]
ωR ∼
= ωR .
Example
Let R = Fp [x, y , z]/(xy , xz, yz). Then R is a one-dimensional FPI
ring which is not Gorenstein.
Proof: One can show that ωR = (y − x, z − x) is a canonical ideal.
[p]
Also, ωR = (x + y + z)p−1 ωR ∼
= ωR .
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
One-dimensional FPI rings
Theorem F
Suppose R is a one-dimensional complete local ring with
algebraically closed residue field. Assume R has at most two
associated primes. Then R is (weakly) FPI if and only if R is
Gorenstein.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
One-dimensional FPI rings
Theorem F
Suppose R is a one-dimensional complete local ring with
algebraically closed residue field. Assume R has at most two
associated primes. Then R is (weakly) FPI if and only if R is
Gorenstein.
Example
Let R = Fp [[t 3 , t 4 , t 5 ]]. Then R is not weakly FPI.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
A question and an example
Question
Suppose R is a local FPI ring. Must R be equidimensional?
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
A question and an example
Question
Suppose R is a local FPI ring. Must R be equidimensional?
Example
Let R = Fp [x, y , z]/(xy , xz). Then R is not FPI.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
A question and an example
Question
Suppose R is a local FPI ring. Must R be equidimensional?
Example
Let R = Fp [x, y , z]/(xy , xz). Then R is not FPI.
Proof: Let E = ER (R/m). One can show by direct computation
that the socle dimension of FR (E ) is 3. Hence, FR (E ) ∼
6 E.
=
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Proof of Theorem B
We can assume (R, m) is local and complete. Let E := ER (R/m)
and for P ∈ Spec R set ER/P := HomR (R/P, E ) = ER/P (R/m).
Using Matlis duality and facts about local cohomology, one can
show that for all q = p e
AnnR FRe (ER/P ) ⊆ P [q] RP ∩ R.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Proof of Theorem B
We can assume (R, m) is local and complete. Let E := ER (R/m)
and for P ∈ Spec R set ER/P := HomR (R/P, E ) = ER/P (R/m).
Using Matlis duality and facts about local cohomology, one can
show that for all q = p e
AnnR FRe (ER/P ) ⊆ P [q] RP ∩ R.
Now let P ∈ AssR R. Then there is an exact sequence
0 → R/P → R. Applying HomR (−, E ), we have an exact sequence
E → ER/P → 0, where E = ER (R/m).
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Proof of Theorem B
We can assume (R, m) is local and complete. Let E := ER (R/m)
and for P ∈ Spec R set ER/P := HomR (R/P, E ) = ER/P (R/m).
Using Matlis duality and facts about local cohomology, one can
show that for all q = p e
AnnR FRe (ER/P ) ⊆ P [q] RP ∩ R.
Now let P ∈ AssR R. Then there is an exact sequence
0 → R/P → R. Applying HomR (−, E ), we have an exact sequence
E → ER/P → 0, where E = ER (R/m). Applying Frobenius e
times, we have
E∼
= FRe (E ) → FRe (R/P) → 0.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Proof of Theorem B
Dualizing again, we have the exactness of
0 → FRe (ER/P )v → R.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Proof of Theorem B
Dualizing again, we have the exactness of
0 → FRe (ER/P )v → R.
Since PER/P = 0, P [q] FRe (ER/P ) = 0 and hence
P [q] FRe (ER/P )v = 0.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Proof of Theorem B
Dualizing again, we have the exactness of
0 → FRe (ER/P )v → R.
Since PER/P = 0, P [q] FRe (ER/P ) = 0 and hence
P [q] FRe (ER/P )v = 0. Thus, FRe (ER/P )v ⊆ HP0 (R) for all e. Hence,
there exists an n such that P n FRe (ER/P ) = 0 for all e.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Proof of Theorem B
Dualizing again, we have the exactness of
0 → FRe (ER/P )v → R.
Since PER/P = 0, P [q] FRe (ER/P ) = 0 and hence
P [q] FRe (ER/P )v = 0. Thus, FRe (ER/P )v ⊆ HP0 (R) for all e. Hence,
there exists an n such that P n FRe (ER/P ) = 0 for all e. But then
(by the preceding slide) P n ⊆ P [q] ∩ R for all q.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Proof of Theorem B
Dualizing again, we have the exactness of
0 → FRe (ER/P )v → R.
Since PER/P = 0, P [q] FRe (ER/P ) = 0 and hence
P [q] FRe (ER/P )v = 0. Thus, FRe (ER/P )v ⊆ HP0 (R) for all e. Hence,
there exists an n such that P n FRe (ER/P ) = 0 for all e. But then
(by the preceding slide) P n ⊆ P [q] ∩ R for all q. Hence P n RP = 0,
which implies ht P = 0.
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln
Thank you!
Tom Marley
The Frobenius Functor and Injective Modules
University of Nebraska-Lincoln