Frobenius and modules of finite flat dimension
Tom Marley
Marcus Webb
University of Nebraska-Lincoln
January 13, 2015
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Frobenius and projective dimension
Throughout R will denote a commutative Noetherian ring of prime
characteristic p.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Frobenius and projective dimension
Throughout R will denote a commutative Noetherian ring of prime
characteristic p.
For e ≥ 1 let R (e) denote the ring R viewed as an R-module via
e
the action r · s := r p s for r ∈ R and s ∈ R (e) .
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Frobenius and projective dimension
Throughout R will denote a commutative Noetherian ring of prime
characteristic p.
For e ≥ 1 let R (e) denote the ring R viewed as an R-module via
e
the action r · s := r p s for r ∈ R and s ∈ R (e) .
Theorem (Peskine-Szpiro, 1973)
If M is a finitely generated R-module and pdR M < ∞ then
(e) , M) = 0 for all i, e ≥ 1. In particular,
TorR
i (R
pdR (e) R (e) ⊗R M < ∞ for all e ≥ 1.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
An observation
Suppose x = x1 , . . . , xr is a regular sequence on R and let I = (x).
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
An observation
Suppose x = x1 , . . . , xr is a regular sequence on R and let I = (x).
Then the Čech complex C (x; R) gives a finite flat resolution of
HIr (R); i.e.,
0 → R → ⊕i Rxi → · · · → Rx1 ···xr → HIr (R) → 0
is exact.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
An observation
Suppose x = x1 , . . . , xr is a regular sequence on R and let I = (x).
Then the Čech complex C (x; R) gives a finite flat resolution of
HIr (R); i.e.,
0 → R → ⊕i Rxi → · · · → Rx1 ···xr → HIr (R) → 0
is exact.
e
Tensoring C (x; R) with R (e) gives the Čech complex C (xp , R (e) ),
e
which is acyclic as xp is a regular sequence on R (e) .
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
An observation
Suppose x = x1 , . . . , xr is a regular sequence on R and let I = (x).
Then the Čech complex C (x; R) gives a finite flat resolution of
HIr (R); i.e.,
0 → R → ⊕i Rxi → · · · → Rx1 ···xr → HIr (R) → 0
is exact.
e
Tensoring C (x; R) with R (e) gives the Čech complex C (xp , R (e) ),
e
which is acyclic as xp is a regular sequence on R (e) .
(e) , H r (R)) = 0 for all i, e ≥ 1.
In particular, TorR
i (R
I
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Questions
Question
(e) , M) = 0 for all i, e > 0?
Suppose fdR M < ∞. Is TorR
i (R
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Questions
Question
(e) , M) = 0 for all i, e > 0?
Suppose fdR M < ∞. Is TorR
i (R
While we’re at it....
Theorem (Herzog, 1974)
Suppose R has finite Krull dimension and M a finitely generated
(e) , M) = 0 for all i > 0 and infinitely many
R-module. If TorR
i (R
e. Then pdR M < ∞.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Questions
Question
(e) , M) = 0 for all i, e > 0?
Suppose fdR M < ∞. Is TorR
i (R
While we’re at it....
Theorem (Herzog, 1974)
Suppose R has finite Krull dimension and M a finitely generated
(e) , M) = 0 for all i > 0 and infinitely many
R-module. If TorR
i (R
e. Then pdR M < ∞.
Question
Does Herzog’s theorem hold for arbitrary modules?
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Answers!
Main Theorem
Let M be an R-module.
(e) , M) = 0 for all i, e > 0 and
(a) If fdR M < ∞ then TorR
i (R
(e)
fdR M = fdR (e) R ⊗R M for all e > 0.
(e) , M) = 0 for all
(b) If R has finite Krull dimension and TorR
i (R
i > 0 and infinitely many e, then pdR M < ∞.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Answers!
Main Theorem
Let M be an R-module.
(e) , M) = 0 for all i, e > 0 and
(a) If fdR M < ∞ then TorR
i (R
(e)
fdR M = fdR (e) R ⊗R M for all e > 0.
(e) , M) = 0 for all
(b) If R has finite Krull dimension and TorR
i (R
i > 0 and infinitely many e, then pdR M < ∞.
If in addition R (1) is finitely generated as an R-module, we have:
(c) If idR M < ∞ then ExtiR (R (e) , M) = 0 for all i, e > 0 and
idR M = idR (e) HomR (R (e) , M) for all e > 0.
(d) If R has finite Krull dimension and ExtiR (R (e) , M) = 0 for all
i > 0 and infinitely many e, then idR M < ∞.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Flat covers
Definition (Enochs, 1981)
A homomorphism φ : F → M is called a flat cover of M if:
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Flat covers
Definition (Enochs, 1981)
A homomorphism φ : F → M is called a flat cover of M if:
1
F is flat.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Flat covers
Definition (Enochs, 1981)
A homomorphism φ : F → M is called a flat cover of M if:
1
F is flat.
2
For any homomorphism ψ : G → M where G is flat there
exists a map g : G → F such that ψ = φg .
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Flat covers
Definition (Enochs, 1981)
A homomorphism φ : F → M is called a flat cover of M if:
1
F is flat.
2
For any homomorphism ψ : G → M where G is flat there
exists a map g : G → F such that ψ = φg .
3
If h : F → F satisfies φ = φh then h is an isomorphism.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Flat covers
Definition (Enochs, 1981)
A homomorphism φ : F → M is called a flat cover of M if:
1
F is flat.
2
For any homomorphism ψ : G → M where G is flat there
exists a map g : G → F such that ψ = φg .
3
If h : F → F satisfies φ = φh then h is an isomorphism.
Theorem (J. Xu, 1995; Bican, El-Bashir, Enochs, 2001)
Flat covers exist!
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Cotorsion modules
Definition (Enochs, 1984)
An R-module is called cotorsion if Ext1R (F , M) = 0 for every flat
R-module F .
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Cotorsion modules
Definition (Enochs, 1984)
An R-module is called cotorsion if Ext1R (F , M) = 0 for every flat
R-module F .
Facts:
The kernel of a flat cover is cotorsion.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Cotorsion modules
Definition (Enochs, 1984)
An R-module is called cotorsion if Ext1R (F , M) = 0 for every flat
R-module F .
Facts:
The kernel of a flat cover is cotorsion.
A flat cover of a cotorsion module is cotorsion.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Cotorsion modules
Definition (Enochs, 1984)
An R-module is called cotorsion if Ext1R (F , M) = 0 for every flat
R-module F .
Facts:
The kernel of a flat cover is cotorsion.
A flat cover of a cotorsion module is cotorsion.
If R is complete then any Artinian or Noetherian R-module is
cotorsion.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Decomposition of flat cotorsion modules
Theorem (Enochs, 1984)
Let F be a flat cotorsion R-module. Then
(a)
F ∼
=
Y
T (q),
q∈Spec R
where T (q) is the completion with respect to the qRq -adic
topology of a free Rq -module G (q).
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Decomposition of flat cotorsion modules
Theorem (Enochs, 1984)
Let F be a flat cotorsion R-module. Then
(a)
F ∼
=
Y
T (q),
q∈Spec R
where T (q) is the completion with respect to the qRq -adic
topology of a free Rq -module G (q).
(b) For q ∈ Spec R the rank of G (q) is determined by F .
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Decomposition of flat cotorsion modules
Theorem (Enochs, 1984)
Let F be a flat cotorsion R-module. Then
(a)
F ∼
=
Y
T (q),
q∈Spec R
where T (q) is the completion with respect to the qRq -adic
topology of a free Rq -module G (q).
(b) For q ∈ Spec R the rank of G (q) is determined by F .
For q ∈ Spec R we denote of the rank of G (q) by π(q, F ).
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Minimal flat resolutions
Definition (Enochs-Xu, 1997)
A minimal flat resolution of M is an acyclic complex (F, ∂) such
that each Fi is flat, Fi = 0 for all i < 0, H0 (F) ∼
= M, and
Fi → coker ∂i+1 is a flat cover for all i.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Minimal flat resolutions
Definition (Enochs-Xu, 1997)
A minimal flat resolution of M is an acyclic complex (F, ∂) such
that each Fi is flat, Fi = 0 for all i < 0, H0 (F) ∼
= M, and
Fi → coker ∂i+1 is a flat cover for all i.
Minimal flat resolutions exist and are unique up to
isomorphism.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Minimal flat resolutions
Definition (Enochs-Xu, 1997)
A minimal flat resolution of M is an acyclic complex (F, ∂) such
that each Fi is flat, Fi = 0 for all i < 0, H0 (F) ∼
= M, and
Fi → coker ∂i+1 is a flat cover for all i.
Minimal flat resolutions exist and are unique up to
isomorphism.
If F is a minimal flat resolution of M then Fi is cotorsion for
i ≥ 1, and F0 is cotorsion if and only if M is.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Minimal flat resolutions
Definition (Enochs-Xu, 1997)
A minimal flat resolution of M is an acyclic complex (F, ∂) such
that each Fi is flat, Fi = 0 for all i < 0, H0 (F) ∼
= M, and
Fi → coker ∂i+1 is a flat cover for all i.
Minimal flat resolutions exist and are unique up to
isomorphism.
If F is a minimal flat resolution of M then Fi is cotorsion for
i ≥ 1, and F0 is cotorsion if and only if M is.
For q ∈ Spec R and i ≥ 1 we let πi (q, M) := π(q, Fi ). We set
π0 (q, M) := π(q, CR (F0 )), where CR (F0 ) is the cotorsion envelope
of F0 . We call the πi (q, M) the Enochs-Xu numbers of M.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Enochs-Xu Theorem
Theorem (Enochs-Xu, 1997)
Let M be an R-module.
1
πi (q, M) = πi (q, CR (M)) for all i, q ∈ Spec R.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Enochs-Xu Theorem
Theorem (Enochs-Xu, 1997)
Let M be an R-module.
1
πi (q, M) = πi (q, CR (M)) for all i, q ∈ Spec R.
2
If (R, m) is local and M is cotorsion then ∂i (Fi ) ⊆ mFi−1
where (F, ∂) is any minimal flat resolution of M.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Enochs-Xu Theorem
Theorem (Enochs-Xu, 1997)
Let M be an R-module.
1
πi (q, M) = πi (q, CR (M)) for all i, q ∈ Spec R.
2
If (R, m) is local and M is cotorsion then ∂i (Fi ) ⊆ mFi−1
where (F, ∂) is any minimal flat resolution of M.
3
If M is cotorsion then for all i ≥ 0 and q ∈ Spec R,
R
πi (q, M) = dimk(q) Tori q (k(q), HomR (Rq , M)).
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Enochs-Xu Theorem
Theorem (Enochs-Xu, 1997)
Let M be an R-module.
1
πi (q, M) = πi (q, CR (M)) for all i, q ∈ Spec R.
2
If (R, m) is local and M is cotorsion then ∂i (Fi ) ⊆ mFi−1
where (F, ∂) is any minimal flat resolution of M.
3
If M is cotorsion then for all i ≥ 0 and q ∈ Spec R,
R
πi (q, M) = dimk(q) Tori q (k(q), HomR (Rq , M)).
Note: Part (1) implies that fdR M = fdR CR (M).
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Peskine and Szpiro’s Acyclicity Lemma
Definition
Let (R, m) be local and M an R-module. We define the depth of
M by
i
depth M := inf{i ≥ 0 | Hm
(M) 6= 0}.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Peskine and Szpiro’s Acyclicity Lemma
Definition
Let (R, m) be local and M an R-module. We define the depth of
M by
i
depth M := inf{i ≥ 0 | Hm
(M) 6= 0}.
Lemma (Peskine-Szpiro, 1973)
Let R be a local ring and consider a bounded complex T of
R-modules: 0 → Ts → Ts−1 → · · · → T0 → 0. Suppose the
following two conditions hold for each i > 0:
1
depth Ti ≥ i;
2
depth Hi (T) = 0 or Hi (T) = 0.
Then Hi (T) = 0 for all i > 0.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Key result: A vanishing result for Enochs-Xu numbers
Theorem
Suppose fdR M < ∞. Then πi (p, M) = 0 for all p ∈ Spec R and
i > depth Rp .
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Key result: A vanishing result for Enochs-Xu numbers
Theorem
Suppose fdR M < ∞. Then πi (p, M) = 0 for all p ∈ Spec R and
i > depth Rp .
Corollary
Suppose (R, m) is local and fdR M < ∞. Suppose F is a minimal
flat resolution of M. Then depthR (e) R (e) ⊗R Fi = ∞ for
i > depth R.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Key result: A vanishing result for Enochs-Xu numbers
Theorem
Suppose fdR M < ∞. Then πi (p, M) = 0 for all p ∈ Spec R and
i > depth Rp .
Corollary
Suppose (R, m) is local and fdR M < ∞. Suppose F is a minimal
flat resolution of M. Then depthR (e) R (e) ⊗R Fi = ∞ for
i > depth R.
Proof.
j
j
Suppose i > depth R. Then Hm
(R (e) ⊗R Fi ) ∼
= Hm (R (e) ) ⊗R Fi = 0
for all j, since Fi is flat and a product of Rp -modules, p 6= m.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Proof of part (a) of the Main Theorem
(e) , M) 6= 0 for some i, e > 0.
Suppose fdR M < ∞ but TorR
i (R
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Proof of part (a) of the Main Theorem
(e) , M) 6= 0 for some i, e > 0.
Suppose fdR M < ∞ but TorR
i (R
(e) , M) ⊆ {m} for all
By localizing, we can assume SuppR TorR
j (R
j > 0, with equality for at least one j.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Proof of part (a) of the Main Theorem
(e) , M) 6= 0 for some i, e > 0.
Suppose fdR M < ∞ but TorR
i (R
(e) , M) ⊆ {m} for all
By localizing, we can assume SuppR TorR
j (R
j > 0, with equality for at least one j.
Let F be a minimal flat resolution of M and G = R (e) ⊗R F.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Proof of part (a) of the Main Theorem
(e) , M) 6= 0 for some i, e > 0.
Suppose fdR M < ∞ but TorR
i (R
(e) , M) ⊆ {m} for all
By localizing, we can assume SuppR TorR
j (R
j > 0, with equality for at least one j.
Let F be a minimal flat resolution of M and G = R (e) ⊗R F.
Then for i > 0, depth Hi (G) = 0 or Hi (G) = 0.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Proof of part (a) of the Main Theorem
(e) , M) 6= 0 for some i, e > 0.
Suppose fdR M < ∞ but TorR
i (R
(e) , M) ⊆ {m} for all
By localizing, we can assume SuppR TorR
j (R
j > 0, with equality for at least one j.
Let F be a minimal flat resolution of M and G = R (e) ⊗R F.
Then for i > 0, depth Hi (G) = 0 or Hi (G) = 0.
If i ≤ depth R then, depth Gi ≥ depth R (e) as Gi is a flat
R (e) -module.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Proof of part (a) of the Main Theorem
(e) , M) 6= 0 for some i, e > 0.
Suppose fdR M < ∞ but TorR
i (R
(e) , M) ⊆ {m} for all
By localizing, we can assume SuppR TorR
j (R
j > 0, with equality for at least one j.
Let F be a minimal flat resolution of M and G = R (e) ⊗R F.
Then for i > 0, depth Hi (G) = 0 or Hi (G) = 0.
If i ≤ depth R then, depth Gi ≥ depth R (e) as Gi is a flat
R (e) -module.
If i > depth R then depth Gi = ∞ by the Corollary.
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Proof of part (a) of the Main Theorem
(e) , M) 6= 0 for some i, e > 0.
Suppose fdR M < ∞ but TorR
i (R
(e) , M) ⊆ {m} for all
By localizing, we can assume SuppR TorR
j (R
j > 0, with equality for at least one j.
Let F be a minimal flat resolution of M and G = R (e) ⊗R F.
Then for i > 0, depth Hi (G) = 0 or Hi (G) = 0.
If i ≤ depth R then, depth Gi ≥ depth R (e) as Gi is a flat
R (e) -module.
If i > depth R then depth Gi = ∞ by the Corollary.
Hence, G = R (e) ⊗R F is acyclic by Peskine and Szpiro’s acyclicity
(e) , M) = 0 for i > 0, a contradiction.
lemma. Thus, TorR
i (R
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln
Thank you!
Tom Marley Marcus Webb
Frobenius and modules of finite flat dimension
University of Nebraska-Lincoln