Dimensions of derived categories and
singularity categories
Ryo Takahashi
Shinshu University/University of Nebraska-Lincoln
October 16, 2011
Joint work with Takuma Aihara
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
2/6
Joint work with Takuma Aihara
Definition
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
2/6
Joint work with Takuma Aihara
Definition
Let T be a triangulated category
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
2/6
Joint work with Takuma Aihara
Definition
Let T be a triangulated category
(e.g. Db (R), Dsg (R) = Db (R)/perf(R), CM(R), . . . ).
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
2/6
Joint work with Takuma Aihara
Definition
Let T be a triangulated category
(e.g. Db (R), Dsg (R) = Db (R)/perf(R), CM(R), . . . ).
Let X , Y ⊆ T and n ≥ 1.
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
2/6
Joint work with Takuma Aihara
Definition
Let T be a triangulated category
(e.g. Db (R), Dsg (R) = Db (R)/perf(R), CM(R), . . . ).
Let X , Y ⊆ T and n ≥ 1.
1
X ∗ Y consists of C ∈ T admitting exact triangles
M → C → N → with M ∈ X and N ∈ Y.
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
2/6
Joint work with Takuma Aihara
Definition
Let T be a triangulated category
(e.g. Db (R), Dsg (R) = Db (R)/perf(R), CM(R), . . . ).
Let X , Y ⊆ T and n ≥ 1.
1
X ∗ Y consists of C ∈ T admitting exact triangles
M → C → N → with M ∈ X and N ∈ Y.
2
⟨X ⟩ is the smallest full subcategory of T containing X and
closed under finite direct sums, direct summands and shifts.
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
2/6
Joint work with Takuma Aihara
Definition
Let T be a triangulated category
(e.g. Db (R), Dsg (R) = Db (R)/perf(R), CM(R), . . . ).
Let X , Y ⊆ T and n ≥ 1.
1
X ∗ Y consists of C ∈ T admitting exact triangles
M → C → N → with M ∈ X and N ∈ Y.
2
⟨X ⟩ is the smallest full subcategory of T containing X and
closed under finite direct sums, direct summands and shifts.
n
z
}|
{
⟨X ⟩n = ⟨⟨X ⟩ ∗ · · · ∗ ⟨X ⟩⟩,
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
2/6
Joint work with Takuma Aihara
Definition
Let T be a triangulated category
(e.g. Db (R), Dsg (R) = Db (R)/perf(R), CM(R), . . . ).
Let X , Y ⊆ T and n ≥ 1.
1
X ∗ Y consists of C ∈ T admitting exact triangles
M → C → N → with M ∈ X and N ∈ Y.
2
⟨X ⟩ is the smallest full subcategory of T containing X and
closed under finite direct sums, direct summands and shifts.
n
z
}|
{
⟨X ⟩n = ⟨⟨X ⟩ ∗ · · · ∗ ⟨X ⟩⟩, ⟨G⟩n := ⟨{G}⟩n for G ∈ T .
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
2/6
Joint work with Takuma Aihara
Definition
Let T be a triangulated category
(e.g. Db (R), Dsg (R) = Db (R)/perf(R), CM(R), . . . ).
Let X , Y ⊆ T and n ≥ 1.
1
X ∗ Y consists of C ∈ T admitting exact triangles
M → C → N → with M ∈ X and N ∈ Y.
2
⟨X ⟩ is the smallest full subcategory of T containing X and
closed under finite direct sums, direct summands and shifts.
n
z
}|
{
⟨X ⟩n = ⟨⟨X ⟩ ∗ · · · ∗ ⟨X ⟩⟩, ⟨G⟩n := ⟨{G}⟩n for G ∈ T .
3
.
The dimension dim T of T is the infimum of r ≥ 0 such
that
T = ⟨G⟩r+1 for some G ∈ T .
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
2/6
Let R be a commutative Noetherian ring.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
3/6
Let R be a commutative Noetherian ring.
Fact
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
3/6
Let R be a commutative Noetherian ring.
Fact
1
dim Db (R) ≥ dim Dsg (R).
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
3/6
Let R be a commutative Noetherian ring.
Fact
1
2
dim Db (R) ≥ dim Dsg (R).
[Rouquier 2008]
If R is a reduced affine algebra, then dim Db (R) ≥ dim R.
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
3/6
Let R be a commutative Noetherian ring.
Fact
1
2
3
dim Db (R) ≥ dim Dsg (R).
[Rouquier 2008]
If R is a reduced affine algebra, then dim Db (R) ≥ dim R.
[Avramov-Iyengar]
If R is local, then dim Dsg (R) ≥ cfrank R − 1.
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
3/6
.
Question
.
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Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
4/6
.
Question
.
.
When dim Db (R) < ∞?
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
4/6
.
Question
.
When dim Db (R) < ∞?
.
Fact
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
4/6
.
Question
.
When dim Db (R) < ∞?
Fact
If R is Artinian, then Db (R) = ⟨R/ rad R⟩ℓℓ(R) , and
dim Db (R) ≤ ℓℓ(R) − 1 < ∞.
.
1
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
4/6
.
Question
.
When dim Db (R) < ∞?
Fact
If R is Artinian, then Db (R) = ⟨R/ rad R⟩ℓℓ(R) , and
dim Db (R) ≤ ℓℓ(R) − 1 < ∞.
[D. Christensen 1998, Krause-Kussin 2006]
If R is regular with dim R < ∞, then Db (R) = ⟨R⟩dim R+1 , and
dim Db (R) ≤ dim R < ∞.
.
1
2
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
4/6
.
Question
.
When dim Db (R) < ∞?
Fact
If R is Artinian, then Db (R) = ⟨R/ rad R⟩ℓℓ(R) , and
dim Db (R) ≤ ℓℓ(R) − 1 < ∞.
[D. Christensen 1998, Krause-Kussin 2006]
If R is regular with dim R < ∞, then Db (R) = ⟨R⟩dim R+1 , and
dim Db (R) ≤ dim R < ∞.
[Rouquier 2008, Keller-Van den Bergh]
If R is essentially of finite type, then dim Db (R) < ∞.
.
.
1
2
3
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
4/6
.
Question
.
When dim Db (R) < ∞?
Fact
If R is Artinian, then Db (R) = ⟨R/ rad R⟩ℓℓ(R) , and
dim Db (R) ≤ ℓℓ(R) − 1 < ∞.
[D. Christensen 1998, Krause-Kussin 2006]
If R is regular with dim R < ∞, then Db (R) = ⟨R⟩dim R+1 , and
dim Db (R) ≤ dim R < ∞.
[Rouquier 2008, Keller-Van den Bergh]
If R is essentially of finite type, then dim Db (R) < ∞.
.
.
1
2
3
Question
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
4/6
.
Question
.
When dim Db (R) < ∞?
Fact
If R is Artinian, then Db (R) = ⟨R/ rad R⟩ℓℓ(R) , and
dim Db (R) ≤ ℓℓ(R) − 1 < ∞.
[D. Christensen 1998, Krause-Kussin 2006]
If R is regular with dim R < ∞, then Db (R) = ⟨R⟩dim R+1 , and
dim Db (R) ≤ dim R < ∞.
[Rouquier 2008, Keller-Van den Bergh]
If R is essentially of finite type, then dim Db (R) < ∞.
.
.
1
2
3
Question
If R is complete local, then dim Db (R) < ∞?
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
4/6
Example
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
5/6
Example
Let R = k[[t4 , t5 , t6 ]] ⊆ k[[t]] = S.
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
5/6
Example
Let R = k[[t4 , t5 , t6 ]] ⊆ k[[t]] = S. We have CR = t8 S ⊆ R.
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
5/6
Example
Let R = k[[t4 , t5 , t6 ]] ⊆ k[[t]] = S. We have CR = t8 S ⊆ R.
f
There is an exact sequence 0 → R −
→ S → S/R → 0,
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
5/6
Example
Let R = k[[t4 , t5 , t6 ]] ⊆ k[[t]] = S. We have CR = t8 S ⊆ R.
f
There is an exact sequence 0 → R −
→ S → S/R → 0, where S/R
is an R/CR -module.
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
5/6
Example
Let R = k[[t4 , t5 , t6 ]] ⊆ k[[t]] = S. We have CR = t8 S ⊆ R.
f
There is an exact sequence 0 → R −
→ S → S/R → 0, where S/R
is an R/CR -module. Fix X ∈ Db (R).
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
5/6
Example
Let R = k[[t4 , t5 , t6 ]] ⊆ k[[t]] = S. We have CR = t8 S ⊆ R.
f
There is an exact sequence 0 → R −
→ S → S/R → 0, where S/R
is an R/CR -module. Fix X ∈ Db (R). There is an exact sequence of
R-complexes
X⊗R f
0 → M → X −−→ X ⊗R S → N → 0,
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
5/6
Example
Let R = k[[t4 , t5 , t6 ]] ⊆ k[[t]] = S. We have CR = t8 S ⊆ R.
f
There is an exact sequence 0 → R −
→ S → S/R → 0, where S/R
is an R/CR -module. Fix X ∈ Db (R). There is an exact sequence of
R-complexes
X⊗R f
0 → M → X −−→ X ⊗R S → N → 0,
which gives exact triangles
M→X→L→
and L → X ⊗R S → N → .
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
5/6
Example
Let R = k[[t4 , t5 , t6 ]] ⊆ k[[t]] = S. We have CR = t8 S ⊆ R.
f
There is an exact sequence 0 → R −
→ S → S/R → 0, where S/R
is an R/CR -module. Fix X ∈ Db (R). There is an exact sequence of
R-complexes
X⊗R f
0 → M → X −−→ X ⊗R S → N → 0,
which gives exact triangles
M→X→L→
and L → X ⊗R S → N → .
We have X ⊗R S ∈ Db (S) = ⟨S⟩2
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
.
October 16, 2011
5/6
Example
Let R = k[[t4 , t5 , t6 ]] ⊆ k[[t]] = S. We have CR = t8 S ⊆ R.
f
There is an exact sequence 0 → R −
→ S → S/R → 0, where S/R
is an R/CR -module. Fix X ∈ Db (R). There is an exact sequence of
R-complexes
X⊗R f
0 → M → X −−→ X ⊗R S → N → 0,
which gives exact triangles
M→X→L→
and L → X ⊗R S → N → .
. R ) = ⟨k⟩2 ,
We have X ⊗R S ∈ Db (S) = ⟨S⟩2 and M, N ∈ Db (R/C
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
5/6
Example
Let R = k[[t4 , t5 , t6 ]] ⊆ k[[t]] = S. We have CR = t8 S ⊆ R.
f
There is an exact sequence 0 → R −
→ S → S/R → 0, where S/R
is an R/CR -module. Fix X ∈ Db (R). There is an exact sequence of
R-complexes
X⊗R f
0 → M → X −−→ X ⊗R S → N → 0,
which gives exact triangles
M→X→L→
and L → X ⊗R S → N → .
. R ) = ⟨k⟩2 ,
We have X ⊗R S ∈ Db (S) = ⟨S⟩2 and M, N ∈ Db (R/C
2
∼
since R/CR = k[[x, y, z]]/(x, y, z) and ℓℓ(R) = 2.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
5/6
Example
Let R = k[[t4 , t5 , t6 ]] ⊆ k[[t]] = S. We have CR = t8 S ⊆ R.
f
There is an exact sequence 0 → R −
→ S → S/R → 0, where S/R
is an R/CR -module. Fix X ∈ Db (R). There is an exact sequence of
R-complexes
X⊗R f
0 → M → X −−→ X ⊗R S → N → 0,
which gives exact triangles
M→X→L→
and L → X ⊗R S → N → .
. R ) = ⟨k⟩2 ,
We have X ⊗R S ∈ Db (S) = ⟨S⟩2 and M, N ∈ Db (R/C
2
∼
since R/CR = k[[x, y, z]]/(x, y, z) and ℓℓ(R) = 2. Hence
L ∈ ⟨S ⊕ k⟩4 ,
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
5/6
Example
Let R = k[[t4 , t5 , t6 ]] ⊆ k[[t]] = S. We have CR = t8 S ⊆ R.
f
There is an exact sequence 0 → R −
→ S → S/R → 0, where S/R
is an R/CR -module. Fix X ∈ Db (R). There is an exact sequence of
R-complexes
X⊗R f
0 → M → X −−→ X ⊗R S → N → 0,
which gives exact triangles
M→X→L→
and L → X ⊗R S → N → .
. R ) = ⟨k⟩2 ,
We have X ⊗R S ∈ Db (S) = ⟨S⟩2 and M, N ∈ Db (R/C
2
∼
since R/CR = k[[x, y, z]]/(x, y, z) and ℓℓ(R) = 2. Hence
L ∈ ⟨S ⊕ k⟩4 , and X ∈ ⟨S ⊕ k⟩6 .
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
5/6
Example
Let R = k[[t4 , t5 , t6 ]] ⊆ k[[t]] = S. We have CR = t8 S ⊆ R.
f
There is an exact sequence 0 → R −
→ S → S/R → 0, where S/R
is an R/CR -module. Fix X ∈ Db (R). There is an exact sequence of
R-complexes
X⊗R f
0 → M → X −−→ X ⊗R S → N → 0,
which gives exact triangles
M→X→L→
and L → X ⊗R S → N → .
. R ) = ⟨k⟩2 ,
We have X ⊗R S ∈ Db (S) = ⟨S⟩2 and M, N ∈ Db (R/C
2
∼
since R/CR = k[[x, y, z]]/(x, y, z) and ℓℓ(R) = 2. Hence
L ∈ ⟨S ⊕ k⟩4 , and X ∈ ⟨S ⊕ k⟩6 . This shows Db (R) = ⟨S ⊕ k⟩6 ,
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
5/6
Example
Let R = k[[t4 , t5 , t6 ]] ⊆ k[[t]] = S. We have CR = t8 S ⊆ R.
f
There is an exact sequence 0 → R −
→ S → S/R → 0, where S/R
is an R/CR -module. Fix X ∈ Db (R). There is an exact sequence of
R-complexes
X⊗R f
0 → M → X −−→ X ⊗R S → N → 0,
which gives exact triangles
M→X→L→
and L → X ⊗R S → N → .
. R ) = ⟨k⟩2 ,
We have X ⊗R S ∈ Db (S) = ⟨S⟩2 and M, N ∈ Db (R/C
2
∼
since R/CR = k[[x, y, z]]/(x, y, z) and ℓℓ(R) = 2. Hence
L ∈ ⟨S ⊕ k⟩4 , and X ∈ ⟨S ⊕ k⟩6 . This shows Db (R) = ⟨S ⊕ k⟩6 ,
and dim Db (R) ≤ 5 < ∞.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
5/6
Theorem (Aihara-T)
.
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
6/6
Theorem (Aihara-T)
Let R be a complete equicharacteristic local ring with perfect residue
field. Then dim Db (R) < ∞.
.
.
Ryo Takahashi (Shinshu Univ./UNL)
Dimensions of derived categories
October 16, 2011
6/6