Weakly proregular sequence and Cech, local cohomology

Weakly proregular sequence and Čech, local cohomology
安藤遼哉
東京理科大学理工学研究科
2021/10/13
Ryoya Ando (Tokyo University of Science) Weakly proregular sequence and Čech, local cohomology 2021/10/13 1 / 29

Research Background
1 Research Background
2 Introduction
3 Čech cohomology and local cohomology
4 Weakly proregular sequence
5 Proof of Schenzel’s theorem
6 Reference

Research Background
• 可換環論，特に非 Noether 環に注目して研究をしています．可換環論は代数幾何学と歴
史的に結びつきの強い分野で（発表者も所属は代数幾何系の研究室です）
，積極的に研究
されているのは Noether 環が中心です．
FAQ
• 非 Noether 環の研究って何をしているの？
• 目的は？
• 応用は？
• …etc.

Research Background
• Q. 非 Noether 環の研究って何をしているの？
I A. 大きく分けて，Noether 環の理論を非 Noether に拡張する研究と，非 Noether でしか起
こり得ない現象を調べる研究があります．
Hamilton and Marley (2007), Kim and Walker (2020), Miller (2008) などが CM 環，
Gorenstein 環などのホモロジカルな性質を拡張して，非 Noether 環上に一般化する研究を
行っています．
２次元以上の付値環は決して Noether 環にはなりません．
また Noether ではない環を含むような環のクラスに Krull 整域（UFD の一般化）があります
（Noether 整域が Krull 整域であることと，整閉整域であることは同値です）
．
少し古いですが，非 Noether 可換環論の話題を集めた本も出ています（Chapman and
Glaz (2000)）
．
今日は Schenzel (2003) による weakly proregular sequence を紹介して，Noether 環における事
実が非 Noether 環に一般化される様子をみていきましょう．そして Schenzel の定理 (Theorem
4.1) の初頭的（？）な証明を発表者の preprint1 (arXiv:2105.07652) に基づいて紹介します．
1Accepted in Moroccan Journal of Algebra and Geometry with Applications.

Introduction
2 Introduction
6 Reference

Introduction
Various cohomologies and homologies are used in (commutative) algebra theory.
Example. 𝐴 : ring (unitary and commutative), 𝐼 : ideal of 𝐴, and 𝑀, 𝑁 ∈ Mod(𝐴).
(Mod(𝐴) : the category of 𝐴-modules, mod(𝐴) : the category of finitely generated 𝐴-modules.
)
• Ext𝑖
𝐴(𝑀, −) ...........Derived functor of Hom𝐴(𝑀, −).
• Tor𝐴
𝑖 (𝑀, −) ...........Derived functor of 𝑀 ⊗𝐴 −.
• 𝐻𝑖
𝐼 (−) ...................Derived functor of lim
−
−
→
Hom𝐴(𝐴/𝐼𝑛, −).
• 𝐻𝑖 ( 𝑓 , −) ...............Koszul homology defined by the 𝐴-linear map 𝑓 : 𝑁 → 𝐴.
• ˇ
𝐻𝑖 (𝑎, −) ................Čech cohomology defined by the sequence 𝑎 = 𝑎1, . . . , 𝑎𝑟 ∈ 𝐴.
• and more!
Derived functor is obtained from a right (or left) exact functor. For example, let 𝐽• be an
injective resolution of 𝑁, then Ext𝑖
(𝑀, 𝑁) B 𝐻𝑖 (Hom(𝑀, 𝐽•)).

Introduction
Why are cohomologies used so much?
One of the reasons for this is that ideal theoretic data can be written in a easier
form for calculation.
Definition 2.1
𝐴 : ring , 𝑀 ∈ Mod(𝐴). 𝑎 ∈ 𝐴 is called 𝑀-regular if ∀𝑥 ≠ 0 ∈ 𝑀, 𝑎𝑥 ≠ 0.
A sequence 𝑎 = 𝑎1, . . . , 𝑎𝑟 ∈ 𝐴 is called an 𝑀-regular sequence if;
• 𝑀/(𝑎1, . . . , 𝑎𝑟 )𝑀 ≠ 0,
• 1 ≤ ∀𝑖 ≤ 𝑟, 𝑎𝑖 is an 𝑀/(𝑎1, . . . , 𝑎𝑖−1)𝑀-regular.

Introduction
Definition 2.2
𝐴 : Noetherian ring, 𝑀 ∈ mod(𝐴) and 𝐼 : ideal with 𝐼𝑀 ≠ 𝑀.
depth𝐼 (𝑀) B sup

𝑟 ≥ 0 ∃
𝑎 = 𝑎1, . . . , 𝑎𝑟 ∈ 𝐼, 𝑎 is an 𝑀-regular sequence.
is called an 𝐼-depth of 𝑀.
Theorem 2.3 (Rees)
Under the above notation, the length of a maximal regular sequence is constant. Also;
depth𝐼 (𝑀) = inf

𝑖 ≥ 0 Ext𝑖
(𝐴/𝐼, 𝑀) ≠ 0 .
This theorem shows that the depth of module is calculatable by using a cohomology!

Čech cohomology and local cohomology
2 Introduction
6 Reference

Definition 3.1
𝐴 : ring , 𝐼 : ideal of 𝐴.
𝐻𝑖
𝐼 (−) : the right derived functor of lim
−
−
→
Hom𝐴(𝐴/𝐼𝑛, −) is called a local cohomology.
Note that there are following isomorphisms,
𝐻𝑖
𝐼 (𝑀) lim
−
−
→
Ext𝑖
(𝐴/𝐼𝑛
, 𝑀)
since taking the inductive limit is an exact functor.

Definition 3.2
𝐴 : ring , 𝑎 = 𝑎1, . . . , 𝑎𝑟 ∈ 𝐴.
{𝑒𝑖} : the standard basis of 𝐴𝑟 .
For each 𝐼 = { 𝑗1, . . . , 𝑗𝑖} (1 ≤ 𝑗1 · · · 𝑗𝑖 ≤ 𝑟), let 𝑎𝐼 = 𝑎𝑗1 · · · 𝑎𝑗𝑖 and 𝑒𝐼 = 𝑒𝑗1 ∧ · · · ∧ 𝑒𝑗𝑖 .
𝐶•(𝑎) : the complex defined by;
𝐶𝑖
(𝑎) B
Õ
#𝐼=𝑖
𝐴𝑎𝐼 𝑒𝐼,
𝑑𝑖
: 𝐶𝑖
(𝑎) → 𝐶𝑖+1
(𝑎); 𝑒𝐼 ↦→
𝑟
Õ
𝑗=1
𝑒𝐼 ∧ 𝑒𝑗 .
It is called a Čech complex.
ˇ
𝐻𝑖 (𝑎) : the cohomology of 𝐶•(𝑎) is called a Čech cohomology.
For 𝑀 ∈ Mod(𝐴), we define 𝐶•(𝑎, 𝑀) B 𝐶•(𝑎) ⊗ 𝑀, ˇ
𝐻𝑖 (𝑎, 𝑀) B 𝐻𝑖 (𝐶•(𝑎, 𝑀)).

Theorem 3.3
𝐴 : Noetherian ring, 𝑎 = 𝑎1, . . . , 𝑎𝑟 ∈ 𝐴 and 𝐼 = (𝑎1, . . . , 𝑎𝑟 ). There are isomorphisms;
𝐻𝑖
𝐼 (𝑀) ˇ
𝐻𝑖
(𝑎, 𝑀)
for any 𝑀 ∈ Mod(𝐴).
What happens if we remove the Noetherian assumption? Can we extend this theorem?
This theorem was extended by Schenzel (2003) by introducing a weakly proregular
sequence.

Weakly proregular sequence
2 Introduction
6 Reference

Theorem 4.1 (Schenzel)
𝐴 : ring, 𝑎 = 𝑎1, . . . , 𝑎𝑟 ∈ 𝐴 and 𝐼 = (𝑎1, . . . , 𝑎𝑟 ).
𝑎 is a weakly proregular sequence ⇐⇒ ∀
𝑖 ≥ 0, ∀
𝑀 ∈ Mod(𝐴), 𝐻𝑖
𝐼 (𝑀) ˇ
𝐻𝑖
(𝑎, 𝑀).
A weakly proregular sequence is defined using the Koszul complex.
Definition 4.2
𝐴 : ring , 𝑎 = 𝑎1, . . . , 𝑎𝑟 ∈ 𝐴. {𝑒𝑖} : the standard basis of 𝐴𝑟 .
𝐾•(𝑎) is the complex defined by ;
𝐾𝑖 (𝑎) =
𝑖
Û
𝐴𝑟
𝑑𝑖 : 𝐾𝑖 (𝑎) → 𝐾𝑖−1(𝑎); 𝑒𝐼 ↦→
𝑖
Õ
𝑘=1
(−1)𝑘+1
𝑎𝑗𝑘 𝑒𝑗1 ∧ · · · ∧ c
𝑒𝑗𝑘 ∧ · · · ∧ 𝑒𝑗𝑖 .
It is called a Koszul (chain) complex.
𝐻𝑖 (𝑎) : the homology of 𝐾•(𝑎) is called a Koszul homology.

𝑎𝑛 : the sequence defined by 𝑎𝑛
1 , . . . , 𝑎𝑛
𝑟 .
Note that by following morphisms, Koszul complexes constitute an inverse system
{𝐾•(𝑎𝑛)}𝑛≥0; 𝜑𝑚𝑛 : 𝐾𝑖 (𝑎𝑚
) → 𝐾𝑖 (𝑎𝑛
); 𝑒𝐼 ↦→ 𝑎𝑚−𝑛
𝐼 𝑒𝐼 (𝑛 ≤ 𝑚).
This induces a morphism between homologies.
Definition 4.3 (Schenzel)
𝐴 : ring , 𝑎 = 𝑎1, . . . , 𝑎𝑟 ∈ 𝐴.
𝑎 is called a weakly proregular sequence if
1 ≤ ∀𝑖 ≤ 𝑟, ∀𝑛 ≥ 0, ∃𝑚 ≥ 𝑛; 𝜑𝑚𝑛 : 𝐻𝑖 (𝑎𝑚) → 𝐻𝑖 (𝑎𝑛) is the zero map.

We will explain that Schenzel’s theorem (Theorem 4.1) is an extension of the Noetherian case.
Definition 4.4 (Greenlees, May)
𝐴 : ring , 𝑎 = 𝑎1, . . . , 𝑎𝑟 ∈ 𝐴.
𝑎 is called a proregular sequence if
1 ≤ ∀𝑖 ≤ 𝑟, ∀𝑛 0, ∃𝑚 ≥ 𝑛; ∀𝑎 ∈ 𝐴, 𝑎𝑎𝑚
𝑖 ∈ (𝑎𝑚
1 , . . . , 𝑎𝑚
𝑖−1) =⇒ 𝑎𝑎𝑚−𝑛
𝑖 ∈ (𝑎𝑛
1 , . . . , 𝑎𝑛
𝑖−1).
The following relations hold;
Regular =⇒ Proregular =⇒ Weakly proregular.
• The first implication is easy. If 𝑎 is a regular sequence, for each 𝑛 0, let 𝑚 = 𝑛.
• The second is proved by calculating a Koszul homology.

Proposition 4.5
𝐴: Noetherian ring , 𝑎 = 𝑎1, . . . , 𝑎𝑟 ∈ 𝐴. 𝑎 is a proregular sequence.
Proof.
Let 𝐽𝑖
𝑚 = ((𝑎𝑚
1 , . . . , 𝑎𝑚
𝑖−1) : 𝑎𝑚
𝑖 𝐴), 𝐼𝑖
𝑛,𝑚 = ((𝑎𝑛
1 , . . . , 𝑎𝑛
𝑖−1) : 𝑎𝑚−𝑛
𝑖 𝐴).
𝑎 is a proregular sequence ⇐⇒ 1 ≤ ∀
𝑖 ≤ 𝑟, ∀
𝑛 0, ∃
𝑚 ≥ 𝑛; 𝐽𝑖
𝑚 ⊂ 𝐼𝑖
𝑛,𝑚.
Fix 1 ≤ ∀𝑖 ≤ 𝑟 and omit from the notation.
Fix 𝑛, {𝐼𝑛,𝑚}𝑚≥𝑛 : ascending chain of ideals ∃𝑚0 ≥ 𝑛; ∀𝑚 ≥ 𝑚0, 𝐼𝑛,𝑚0 = 𝐼𝑛,𝑚.
Let 𝑚 B 𝑚0 + 𝑛, then ∀𝑎 ∈ 𝐽𝑚0 , 𝑎𝑎𝑚−𝑛
𝑖 = 𝑎𝑎𝑚0
𝑖 ∈ (𝑎𝑚0
1 , . . . , 𝑎𝑚0
𝑖−1) ⊂ (𝑎𝑛
1 , . . . , 𝑎𝑛
𝑖−1).
So 𝐽𝑚0 ⊂ 𝐼𝑛,𝑚0 .

Corollary 4.6 (ICYMI : Theorem 3.3)
𝐴 : Noetherian ring, 𝑎 = 𝑎1, . . . , 𝑎𝑟 ∈ 𝐴 and 𝐼 = (𝑎1, . . . , 𝑎𝑟 ). There are isomorphisms;
𝐻𝑖
𝐼 (𝑀) ˇ
𝐻𝑖
(𝑎, 𝑀)
for any 𝑀 ∈ Mod(𝐴).
Another proof of Theorem 3.3.
By above proposition, 𝑎 is proregular. 𝑎 is weakly proregular.
Then according to Schenzel’s theorem, 𝐻𝑖
𝐼 (𝑀) ˇ
𝐻𝑖 (𝑎, 𝑀).

Proof of Schenzel’s theorem
2 Introduction
6 Reference

Why is a weakly proregular sequence defined by using a Koszul homology?
A Čech cohomology can be written by using a Koszul cohomology!
𝐾•(𝑎) B Hom(𝐾•(𝑎), 𝐴). For 𝑀 ∈ Mod(𝐴), 𝐾•(𝑎, 𝑀) B Hom(𝐾•(𝑎), 𝑀) = 𝐾•(𝑎) ⊗ 𝑀.
𝐾•(𝑎) : · · · 𝐾1(𝑎) 𝐾0(𝑎) 0
𝐾•(𝑎) : 0 𝐾0(𝑎) 𝐾1(𝑎) · · ·
Hom(−,𝐴)
The opposition of morphism induces an inductive system {𝐾•(𝑎𝑛)}𝑛≥0;
𝜑𝑛𝑚
: 𝐾𝑖
(𝑎𝑛
) → 𝐾𝑖
(𝑎𝑚
); (𝑒𝐼)∗
↦→ 𝑎𝑚−𝑛
𝐼 (𝑒𝐼)∗
.

Proposition 5.1
𝐴 : ring, 𝑎 = 𝑎1, . . . , 𝑎𝑟 ∈ 𝐴 and 𝑀 ∈ Mod(𝐴). Then;
ˇ
𝐻𝑖
(𝑎, 𝑀) lim
−
−
→
𝐻𝑖
(𝑎𝑛
, 𝑀).
Sketch of the proof.
𝜑𝑖 : 𝐾𝑖 (𝑎) → 𝐶𝑖 (𝑎); (𝑒𝐼)∗ ↦→ (1/𝑎𝐼)𝑒𝐼 is a morphism of complexes.
So we get 𝜑•
𝑛 : 𝐾•(𝑎𝑛) → 𝐶•(𝑎𝑛) = 𝐶•(𝑎). It induces 𝜑 : lim
−
−
→
𝐾•(𝑎𝑛) → 𝐶•(𝑎) and this is an
isomorphism.
· · · 𝐾•(𝑎𝑛) 𝐾•(𝑎𝑚) · · · lim
−
−
→
𝐾•(𝑎𝑛)
𝐶•(𝑎)
𝜑•
𝑛
𝜑𝑛𝑚
𝜑•
𝑚
𝜑
Then ; lim
−
−
→
𝐻𝑖 (𝑎𝑛, 𝑀) = 𝐻𝑖 (lim
−
−
→
𝐾•(𝑎𝑛) ⊗ 𝑀) 𝐻𝑖 (𝐶•(𝑎𝑛) ⊗ 𝑀) = ˇ
𝐻𝑖 (𝑎𝑛, 𝑀).


Remark 5.2
By Proposition 5.1,
ˇ
𝐻𝑖
(𝑎, 𝑀) lim
−
−
→
𝐻𝑖
(𝑎𝑛
, 𝑀).
By the definition,
𝐻𝑖
𝐼 (𝑀) lim
−
−
→
Ext𝑖
(𝐴/𝐼𝑛
, 𝑀).
Schenzel’s theorem holds if 𝐻𝑖 (𝑎𝑛, 𝑀) Ext𝑖
(𝐴/𝐼𝑛, 𝑀), which is true when 𝑎 is a
regular sequence.
However, if 𝑎 is not regular, it may not work. So we will need to find an another way.
We will show a way to use the 𝛿-functor.

𝐴: ring , 𝐼 : ideal of 𝐴. Let Γ𝐼 (𝑀) B

𝑥 ∈ 𝑀 ∃𝑛 ≥ 0; 𝐼𝑛𝑥 = 0 .
The functor Γ𝐼 (−) connects a local cohomology and a Čech cohomology.
Lemma 5.3
𝐴 : ring, 𝑎 = 𝑎1, . . . , 𝑎𝑟 ∈ 𝐴, 𝐼 = (𝑎1, . . . , 𝑎𝑟 ) : ideal of 𝐴 and 𝑀 ∈ Mod(𝐴).
𝐻0
𝐼 (𝑀) Γ𝐼 (𝑀) ˇ
𝐻0
(𝑎, 𝑀).
Proof.
• First isomorphism : 𝐻0
𝐼 (𝑀) = lim
−
−
→
Hom(𝐴/𝐼𝑛, 𝑀), Hom(𝐴/𝐼𝑛, 𝑀) {𝑥 ∈ 𝑀 | 𝐼𝑛𝑥 = 0} .
• ˇ
𝐻0(𝑎, 𝑀) is the kernel of (𝑀 →
É𝑟
𝑖=1 𝑀𝑎𝑖 𝑒𝑖; 𝑥 ↦→ (𝑥/1)𝑒𝑖).
∀𝑥 ∈ ˇ
𝐻0(𝑎, 𝑀), 1 ≤ ∀𝑖 ≤ 𝑟, ∃𝑛𝑖 ≥ 0; 𝑎𝑛𝑖
𝑖 𝑥 = 0. i.e. 𝑥 ∈ Γ𝐼 (𝑀).
Similarly Γ𝐼 (𝑀) ⊂ ˇ
𝐻0(𝑎, 𝑀). ˇ
𝐻0(𝑎, 𝑀) = Γ𝐼 (𝑀) as a submodule of 𝑀.


Definition 5.4
𝒜, ℬ : Abelian categories.
𝑇• B {𝑇𝑖 : 𝒜 → ℬ}𝑖≥0 : family of additive functors.
𝑇• is called a 𝛿-functor if ;
• For each exact sequence 0 → 𝐴1 → 𝐴2 → 𝐴3 → 0 in 𝒜, ∃𝛿𝑖 : 𝑇𝑖 (𝐴3) → 𝑇𝑖+1(𝐴1);
0 𝑇0(𝐴1) 𝑇0(𝐴2) 𝑇0(𝐴3) · · · 𝑇𝑖 (𝐴1) 𝑇𝑖 (𝐴2) 𝑇𝑖 (𝐴3) · · ·
𝛿0 𝛿𝑖−1 𝛿𝑖
is exact.
• It transfers a commutative diagram to a commutative diagram.
𝛿𝑖 is called a connecting morphism.

The 𝛿-functor is a generalisation of the derived functor.
It is also useful for proving that the family of functors are form a derived functor!
Definition 5.5
𝒜, ℬ : Abelian categories, 𝐹 : 𝒜 → ℬ : additive functor.
𝐹 is called effaceable if ∀ 𝐴 ∈ 𝒜, ∃ 𝑀 ∈ 𝒜; ∃𝑢 : 𝐴 → 𝑀 : injection; 𝐹(𝑢) = 0.
Proposition 5.6
𝒜, ℬ : Abelian categories, 𝒜 has enough injectives. 𝑇• = {𝑇𝑖}𝑖≥0 : 𝛿-functor.
∀𝑖 0,𝑇𝑖 is effaceable. Then;
• 𝑇0 is left-exact.
• ∀𝑖 ≥ 0,𝑇𝑖 𝑅𝑖𝑇0 (up to unique isomorphism).

Proposition 5.7
ˇ
𝐻•(𝑎, −) is a 𝛿-functor with ˇ
𝐻0(𝑎, −) 𝐻0
𝐼 (−).
0 → 𝑀1 → 𝑀2 → 𝑀3 → 0 : exact sequence of Mod(𝐴).
𝐶•(𝑎, 𝑀) = 𝐶•(𝑎) ⊗ 𝑀 and 𝐶𝑖 (𝑎) is flat. We obtain an exact sequence of
complexes by taking tensor products.
0 𝐶•(𝑎, 𝑀1) 𝐶•(𝑎, 𝑀2) 𝐶•(𝑎, 𝑀3) 0 .
So there are connecting morphisms.

Proposition 5.8
𝐴: ring, 𝑎 = 𝑎1, . . . , 𝑎𝑟 ∈ 𝐴.
𝑎 is a weakly proregular seqence ⇐⇒ ˇ
𝐻•
(𝑎, −) is an effaceable 𝛿-functor.
It is enough to check each injective module 𝐽, ˇ
𝐻𝑖 (𝑎, 𝐽) = 0 (∀𝑖 0).
Use Proposition 5.1. i.e. ˇ
𝐻𝑖 (𝑎, 𝑀) lim
−
−
→
𝐻𝑖 (𝑎𝑛, 𝑀).
Calculate the Koszul (co)homology! (Note that 𝐻𝑖 (𝑎𝑛, 𝐽) Hom(𝐻𝑖 (𝑎𝑛), 𝐽).)

Theorem 5.9 (ICYMI : Schenzel’s theorem)
𝐴 : ring, 𝑎 = 𝑎1, . . . , 𝑎𝑟 ∈ 𝐴 and 𝐼 = (𝑎1, . . . , 𝑎𝑟 ).
𝑖 ≥ 0, ∀
𝑀 ∈ Mod(𝐴), 𝐻𝑖
𝐼 (𝑀) ˇ
𝐻𝑖
(𝑎, 𝑀).
Elementary proof of Schenzel’s theorem. A(2021).
It is a combination of what has been said so far.
𝐻0
𝐼 (−) Γ𝐼 (𝑀) ˇ
𝐻0
(𝑎, −). (Lem. 5.3)
𝑎 is a weakly proregular sequence ⇐⇒ ˇ
𝐻•
(𝑎, −) is an effaceable 𝛿-functor. (Prop. 5.8)
𝑖 ≥ 0, ˇ
𝐻𝑖
(𝑎, −) 𝐻𝑖
𝐼 (−) = 𝑅𝑖
Γ𝐼 (−). (Prop. 5.6)


Reference
Reference
[And21] R. Ando (2021) “A note on weakly proregular sequences”, Accepted in Moroccan Journal of Algebra and
Geometry with Applications, arXiv:2105.07652.
[CG00] S. T. Chapman and S. Glaz eds. (2000) Non-Noetherian Commutative Ring Theory : Springer.
[GM92] J. P. C. Greenlees and J. P. May (1992) “Derived functors of I-adic completion and local homology”, Journal
of Algebra, Vol. 149, No. 2, pp. 438–453, DOI: 10.1016/0021-8693(92)90026-I.
[HM07] T. D. Hamilton and T. Marley (2007) “Non-Noetherian Cohen–Macaulay rings”, Journal of Algebra, Vol. 307,
No. 1, pp. 343–360, DOI: 10.1016/j.jalgebra.2006.08.003.
[KW20] Y. Kim and A. Walker (2020) “A note on Non-Noetherian Cohen–Macaulay rings”, Proc. Amer. Math. Soc.,
Vol. 148, No. 3, pp. 1031–1042, DOI: 10.1090/proc/14836, arXiv:1812.05079.
[Mil08] L. M. Miller (2008) “A Theory of Non-Noetherian Gorenstein Rings”, Ph.D. dissertation, University of
Nebraska at Lincoln.
[Sch03] P. Schenzel (2003) “Proregular sequences, local cohomology, and completion”, Math. Scand., Vol. 92, No. 2,
pp. 161–180, DOI: 10.7146/math.scand.a-14399.

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