Annihilation of Cohomology, Generation of Modules and Finiteness of Derived Dimension

Bahlekeh, Abdolnaser; Hakimian, Ehsan; Salarian, Shokrollah; Takahashi, Ryo

doi:10.1093/qmath/haw015

Cited by 7 publications

(7 citation statements)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this subsection, we show that we can drop the completeness assumption. See also [BHST16]. Our result follows from their Theorem 4.5 but we give a different proof.…”

Section: Completion Of a One Dimensional Gorenstein Ringmentioning

confidence: 72%

See 1 more Smart Citation

The Cohomology Annihilator of a Curve Singularity

Esentepe

2018

Preprint

View full text Add to dashboard Cite

The aim of this paper is to study the theory of cohomology annihilators over commutative Gorenstein rings. We adopt a triangulated category point of view and study the annihilation of stable category of maximal Cohen-Macaulay modules. We prove that in dimension one the cohomology annihilator ideal and the conductor ideal coincide under mild assumptions. We present a condition on a ring homomorphism between Gorenstein rings which allows us to carry the cohomology annihilator of the domain to that of the codomain. As an application, we generalize the Milnor-Jung formula for algebraic curves to their double branched covers. We also show that the cohomology annihilator of a Gorenstein local ring is contained in the cohomology annihilator of its Henselization and in dimension one the cohomology annihilator of its completion. Finally, we investigate a relation between the cohomology annihilator of a Gorenstein ring and stable annihilators of its noncommutative resolutions.

show abstract

“…In this subsection, we show that we can drop the completeness assumption. See also [BHST16]. Our result follows from their Theorem 4.5 but we give a different proof.…”

Section: Completion Of a One Dimensional Gorenstein Ringmentioning

confidence: 72%

“…For any local ring R, its Henselization R h is the smallest Henselian ring extending R. In this subsection, we show that the cohomology annihilator of a Gorenstein local ring is contained in the cohomology annihilator of its Henselization. See also [BHST16].…”

Section: Henselizationmentioning

confidence: 99%

The Cohomology Annihilator of a Curve Singularity

Esentepe

2018

Preprint

View full text Add to dashboard Cite

show abstract

“…When R is a Gorenstein local ring and dim D sg (R) < ∞, (4) was proved by Bahlekeh, Hakimian, Salarian, and Takahashi [1,Theorem 3.3].…”

Section: 1mentioning

confidence: 99%

Annihilators and dimensions of the singularity category

Liu¹

2022

Preprint

View full text Add to dashboard Cite

Let R be a commutative noetherian ring. We prove that if R is either an equidimensional finitely generated algebra over a perfect field, or an equidimensional equicharacteristic complete local ring with a perfect residue field, then the annihilator of the singularity category of R coincides with the Jacobian ideal of R up to radical. We establish a relation between the annihilator of the singularity category of R and the cohomological annihilator of R under some mild assumptions. Finally, we give an upper bound for the dimension of the singularity category of an equicharacteristic excellent local ring with isolated singularity. This extends a result of Dao and Takahashi to non Cohen-Macaulay rings.

show abstract

“…The reader might also wish to look at Lunts [22,Theorem 6.3] for a different approach to the proof. If we specialize the result of Rouquier, extended by Keller and Van den Bergh, to the case where X = Spec(R) is an affine scheme, we learn that D b (R-mod) is regular whenever R is of finite type over a field k. In recent years there has been interest among commutative algebraists to understand this better: the reader is referred to Aihara and Takahashi [1], Iyengar and Takahashi [14], and Bahlekeh, Hakimian, Salarian and Takahashi [2] for a sample of the literature. There is also a connection with the concept of the radius of the (abelian) category of modules over R; see Dao and Takahashi [8,9] and Iyengar and Takahashi [14].…”

Section: And Corol-mentioning

confidence: 99%

Strong generators in $D^{perf}(X)$ and $D^b_{coh}(X)$

Neeman¹

2017

Preprint

View full text Add to dashboard Cite

We solve two open problems: first we prove a conjecture of Bondal and Van den Bergh, showing that the category D perf (X) is strongly generated whenever X is a quasicompact, separated scheme, admitting a cover by open affine subsets Spec(Ri) with each Ri of finite global dimension. We also prove that, for a noetherian scheme X of finite type over an excellent scheme of dimension ≤ 2, the derived category D b coh (X) is strongly generated. The known results in this direction all assumed equal characteristic, we have no such restriction.The method is interesting in other contexts: our key lemmas turn out to give a simple proof that, if f : X −→ Y is a separated morphism of quasicompact, quasiseparated schemes such that Rf * : Dqc(X) −→ Dqc(Y ) takes perfect complexes to complexes of bounded-below Tor-amplitude, then f must be of finite Tor-dimension.

show abstract

Annihilation of Cohomology, Generation of Modules and Finiteness of Derived Dimension

Cited by 7 publications

References 48 publications

The Cohomology Annihilator of a Curve Singularity

The Cohomology Annihilator of a Curve Singularity

Annihilators and dimensions of the singularity category

Strong generators in $D^{perf}(X)$ and $D^b_{coh}(X)$

Contact Info

Product

Resources

About