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Annihilation of cohomology, generation of modules and finiteness of derived dimension
Commutative Algebra · Submitted 23 Apr 2015
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove that the cohomology annihilator $\ca(R)$ of $R$ is $\m$-primary if and only if for some $n\ge0$ the $n$-th syzygies in $\mod R$ are constructed from syzygies of $k$ by taking direct sums/summands and a fixed number of extensions.
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The abstract, sentence by sentence
Let $(R,\m,k)$ be a commutative noetherian local ring of Krull dimension $d$. We prove that the cohomology annihilator $\ca(R)$ of $R$ is $\m$-primary if and only if for some $n\ge0$ the $n$-th syzygies in $\mod R$ are constructed from syzygies of $k$ by taking direct sums/summands and a fixed number of extensions. These conditions yield that $R$ is an isolated singularity such that the bounded derived category $\db(R)$ and the singularity category $\ds(R)$ have finite dimension, and the converse holds when $R$ is Gorenstein. We also show that the modules locally free on the punctured spectrum are constructed from syzygies of finite length modules by taking direct sums/summands and $d$ extensions. This result is exploited to investigate several ascent and descent problems between $R$ and its completion $\widehat R$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $(R,\m,k)$ be a commutative noetherian local ring of Krull dimension $d$. | background 1.00 | 0.05 | 0.01 | asserted 0.83 |
| s1 | We prove that the cohomology annihilator $\ca(R)$ of $R$ is $\m$-primary if and only if for some $n\ge0$ the $n$-th syzygies in $\mod R$ are constructed from syzygies of $k$ by taking direct sums/summands and a fixed number of extensions. | result 0.98 | 0.95 | 0.62 | asserted 1.00 |
| s2 | These conditions yield that $R$ is an isolated singularity such that the bounded derived category $\db(R)$ and the singularity category $\ds(R)$ have finite dimension, and the converse holds when $R$ is Gorenstein. | result 0.55 | 0.85 | 0.47 | asserted 1.00 |
| s3 | We also show that the modules locally free on the punctured spectrum are constructed from syzygies of finite length modules by taking direct sums/summands and $d$ extensions. | result 0.98 | 0.93 | 0.11 | asserted 1.00 |
| s4 | This result is exploited to investigate several ascent and descent problems between $R$ and its completion $\widehat R$. | contribution 0.85 | 0.91 | 0.05 | asserted 0.99 |
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Original abstract
Let $(R,\m,k)$ be a commutative noetherian local ring of Krull dimension $d$. We prove that the cohomology annihilator $\ca(R)$ of $R$ is $\m$-primary if and only if for some $n\ge0$ the $n$-th syzygies in $\mod R$ are constructed from syzygies of $k$ by taking direct sums/summands and a fixed number of extensions. These conditions yield that $R$ is an isolated singularity such that the bounded derived category $\db(R)$ and the singularity category $\ds(R)$ have finite dimension, and the converse holds when $R$ is Gorenstein. We also show that the modules locally free on the punctured spectrum are constructed from syzygies of finite length modules by taking direct sums/summands and $d$ extensions. This result is exploited to investigate several ascent and descent problems between $R$ and its completion $\widehat R$.
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In recent years there has been interest among commutative algebraists to understand this better: the reader is referred to Aihara and Takahashi [ref] , Iyengar and Takahashi [ref] , and Bahlekeh, Hakimian, Salarian and Takahashi [ref] for a sample of the literature.
Strong generators in $D^{perf}(X)$ and $D^b_{coh}(X)$ · Introduction · 0.98
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