Publication Date:
2016-09-17
Description:
Let $(R,\mathfrak {m},k)$ be a commutative noetherian local ring of Krull dimension $d$ . We prove that the cohomology annihilator $\mathsf {ca}\,(R)$ of $R$ contains a power of $\mathfrak {m}$ if and only if, for some $n\ge 0,$ the $n$ th syzygies in $\mathsf {mod}\,R$ 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 $\mathsf {D^b}\,(R)$ and the singularity category $\mathsf {D_{sg}}\,(R)$ 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 $\hat R$ .
Print ISSN:
0033-5606
Electronic ISSN:
1464-3847
Topics:
Mathematics
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