On the ubiquity of uniformly dominant local rings
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| Format: | Preprint |
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2026
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| _version_ | 1866910049340751872 |
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| author | Kobayashi, Toshinori Takahashi, Ryo |
| author_facet | Kobayashi, Toshinori Takahashi, Ryo |
| contents | Let R be a d-dimensional Cohen-Macaulay complete local ring with infinite residue field k. The dominant index $\operatorname{dx}(R)$ is by definition the least number of extensions necessary to build k in the singularity category $\operatorname{D^{sg}}$ out of each nonzero object, up to finite direct sums, direct summands and shifts. The local ring R is called uniformly dominant if $\operatorname{dx}(R)$ is finite. In this paper, we prove that R is uniformly dominant with $\operatorname{dx}(R)\le6d+5$ if R has codimension 2 and is not a complete intersection. Also, we show that R is uniformly dominant with $\operatorname{dx}(R)\le d+1$ if R is Burch, and with $\operatorname{dx}(R)\le d$ if R is either a quasi-fiber product ring, or has multiplicity at most 5 and is not Gorenstein. A result on hypersurfaces by Ballard, Favero and Katzarkov is recovered, and results on Burch rings and quasi-fiber product rings by Takahashi are refined. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_10810 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the ubiquity of uniformly dominant local rings Kobayashi, Toshinori Takahashi, Ryo Commutative Algebra Representation Theory 13D09, 13C60, 13H10 Let R be a d-dimensional Cohen-Macaulay complete local ring with infinite residue field k. The dominant index $\operatorname{dx}(R)$ is by definition the least number of extensions necessary to build k in the singularity category $\operatorname{D^{sg}}$ out of each nonzero object, up to finite direct sums, direct summands and shifts. The local ring R is called uniformly dominant if $\operatorname{dx}(R)$ is finite. In this paper, we prove that R is uniformly dominant with $\operatorname{dx}(R)\le6d+5$ if R has codimension 2 and is not a complete intersection. Also, we show that R is uniformly dominant with $\operatorname{dx}(R)\le d+1$ if R is Burch, and with $\operatorname{dx}(R)\le d$ if R is either a quasi-fiber product ring, or has multiplicity at most 5 and is not Gorenstein. A result on hypersurfaces by Ballard, Favero and Katzarkov is recovered, and results on Burch rings and quasi-fiber product rings by Takahashi are refined. |
| title | On the ubiquity of uniformly dominant local rings |
| topic | Commutative Algebra Representation Theory 13D09, 13C60, 13H10 |
| url | https://arxiv.org/abs/2603.10810 |