Furkejuvvon:
| Váldodahkki: | |
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| Materiálatiipa: | Preprint |
| Almmustuhtton: |
2025
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| Fáttát: | |
| Liŋkkat: | https://arxiv.org/abs/2505.00521 |
| Fáddágilkorat: |
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| _version_ | 1866915804821323776 |
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| author | Lyu, Shiji |
| author_facet | Lyu, Shiji |
| contents | This is a two-part article. In the first part, we study an alternative notion to Nagata rings. A Nagata ring is a Noetherian ring $R$ such that every finite $R$-algebra that is an integral domain has finite normalization. We replace the normalization by an $(S_2)$-ification, study new phenomena, and prove parallel results. In particular, we show a Nagata domain has a finite $(S_2)$-ification. In the second part, we study the local lifting problem. We show that for a semilocal Noetherian ring $R$ that is $I$-adically complete for an ideal $I$, if $R/I$ has $(S_k)$ (resp. Cohen--Macaulay, Gorenstein, lci) formal fibers, so does $R$. As a consequence, we show if $R/I$ is a quotient of a Cohen--Macaulay ring, so is $R$. We also discuss difficulties in lifting geometrically $(R_k)$ formal fibers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00521 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $(S_2)$-ifications, semi-Nagata rings, and the lifting problem Lyu, Shiji Commutative Algebra This is a two-part article. In the first part, we study an alternative notion to Nagata rings. A Nagata ring is a Noetherian ring $R$ such that every finite $R$-algebra that is an integral domain has finite normalization. We replace the normalization by an $(S_2)$-ification, study new phenomena, and prove parallel results. In particular, we show a Nagata domain has a finite $(S_2)$-ification. In the second part, we study the local lifting problem. We show that for a semilocal Noetherian ring $R$ that is $I$-adically complete for an ideal $I$, if $R/I$ has $(S_k)$ (resp. Cohen--Macaulay, Gorenstein, lci) formal fibers, so does $R$. As a consequence, we show if $R/I$ is a quotient of a Cohen--Macaulay ring, so is $R$. We also discuss difficulties in lifting geometrically $(R_k)$ formal fibers. |
| title | $(S_2)$-ifications, semi-Nagata rings, and the lifting problem |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2505.00521 |