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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.16421 |
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| _version_ | 1866915892887027712 |
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| author | De Stefani, Alessandro Rossi, Maria Evelina Varbaro, Matteo |
| author_facet | De Stefani, Alessandro Rossi, Maria Evelina Varbaro, Matteo |
| contents | Let $(R,\mathfrak{m})$ be a complete local ring, and $G={\rm gr}_{\mathfrak{m}}(R)$ be its associated graded ring. We introduce a homogenization technique which allows to relate $G$ to the special fiber and $R$ to the generic fiber of a "Gröbner-like" deformation. Using this technique we prove sharp results concerning the connectedness of $R$ and $G$. We also construct a family of local domains which fail to satisfy Abhyankar's inequality for the Hilbert-Samuel multiplicity. However, we prove a version of the inequality which holds when $R$ is connected in codimension one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16421 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | From a local ring to its associated graded algebra De Stefani, Alessandro Rossi, Maria Evelina Varbaro, Matteo Commutative Algebra Let $(R,\mathfrak{m})$ be a complete local ring, and $G={\rm gr}_{\mathfrak{m}}(R)$ be its associated graded ring. We introduce a homogenization technique which allows to relate $G$ to the special fiber and $R$ to the generic fiber of a "Gröbner-like" deformation. Using this technique we prove sharp results concerning the connectedness of $R$ and $G$. We also construct a family of local domains which fail to satisfy Abhyankar's inequality for the Hilbert-Samuel multiplicity. However, we prove a version of the inequality which holds when $R$ is connected in codimension one. |
| title | From a local ring to its associated graded algebra |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2406.16421 |