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arXiv · 2406.16421

From a local ring to its associated graded algebra

Alessandro De Stefani·Maria Evelina Rossi·Matteo Varbaro
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Abstract

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 "Gr\"obner-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.

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Alessandro De Stefani, Maria Evelina Rossi, Matteo Varbaro. 2024-06-24. From a local ring to its associated graded algebra. https://arxiv.org/abs/2406.16421

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