From a local ring to its associated graded algebra
Abstract
Abstract Let $$(R,\mathfrak {m})$$ ( R , m ) be a complete local ring, and $$G=\textrm{gr}_{\mathfrak {m}}(R)$$ G = gr 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.
Fields of Study & Subjects
Advanced Topics in Algebra
Rings
Modules
and Algebras
Algebraic structures and combinatorial models
Publication details
- Journal / source
- Mathematische Annalen
- Publisher
- Springer Science and Business Media LLC
- Volume
- 395
- Issue
- 3
- Publication date
- 7 May 2026
- Language
- EN