Let (R,m) be a complete local ring, and G=grm(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.
From a local ring to its associated graded algebra
De Stefani A.;Rossi M. E.;Varbaro M.
2026-01-01
Abstract
Let (R,m) be a complete local ring, and G=grm(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.
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