On the associated graded ring of a local Cohen-Macaulay ring

Equations defining rational singularities

“…But a recent result of J. Sally [18] implies that the tangent cone of R is also CohenMacaulay (because emb dim = mult + dim-1), hence the same analysis applies.…”

On Buchsbaum rings obtained by gluing

“…Let A be a Buchsbaum local ring with maximal ideal m. Then In case the field A\m is infinite, the converse is also true. The proof of these facts is essentially the same as that of J. Sally [7], Theorem 1. [10], Theorem 1.3 for the detail.)…”

Sally modules and associated graded rings

“…., e d are the normalized coefficients of the Hilbert-Samuel polynomial of I. Pioneering work on the interplay described in (c) was made by Judith Sally in a sequence of papers [22,23,24,25,26,27,28]. A major recognition of her important contribution came with the introduction of the Sally module (see [31,32]).…”

“…A recurring theme in the work of J. Sally is the discovery of conditions on the multiplicity e of the local ring (R, m) that assure that gr m (R) is Cohen-Macaulay. By [1], the multiplicity e of R satisfies the inequality e ≥ µ(m) − d + 1, where µ(m) denotes the minimal number of generators of m. More precisely, the closed formula is: e = µ(m) − d + 1 + λ(m 2 /Jm), where J is a minimal reduction of m. If λ(m 2 /Jm) = 0, i.e., R has minimal multiplicity, J. Sally proved in [22] that gr m (R) is always Cohen-Macaulay. After this case was settled it was natural to investigate the case in which λ(m 2 /Jm) = 1, i.e., e = µ(m) − d + 2.…”