Inverse Systems of Local Rings

The weak Lefschetz property for Artinian Gorenstein algebras of codimension three

“…We give now some definitions and results taken from a paper by Maeno and Watanabe [16] and from a recent paper by Gondim and Zappalá [9]. The general facts on the Macaulay's inverse system can be seen in [8]. Now we regard R as an R-module via the operation "•" defined by…”

How to determine a curve singularity

“…In this section, we establish a Macaulay-like duality for the family of sub-k-algebras B of = k[[t]] of finite codimension. For the classical Macaulay's duality, see [20], [14], and for the generalization to higher dimension of Macaulay's duality, see [15]. Recall that Macaulay's duality is a particular case of Matlis' duality (see [4]).…”

“…As in the Artin case, we can relate the canonical module with the inverse system. In that case, we have that if I is an Artinian ideal, then I ⊥ ≅ E R/I (k) ≅ ω R/I (see [4,14]). In the case of branches, we can determine the "negative" part of the canonical module.…”

“…For this purpose, we establish a Macaulay-like duality between finite codimension sub-k-algebras B of and finite codimension k-vector subspaces B ⊥ , so-called algebra-forming vector spaces, of the polynomial ring Δ. The polynomial ring Δ acts on by differentiation as in Macaulay's duality (see [14][15][16]20]). At the end 2 J .…”