let r be a right gf-closed ring with finite left and right gorenstein global dimension. we prove that if i is an ideal of r such that r/i is a semi-simple ring, then the gorensntein flat dimensnion of r/i as a right r-module and the gorensntein injective dimensnnion of r/i as a left r-module are identical. in particular, we show that for a simple module s over a commutative gorensntein ring r, ...

In a recent paper [17] Miro-Roig, Mezzetti and Ottaviani highlight the link between rational varieties satisfying a Laplace equation and artinian ideals failing the Weak Lefschetz Property. Continuing their work we extend this link to the more general situation of artinian ideals failing the Strong Lefschetz Property. We characterize the failure of the SLP (which includes WLP) by the existence ...

We obtain a complete structural characterization of Cohn-Leavitt algebras over no-exit objects as graded involutive algebras. Corollaries of this result include graph-theoretic conditions characterizing when a Leavitt path algebra is a directed union of (graded) matricial algebras over the underlying field and over the algebra of Laurent polynomials and when the monoid of isomorphism classes of...

In [17] the authors highlight the link between rational varieties satisfying a Laplace equation and artinian ideals that fail the Weak Lefschetz property. Continuing their work we extend this link to the more general situation of artinian ideals failing the Strong Lefschetz Property. We characterize the failure of SLP (that includes WLP) by the existence of special singular hypersurfaces (cones...

Since the 1970’s, great interest has been taken in the study of pure O-sequences, which, due to Macaulay’s theory of inverse systems, have a bijective correspondence to the Hilbert functions of Artinian level monomial algebras. Much progress has been made in classifying these according to their shape. Macaulay’s theorem immediately gives us that all Artinian algebras in two variables have unimo...

We'll review the study of Galois deformations. Here's the setup. Let G be a pro nite group, and let ρ : G→ GLn(k) be a representation of G over a nite eld k of characteristic p. Let Λ be a complete DVR with residue eld k, and let CΛ denote the category whose objects are artinian local Λ-algebras with residue eld k, and whose morphisms are local homomorphisms. Let ĈΛ denote the category of compl...

An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of Artinian Gorenstein algebras with the Weak Lefschetz Property, a property shared by most Artinian Gorenstein algebras. Starting with an arbitrary SI-sequence, we construct a reduced, arithmeti...

We obtain some methods to construct a (strongly) proper resolution (resp. coproper coresolution) of one end term in a short exact sequence from that of the other two terms. By using this method, we prove that for a left and right Noetherian ring R, RR satisfies the Auslander condition if and only if so does every flat left R-module, if and only if the injective dimension of the ith term in a mi...

The paper uses a new approach to investigate prime submodules and minimal prime submodules of certain modules such as Artinian and torsion modules. In particular, we introduce a concrete formula for the radical of submodules of Artinian modules.

It is unknown if an Artinian level O-sequence of codimension 3 and type r (≥ 2) is unimodal, while it is known that any Gorenstein O-sequence of codimension 3 is unimodal. We show that some Artinian non-unimodal O-sequence of codimension 3 cannot be level. We also find another non-level case: if some Artinian algebra A of codimension 3 has the Hilbert function H : h0 h1 · · · hd−1 hd · · · hd }...