In this paper we study the Weak Lefschetz property of two classes of standard graded Artinian Gorenstein algebras associated in a natural way to the Apéry set of numerical semigroups. To this aim we also prove a general result about the transfer of the Weak Lefschetz property from an Artinian Gorenstein algebra to its quotients modulo a colon ideal.

Homology theory relative to classes of objects other than those of projective or injective objects in abelian categories has been widely studied in the last years, giving a special relevance to Gorenstein homological algebra. We prove the existence of Gorenstein flat precovers in any locally finitely presented Grothendieck category in which the class of flat objects is closed under extensions, the existence of Gorenstein injective preenvelopes in any locally noetherian Grothendieck category in which the class of all Gorenstein injective objects is closed under direct products, and the existence of special Gorenstein injective preenvelopes in locally noetherian Grothendieck categories with a generator lying in the left orthogonal class to that of Gorenstein injective objects.

In this paper we study generalized Gorenstein Arf rings: a class of one-dimensional Cohen–Macaulay local Arf rings that is strictly contained in the class of Gorenstein rings. We obtain new characterizations and examples of Arf rings, and give applications of our argument to numerical semigroup rings and certain idealizations. In particular, we generalize a beautiful result of Barucci and Fröberg concerning Arf numerical semigroup rings.

In this paper we prove new embedding results for compactly supported deformations of CR submanifolds of Cn+d: We show that if M is a 2-pseudoconcave CR submanifold of type (n,d) in Cn+d, then any compactly supported CR deformation stays in the space of globally CR embeddable in Cn+d manifolds. This improves an earlier result, where M was assumed to be a quadratic 2-pseudoconcave CR submanifold of Cn+d. We also give examples of weakly 2-pseudoconcave CR manifolds admitting compactly supported CR deformations that are not even locally CR embeddable.

We prove a structure theorem for any n-rectifiable set E⊂Rn+1,n≥1, satisfying a weak version of the lower ADR condition, and having locally finite Hn (n-dimensional Hausdorff) measure. Namely, that Hn-almost all of E can be covered by a countable union of boundaries of bounded Lipschitz domains contained in Rn+1∖E. As a consequence, for harmonic measure in the complement of such a set E, we establish a non-degeneracy condition which amounts to saying that Hn|E is “absolutely continuous” with respect to harmonic measure in the sense that any Borel subset of E with strictly positive Hn measure has strictly positive harmonic measure in some connected component of Rn+1∖E. We also provide some counterexamples showing that our result for harmonic measure is optimal. Moreover, we show that if, in addition, a set E as above is the boundary of a connected domain Ω⊂Rn+1 which satisfies an infinitesimal interior thickness condition, then Hn|∂Ω is absolutely continuous (in the usual sense) with respect to harmonic measure for Ω. Local versions of these results are also proved: if just some piece of the boundary is n-rectifiable then we get the corresponding absolute continuity on that piece. As a consequence of this and recent results in “Rectifiability of harmonic measure” [Geom. Funct. Anal. 26 (2016), 703–728], we can decompose the boundary of any open connected set satisfying the previous conditions in two disjoint pieces: one that is n-rectifiable where Hausdorff measure is absolutely continuous with respect to harmonic measure and another purely n-unrectifiable piece having vanishing harmonic measure.