arXiv · 2610.41443
In this paper, we introduce and study the notion of 2-nil hyperideals of multiplicative hyperring.
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Mahdi Anbarloei. 2026-10-30. 2-nil Hyperideals of multiplicative hyperrings. https://arxiv.org/abs/2610.41443
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Matlis proved a lot of homological properties of the fraction field of an integral domain. In this paper, we simplify and extend some of them from 1-dimensional (resp. rank one) cases to the higher dimensional (resp. finite rank) cases. For example, we study the weakly co-torsion property of Ext$(-,\sim)$, and use it to present splitting criteria. These are equipped with several applications. For instance, we compute the projective dimension of $\widehat{R}$ and present some non-noetherian versions of Grothendieck's localization problem. We construct a new class of co-Hopfian modules and extend Matlis' decomposability problem to higher ranks. In particular, this paper deals with the basic properties of Matlis' quadric $(Q,Q/R, \widehat{R},\overset{\sim}R).$
We study the following general situation. Let $R\to S$ be a finite flat morphism of regular rings of zero (resp. prime) mixed or (equi)-characteristic. For $P\in Spec R$, put $A=R/P, C=S/PS,$ and $B=S/\sqrt{PS}=C_{\mathrm{red}}.$ The basic question is whether a property of $B$ forces the same property of $A$. The main point is that ordinary finite-flat descent applies naturally to $A\to C$, whereas the passage $C\to C_{\mathrm{red}}$ may (not) destroy nilpotent information.
Over a field of characteristic zero, we characterize the Hilbert functions that determine the degree reverse lexicographic generic initial ideal among Artinian quotients of embedding dimension $n\ge2$ with the $(n-2)$-strong Lefschetz property. The criterion is a finite arithmetic test on successive positive differences of the Hilbert function. It uses recursively defined integer vectors and requires no construction of a monomial ideal. When uniqueness holds, the same vectors recover the common generic initial ideal degree by degree. The proof reduces uniqueness successively to uniqueness of lower sets of specified cardinalities in Borel orders. In four variables, the test simplifies to avoidance of explicitly computed integer intervals. We recover the quasi-symmetry criterion of Harima and Wachi and give infinite families beyond that criterion in every number of variables at least four.