arXiv subjects

So Nakamura

Publications and source records attributed to So Nakamura.

5 recordsLinked to original sources

arXiv2026

$N$-Koszul algebras of finite global dimension for $N\geq 3$

Let $N\geq 3$. The class of $N$-Koszul AS regular algebras, or more generally, that of $N$-Koszul AS Gorenstein algebras, has attracted much attention from algebraists. Nevertheless, there have been no known examples of $N$-Koszul AS regular algebras of finite global dimension other than the ones of global dimension $3$. A recent work by Kabbaj showed that, such an $N$-Koszul algebra $A$ of finite global dimension has to have a large global dimension and that $N$ has to be prime, under the assumptions that $(1)$ $A$ has a Hilbert series of weighted polynomial rings and that $(2)$ the trivial $A$-module $\mathbb{K}$ has a finite free resolution. All AS regular algebras satisfy the latter assumption and are expected to do the former as well. In this paper, we prove that such an $N$-Koszul algebra $A$ must be one of the known $3$-Koszul AS regular algebras of global dimension $3$ if the order of the pole of its Hilbert series $h_A(t)$ at $t=1$ is greater than $\frac{21d+1}{22}$, where $d$ is the global dimension of $A$. As a corollary, we prove that any $N$-Koszul AS regular algebra $A$ must be one of the known $3$-Koszul AS regular algebras of global dimension $3$ if $A$ has a Hilbert series of weighted polynomial rings and if the GK dimension of $A$ coincides with the global dimension of $A$, both of which have been conjectured to hold for any AS regular algebras.

math.RA↗

arXiv2026

Quotients of mosaics and related hyperstructures

This is a thorough study of quotients of hyperstructures that generalize hypergroups, namely mosaics and semimosaics. The quotients in these categories generalize those studied previously in the literature on hypergroups. We describe the effective congruences in these categories by characterizing them in terms of their underlying equivalence relation. This characterization is applied to provide new methods of constructing quotient objects modulo the action of endomorphisms, as well as to study explicit quotient mosaics of some small groups. We also show that the category of mosaics has a natural proto-exact structure.

math.CT↗

arXiv2025

A ringed-space-like structure on coalgebras for noncommutative algebraic geometry

Inspired by the perspective of Reyes' noncomutative spectral theory, we attempt to develop noncommutative algebraic geometry by introducing ringed coalgebras, which can be thought of as a noncommutative generalization of schemes over a field $k$. These objects arise from fully residually finite-dimensional(RFD) algebras introduced by Reyes and from schemes locally of finite type over $k$. The construction uses the Heyneman-Sweedler finite dual coalgebra and the Takeuchi underlying coalgebra. When $k$ is algebraically closed, the formation of ringed coalgebras gives a fully faithful functor out of the category of fully RFD algebras, as well as a fully faithful functor out of the category of schemes locally of finite type. The restrictions of these two functors to the category of (commutative) finitely generated algebras are isomorphic. Finally, we introduce modules over ringed coalgebras and show that the category of finitely generated modules on a fully RFD algebra and that of coherent sheaves on a scheme locally of finite type can, if $k$ is algebraically closed, be fully faithfully embedded into the category of modules over the corresponding ringed coalgebras.

math.RA↗

arXiv2024

Projective Geometries and Simple Pointed Matroids as $\mathbb{F}_1$-modules

We describe a fully faithful embedding of projective geometries, given in terms of closure operators, into $\mathbb{F}_1$-modules, in the sense of Connes and Consani. This factors through a faithful functor out of simple pointed matroids. This follows from our construction of a fully faithful embedding of weakly unital, commutative hypermagmas into $\fun$-modules. This embedding is of independent interest as it generalizes the classical Eilenberg-MacLane embedding for commutative monoids and recovers Segal's nerve construction for commutative partial monoids. For this reason, we spend some time elaborating its structure.

math.CT↗

arXiv2023

Categories of hypermagmas, hypergroups, and related hyperstructures

In order to diagnose the cause of some defects in the category of canonical hypergroups, we investigate several categories of hyperstructures that generalize hypergroups. By allowing hyperoperations with possibly empty products, one obtains categories with desirable features such as completeness and cocompleteness, free functors, regularity, and closed monoidal structures. We show by counterexamples that such constructions cannot be carried out within the category of canonical hypergroups. This suggests that (commutative) unital, reversible hypermagmas -- which we call mosaics -- form a worthwhile generalization of (canonical) hypergroups from the categorical perspective. Notably, mosaics contain pointed simple matroids as a subcategory, and projective geometries as a full subcategory.

math.CT↗