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Norihiko Minami

Publications and source records attributed to Norihiko Minami.

7 recordsLinked to original sources
arXiv2022

Generalized Lüroth problems, hierarchized I: SBNR -- stably birationalized unramified sheaves and lower retract rationality

This is the first of a series of papers, where we investigate hierarchies of generalized {L}üroth problems on the hierarchy of rationality, starting with the obvious hierarchy between the rationality and the ruledness. Our primary goal here was to construct very general necessary conditions for a smooth, not necessary proper, scheme of finite type over the perfect base field $k$ to be "retract $(-i)$-rational". We achieve this goal by constructing "stably birationalized Nisnevich subsheaf" $S_{sb}$ inside any Morel's unramified sheaf $S,$ where $S_{sb}$ coincides with $S$ on proper smooth $k$-schemes of finite type. Such a stably birationalized Nisnevich subsheaf $S_{sb}$ sheds a new light on the familiar irrational examples of Artin-Mumford, Saltman, Colliot-Thélène-Ojanguren, Bogomolov, Peyre, Colliot-Thélène-Voisin, and many other retract irrational classifying space $BG$ examples presented as counterexamples to the Noether problem of the complex number base field case for a finite group $G.$ In fact, for all of these examples, the game is not over from our hierarchical perspective! A consequence of our construction of $S_{sb}$ is the stably birational invariance of an arbitrary unramified sheaf $S$ on proper smooth $k$-schemes of finite type. This in particular implies that, for any generalized motivic cohomology theory, its naively defined unramified (resp. stably birationalized) gemeralized motivic cohomology theory is stably birational invariant on smooth proper $k$-schemes of finite type (resp. smooth $k$-schemes of finite type). In the course of constructing $S_{sb},$ we have also shown a general local uniformization theorem of the first kind for arbitrary geometric valuations.

math.AG↗
arXiv2019

From Ohkawa to strong generation via approximable triangulated categories -- a variation on the theme of Amnon Neeman's Nagoya lecture series

This survey stems from Amnon Neeman's lecture series at Ohakawa's memorial workshop. Starting with Ohakawa's theorem, this survey intends to supply enough motivation, background and technical details to read Neeman's recent papers on his "approximable triangulated categories" and his $D_{coh}^b(X)$ strong generation sufficient criterion via de Jong's regular alteration, even for non-experts. At the same time, the author, who happens to be a coorganizer of this workshop and an editor of the follow-up proceedings to be published as a series in Springer Proceedings in Mathematics and Statistics, repeatedly mentioned rich mathematical interaction with other papers in the proceedings, whenever appropriate.

math.AG↗
arXiv2009

On the random variable $\N \ni l \mapsto \gcd(l,n_1) \gcd(l, n_2) ... \gcd(l, n_k) \in \N$

We compute the "moments" and its continuous analogue of the random variable $\N \ni l \mapsto \gcd(l,n_1) \gcd(l, n_2) ... \gcd(l, n_k) \in \N$ by a purely elementary method. This generalizes a result of Deitmar-Koyama-Kurokawa, which computed its "average" using some analysis involving L-function. We show this average is nothing but the invariant $μ(A) := \sum_{a\in A} \frac{1}{| a |}$ for a finite abelian group $A = \prod_{j=1)^k Z/n_j$. In ArXiv-0910.3879v1, this invariant plays an important role in the Soulé type zeta functions for Noetherian $F_1$-schemes in the sense of Connes-Consani.

math.NT↗
arXiv2009

Meromorphicity of some deformed multivariable zeta functions for $F_1$-schemes

Motivated by recent work of Deitmar-Koyama-Kurokawa, Kurokawa-Ochiai, Connes-Consani, and the author, we define some multivariable deformed zeta functions of Hurwitz-Igusa type for a Noetherian $\F_1$-scheme $X$ in the sense of Connes-Consani. Our zeta functions generalize both the zeta functions studied by Deitmar-Koyama-Kurokawa, Kurokawa-Ochiai, and the log derivative of the modified Soulé type zeta function Connes-Consani. We give an explicit presentation for these zeta functions using the Hurwitz zeta functions, and so, we can derive its meromorphicity. When restricted to the log derivative of the modified Soulé type zeta functions, we find our invariant $μ(A)$ for a finite abelian group $A$, introduced in ArXiv-0907.0918v2, plays an extremely important role in the Soulé type zeta functions.

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