arXiv subjects

Tsutomu Nakamura

Publications and source records attributed to Tsutomu Nakamura.

9 recordsLinked to original sources

arXiv2024

Tilting complexes and codimension functions over commutative noetherian rings

In the derived category of a commutative noetherian ring, we explicitly construct a silting object associated with each sp-filtration of the Zariski spectrum satisfying the "slice" condition. Our new construction is based on local cohomology and it allows us to study when the silting object is tilting. For a ring admitting a dualizing complex, this occurs precisely when the sp-filtration arises from a codimension function on the spectrum. In the absence of a dualizing complex, the situation is more delicate and the tilting property is closely related to the condition that the ring is a homomorphic image of a Cohen-Macaulay ring. We also provide dual versions of our results in the cosilting case.

math.AC↗

arXiv2022

Indecomposable pure-injective objects in stable categories of Gorenstein-projective modules over Gorenstein orders

We give a result of Auslander-Ringel-Tachikawa type for Gorenstein-projective modules over a complete Gorenstein order. In particular, we prove that a complete Gorenstein order is of finite Cohen-Macaulay representation type if and only if every indecomposable pure-injective object in the stable category of Gorenstein-projective modules is compact.

math.AC↗

arXiv2021

Flat cotorsion modules over Noether algebras

For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs' result for a commutative noetherian ring. As a consequence, we show that pointwise Matlis duality gives a bijective correspondence between the isoclasses of indecomposable injective left modules and the isoclasses of indecomposable flat cotorsion right modules. This correspondence is an explicit realization of Herzog's homeomorphism induced from elementary duality of Ziegler spectra.

math.RT↗

arXiv2020

Telescope conjecture for homotopically smashing t-structures over commutative noetherian rings

We show that any homotopically smashing t-structure in the derived category of a commutative noetherian ring is compactly generated. This generalizes the validity of the telescope conjecture for commutative noetherian rings due to Neeman. As another consequence, we obtain a cofinite type result for pure-injective cosilting objects. We also give a formulation of telescope conjecture for homotopically smashing t-structures in underlying triangulated categories of certain Grothendieck derivators.

math.AC↗

arXiv2020

Minimal semi-flat-cotorsion replacements and cosupport

Over a commutative noetherian ring $R$ of finite Krull dimension, we show that every complex of flat cotorsion $R$-modules decomposes as a direct sum of a minimal complex and a contractible complex. Moreover, we define the notion of a semi-flat-cotorsion complex as a special type of semi-flat complex, and provide functorial ways to construct a quasi-isomorphism from a semi-flat complex to a semi-flat-cotorsion complex. Consequently, every $R$-complex can be replaced by a minimal semi-flat-cotorsion complex in the derived category over $R$. Furthermore, we describe structure of semi-flat-cotorsion replacements, by which we recover classic theorems for finitistic dimensions. In addition, we improve some results on cosupport and give a cautionary example. We also explain that semi-flat-cotorsion replacements always exist and can be used to describe the derived category over any associative ring.

math.AC↗

arXiv2019

Cosupports and minimal pure-injective resolutions of affine rings

We prove that any affine ring $R$ over a field $k$ has full cosupport, i.e., the cosupport of $R$ is equal to ${\rm Spec}\, R$. Using this fact, we give a complete description of all terms in a minimal pure-injective resolution of $R$, provided that $|k|=\aleph_1$ and ${\rm dim}\, R\geq 2$, or $|k|\geq \aleph_1$ and ${\rm dim}\, R=2$. As a corollary, we obtain a partial answer to a conjecture by Gruson.

math.AC↗

arXiv2019

Little dimension and the improved new intersection theorem

Let $R$ be a commutative noetherian local ring. We define a new invariant for $R$-modules which we call the little dimension. Using it, we extend the improved new intersection theorem.

math.AC↗

arXiv2018

Localization functors and cosupport in derived categories of commutative Noetherian rings

Let $R$ be a commutative Noetherian ring. We introduce the notion of localization functors $λ^W$ with cosupports in arbitrary subsets $W$ of $\text{Spec}\, R$; it is a common generalization of localizations with respect to multiplicatively closed subsets and left derived functors of ideal-adic completion functors. We prove several results about the localization functors $λ^W$, including an explicit way to calculate $λ^W$ by the notion of Cech complexes. As an application, we can give a simpler proof of a classical theorem by Gruson and Raynaud, which states that the projective dimension of a flat $R$-module is at most the Krull dimension of $R$. As another application, it is possible to give a functorial way to replace complexes of flat $R$-modules or complexes of finitely generated $R$-modules by complexes of pure-injective $R$-modules.

math.AC↗

arXiv2017

A local duality principle in derived categories of commutative Noetherian rings

Let R be a commutative Noetherian ring. We introduce the notion of colocalization functors with supports in arbitrary subsets of Spec R, which is a natural generalization of right derived functors of section functors with supports in specialization-closed subsets. We prove that the local duality theorem and the vanishing theorem of Grothendieck type hold for colocalization functors.

math.AC↗