The Injective Spectrum of a Right Noetherian Ring II: Sheaves and Torsion Theories

Integrality of noetherian Grothendieck categories

We introduce the notion of integrality of Grothendieck categories as a simultaneous generalization of the primeness of noncommutative noetherian rings and the integrality of locally noetherian schemes. Two different spaces associated to a Grothendieck category yield respective definitions of integrality, and we prove the equivalence of these definitions using a Grothendieck-categorical version of Gabriel's correspondence, which originally related indecomposable injective modules and prime two-sided ideals for noetherian rings. The generalization of prime two-sided ideals is also used to classi…

Cotilting sheaves over weighted noncommutative regular projective curves

We consider the category $\operatorname{Qcoh}\mathbb{X}$ of quasicoherent sheaves where $\mathbb{X}$ is a weighted noncommutative regular projective curve over a field $k$. This category is a hereditary, locally noetherian Grothendieck category. We classify all indecomposable pure-injective sheaves and all cotilting sheaves of slope $\infty$. In the cases of nonnegative orbifold Euler characteristic this leads to a classification of pure-injective indecomposable sheaves and a description of all large cotilting sheaves in $\operatorname{Qcoh}\mathbb{X}$.

Injective Semimodules - Revisited

Injective modules play an important role in characterizing different classes of rings (e.g. Noetherian rings, semisimple rings). Some semirings have no non-zero injective semimodules (e.g. the semiring of non-negative integers). In this paper, we study some of the basic properties of the so called e-injective semimodules introduced by the first author using a new notion of exact sequences of semimodules. We clarify the relationships between the injective semimodules, the e-injective semimodule, and the i-injective semimodules through several implications, examples and counter examples. Moreove…

反合凝聚层的反模在局部Noetherian形式方案上

Contraherent cosheaves of contramodules on Noetherian formal schemes

本文定义了局部Noetherian形式方案上反合凝聚层反模的精确范畴，以及局部反合凝聚层反模（相对于给定的开覆盖）的精确范畴，并构造了局部反合凝聚层反模在局部Noetherian形式方案映射下的直接像和逆像函子。

Flabby and injective objects in toposes

We introduce a general notion of flabby objects in elementary toposes and study their basic properties. In the special case of localic toposes, this notion reduces to the common notion of flabby sheaves, yielding a site-independent characterization of flabby sheaves. Continuing a line of research started by Roswitha Harting, we use flabby objects to show that an internal notion of injective objects coincides with the corresponding external notion, in stark contrast with the situation for projective objects. We show as an application that higher direct images can be understood as internal cohom…

Classification of categorical subspaces of locally noetherian schemes

We classify the prelocalizing subcategories of the category of quasi-coherent sheaves on a locally noetherian scheme. In order to give the classification, we introduce the notion of a local filter of subobjects of the structure sheaf. The essential part of the argument is given as results on a Grothendieck category with certain properties. We also classify the localizing subcategories, the closed subcategories, and the bilocalizing subcategories in terms of filters.

On stable-projective and injective-costable decompositions of modules

It is proved that, for a left hereditary ring, an arbitrary left module has a representation in the form of the direct sum of a stable left module and indecomposable projective left modules (if and only if an arbitrary left module has a representation in the form of the direct sum of a stable left module and a projective left module) if and only if the ring is left perfect and right coherent. In that case, the above-mentioned representations are unique up to isomorphism; the latter representation is also functorial. The essential ingredient in the proofs of the above-mentioned statements is a …

非交换支集、局部上同调与谱序列

Noncommutative supports, local cohomology and spectral sequences

本文在 Artin-Zhang 非交换代数几何框架中，利用初等对象及其内射包建立支集与相伴素理想理论，并由此导出局部上同调相关导出函子的谱序列及广义 Nagata 理想变换。

突变与Gabriel谱

Mutation and the Gabriel spectrum

本文研究了在紧生成三角范畴中纯投射余丝紧对象的突变，证明右突变在特定拓扑下诱导同胚，并在交换Noetherian环的导出范畴中证明其为开双射，同时揭示了模范畴的唯一性。

The stable module category of a general ring

For any ring R we construct two triangulated categories, each admitting a functor from R-modules that sends projective and injective modules to 0. When R is a quasi-Frobenius or Gorenstein ring, these triangulated categories agree with each other and with the usual stable module category. Our stable module categories are homotopy categories of Quillen model structures on the category of R-modules. These model categories involve generalizations of Gorenstein projective and injective modules that we derive by replacing finitely presented modules by modules of type FP-infinity. Along the way, we …

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.

拟凝聚挠层、半导出范畴与半张量积：ind-概形上拟凝聚层的半无限代数几何

Quasi-coherent torsion sheaves, the semiderived category, and the semitensor product: Semi-infinite algebraic geometry of quasi-coherent sheaves on ind-schemes

本文在带平坦仿射态射到具有对偶化复形的ind-诺特ind-概形的ind-概形上，构造了拟凝聚挠层半导出范畴的半无限张量结构，其中半张量积混合了基底余张量积与纤维导出张量积，对偶化复形的逆像为单位对象，部分实现了半无限代数几何纲领。