Abhishek Banerjee
Let $\mathcal A$ be a locally noetherian Grothendieck category. In this paper, we study subcategories of $\mathcal A$ using subsets of the Rosenberg spectrum $\mathfrak Spec(\mathcal A)$. Along the way, we also develop results in local algebra with respect to the category $\mathcal A$ that we believe to be of independent interest.
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For an essentially small triangulated category $\mathcal{T}$, we introduce the notion of prime thick subcategories and define the spectrum of $\mathcal{T}$, which shares the basic properties with the spectrum of a tensor triangulated category introduced by Balmer. We mainly focus on triangulated categories that appear in algebraic geometry such as the derived and the singularity categories of a noetherian scheme $X$. We prove that certain classes of thick subcategories are prime thick subcategories of these triangulated categories. Furthermore, we use this result to show that certain subspaces…
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.
In this paper, we introduce the atom spectrum of an abelian category as a topological space consisting of all the equivalence classes of monoform objects. In terms of the atom spectrum, we give a classification of Serre subcategories of an arbitrary noetherian abelian category. Moreover we show that the atom spectrum of a locally noetherian Grothendieck category is homeomorphic to its Ziegler spectrum.
We study the derived categories of small categories over commutative noetherian rings. Our main result is a parametrization of the localizing subcategories in terms of the spectrum of the ring and the localizing subcategories over residue fields. In the special case of representations of Dynkin quivers over a commutative noetherian ring we give a complete description of the localizing subcategories of the derived category, a complete description of the thick subcategories of the perfect complexes and show the telescope conjecture holds. We also present some results concerning the telescope con…
Atom spectra of symmetric monoidal abelian categories and classification of subcategories
本文扩展了在有限生成模范畴中余数类和余数自由类的分类结果，引入轨道原子谱工具，证明在自然张量条件下，若干子范畴坍缩为Serre子范畴或余数自由类，并分类了与张量结构兼容的余数自由类。
We define the extension group between an atom and an object in a locally noetherian Grothendieck category as a module over a skew field. We show that the dimension of the i-th extension group between an atom and an object coincides with the i-th Bass number of the object with respect to the atom. As an application, we give a bijection between the E-stable subcategories closed under arbitrary direct sums and direct summands and the subsets of the atom spectrum and show that such subcategories are also closed under extensions, kernels of epimorphisms, and cokernels of monomorphisms. We show some…
For an exact category having enough projective objects, we establish a bijection between thick subcategories containing the projective objects and thick subcategories of the stable derived category. Using this bijection we classify thick subcategories of finitely generated modules over local complete intersections and produce generators for the category of coherent sheaves on a separated noetherian scheme with an ample family.
The Grothendieck monoid of an exact category is a monoid version of the Grothendieck group. We use it to classify Serre subcategories of an exact category and to reconstruct the topology of a noetherian scheme. We first construct bijections between (i) the set of Serre subcategories of an exact category, (ii) the set of faces of its Grothendieck monoid, and (iii) the monoid spectrum of its Grothendieck monoid. By using (ii), we classify Serre subcategories of exact categories related to a finite dimensional algebra and a smooth projective curve. In particular, we determine the Grothendieck mon…
For any graded commutative noetherian ring, where the grading group is abelian and where commutativity is allowed to hold in a quite general sense, we establish an inclusion-preserving bijection between, on the one hand, the twist-closed localizing subcategories of the derived category, and, on the other hand, subsets of the homogeneous spectrum of prime ideals of the ring. We provide an application to weighted projective schemes.
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…
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。
Classifying localizing subcategories of a locally coherent category
本文通过原子谱的开子集分类局部凝聚Grothendieck范畴的有限型局部化子范畴，并证明若原子谱相等则范畴局部诺特，最后应用于交换凝聚环。
Colocalizing subcategories on differentially graded algebras
本文研究有界非正交换微分分次代数A的导出范畴D(A)中的余局部化子范畴，当D(A)由对应H^0(A)的剩余域的DG模生成时，证明其局部化与余局部化子范畴与Spec(H^0(A))的子集一一对应，推广了Neeman的经典结果。
IE-closed subcategories of commutative rings are torsion-free classes
本文证明交换noetheian环上有限生成模范畴中在像和扩张下封闭的子范畴（IE闭子范畴）必在子模下封闭且为挠自由类，并完全刻画了其成为Serre子范畴的右noetherian环条件。
Ideal category of a Noetherian ring
本文定义了带1的Noetherian环$R$的左[右]理想范畴$\mathbb{L}_R$ [$\mathbb{R}_R$]，证明其为带零对象的预加性范畴及$R$-模范畴的满子范畴，并满足子对象与态射分解性质。
Let $\mathscr{A}$ be an abelian category having enough projective and injective objects, and let $\mathscr{T}$ be an additive subcategory of $\mathscr{A}$ closed under direct summands. A known assertion is that in a short exact sequence in $\mathscr{A}$, the $\mathscr{T}$-projective (respectively, $\mathscr{T}$-injective) dimensions of any two terms can sometimes induce an upper bound of that of the third term by using the same comparison expressions. We show that if $\mathscr{T}$ contains all projective (respectively, injective) objects of $\mathscr{A}$, then the above assertion holds true if…
Classification of localizing subcategories along t-structures
本研究探讨带t-结构的稳定∞-范畴、预稳定∞-范畴与阿贝尔范畴中局部化子范畴的对应关系，证明弱局部化子范畴的双射结论并推广已有结果，还将Lurie的对应提升至稳定范畴。
Spectrum of an abelian category via premonoform objects
本文针对阿贝尔范畴构造了由（诺特）预单形对象等价类组成的拓扑空间${\rm (n)PSpec}\cA$，实现了对挠类、Serre子范畴等的分类，且交换诺特环下$\nPSpec A$与$\Spec A$同胚，还建立了其闭子集与原子谱开子集的对应关系。
Let S be a commutative ring with topologically noetherian spectrum and let R be the absolutely flat approximation of S. We prove that subsets of the spectrum of R parametrise the localising subcategories of D(R). Moreover, we prove the telescope conjecture holds for D(R). We also consider unbounded derived categories of absolutely flat rings which are not semi-artinian and exhibit an example of a cohomological Bousfield class that is not a Bousfield class.
We prove that the Balmer spectrum of a tensor triangulated category is homeomorphic to the Zariski spectrum of its graded central ring, provided the triangulated category is generated by its tensor unit and the graded central ring is noetherian and regular in a weak sense. There follows a classification of all thick subcategories, and the result extends to the compactly generated setting to yield a classification of all localizing subcategories as well as the analog of the telescope conjecture. This generalizes results of Shamir for commutative ring spectra.
We formulate a general abstract criterion for verifying the local-to-global principle for a rigidly-compactly generated tensor triangulated category. Our approach is based upon an inductive construction using dimension functions. Using our criterion we give a new proof of the theorem that the local-to-global principle holds for such categories when they have a model and the spectrum of the compacts is noetherian. As further applications we give a new set of conditions on the spectrum of the compacts that guarantee the local-to-global principle holds and use this to classify localising subcateg…
A Categorical Approach to Finiteness Conditions
本文引入统一Noetherian性、Artinian性与拓扑紧致性的一般范畴框架，利用τ-紧致性与原模性证明相关对象类在子对象、商与扩张下的闭包性质，并应用于Abel范畴构成正合子范畴。
In recent years, Benson, Iyengar and Krause have developed a theory of stratification for compactly generated triangulated categories with an action of a graded commutative Noetherian ring. Stratification implies a classification of localizing and thick subcategories in terms of subsets of the prime ideal spectrum of the given ring. In this paper two stratification results are presented: one for the derived category of a commutative ring-spectrum with polynomial homotopy and another for the derived category of cochains on certain spaces. We also give the stratification of cochains on a space a…
The center $Z(\mathcal{A})$ of an abelian category $\mathcal{A}$ is the endomorphism ring of the identity functor on that category. A localizing subcategory of a Grothendieck category $\mathcal{C}$ is said to be stable if it is stable under essential extensions. The set $\mathbf{L}^{st}(\mathcal{C})$ of stable localizing subcategories of $\mathcal{C}$ is partially ordered under reverse inclusion. We show $\mathcal{L} \mapsto Z(\mathcal{C}/\mathcal{L})$ defines a sheaf of commutative rings on $\mathbf{L}^{st}(\mathcal{C})$ with respect to finite coverings. When $\mathcal{C}$ is assumed to be lo…
For a separated Noetherian scheme $X$ with an ample family of line bundles and a non-zero-divisor $W\inΓ(X,L)$ of a line bundle $L$ on $X$, we classify certain thick subcategories of the derived matrix factorization category ${\rm DMF}(X,L,W)$ of the Landau-Ginzburg model $(X,L,W)$. Furthermore, by using the classification result and the theory of Balmer's tensor triangular geometry, we show that the spectrum of the tensor triangulated category $({\rm DMF}(X,L,W), \otimes^{\frac{1}{2}})$ is homeomorphic to the relative singular locus ${\rm Sing}(X_0/X)$, introduced in this paper, of the zero s…
Gabriel Spectrum of Persistence Categories
本文用纯拓扑方法确定了向量空间层范畴的 Gabriel 谱，证明其同胚于索伯化空间的 Skula 拓扑，并应用于持久性理论，得到持久模范畴的 Gabriel 谱的具体计算。
A torsion theoretic interpretation for sheaves of modules and Grothendieck topologies on directed categories
该研究从挠论角度刻画模层，分析有向范畴上Grothendieck拓扑的结构，分类特殊诺特EI范畴的拓扑并推广表示的局部自内射性质，给出群表示论应用方向。
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