arXiv subjects

Kabeer Manali Rahul

Publications and source records attributed to Kabeer Manali Rahul.

11 recordsLinked to original sources

arXiv2026

Fiberwise criteria for Fourier--Mukai equivalences

We study the behavior of integral transforms under base change. In particular, we establish a yoga of local algebra and fibers to test for derived equivalences or fully faithfulness via integral transforms. This generalizes a result of Orlov to singular varieties and strengthens several results in the literature by allowing arbitrary base fields. Additionally, it provides new insight into fibrations and their singularities in arithmetic settings (e.g.\ projective and flat schemes over a DVR).

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arXiv2026

Categorical characterizations of regularity for algebraic stacks

For a Noetherian scheme $X$ of finite Krull dimension, Neeman recently established two characterizations of the regularity of $X$ using strong generators and bounded $t$-structures on $\operatorname{Perf}(X)$. In this note, we obtain variants of Neeman's results for large classes of Noetherian algebraic stacks. An important intermediate step is the fact that $X$ is regular if and only if $\operatorname{Perf}(X)=D_{\operatorname{coh}}^b(X)$, which we establish for Noetherian algebraic stacks. Our approach also yields a criterion for the existence of classical generators for the bounded derived categories of coherent sheaves on algebraic stacks, generalizing previous results for commutative rings and schemes.

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arXiv2026

Coherent sheaves in logarithmic geometry

This paper introduces an abelian category of logarithmic coherent sheaves that arranges coherent sheaves across all expansions and root stacks of a simple normal crossing degeneration. Formally, logarithmic coherent sheaves are coherent sheaves in the full logarithmic étale topology. We develop a suite of tools that reduces the evaluation of the basic functors of homological algebra to the conventional calculation on a computable logarithmic alteration. A second paper will establish good properties of the associated logarithmic derived category. We thus offer a unified perspective on logarithmic moduli spaces of coherent sheaves: The logarithmic Quot spaces motivated by Maulik and Ranganathan's logarithmic Donaldson--Thomas theory, the logarithmic Picard group constructed by Molcho and Wise, and moduli spaces of logarithmic parabolic sheaves as developed by Borne, Talpo, and Vistoli. In establishing the connection with logarithmic Picard groups, we offer a new interpretation of chip firing as the combinatorial shadow to a logarithmic version of S-equivalence.

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arXiv2026

A note on quasi-perfect morphisms

This note is concerned with quasi-perfect morphisms between Noetherian algebraic spaces. In particular, we study the local behavior of quasi-perfect proper morphisms. We show that quasi-perfectness of a proper morphism can be detected at the étale local rings of points of the target, as well as their completions and (strict) Henselizations. As a corollary, we obtain that the locus of points where a proper morphism is quasi-perfect is Zariski open.

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arXiv2025

Integral transforms on singularity categories for Noetherian schemes

This work studies conditions under which integral transforms induce exact functors on singularity categories between schemes that are proper over a Noetherian base scheme. A complete characterization for this behavior is provided, which extends earlier work of Ballard and Rizzardo. We leverage a description of the bounded derived category of coherent sheaves as finite cohomological functors on the category of perfect complexes, which is an application of Neeman's approximable triangulated categories, to reduce arguments to an affine local setting. Moreover, we study adjoints of such functors, extend a result of Olander to varieties with mild singularities, and provide an obstruction for derived equivalences between singular varieties.

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arXiv2025

Representability theorems via metric techniques

We prove new Brown representability theorems for triangulated categories using metric techniques as introduced in the work of Neeman. In the setting of algebraic geometry, this gives us new representability theorems for homological and cohomological functors on the bounded derived category of coherent sheaves. To prove this result, we introduce a generalisation of the notion of an approximable triangulated category.

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arXiv2025

Admissible subcategories and metric techniques

In this work, we provide a way of constructing new semiorthogonal decompositions using metric techniques (à la Neeman). Given a semiorthogonal decomposition on a category with a special kind of metric, which we call a compressible metric, we can construct new semiorthogonal decomposition on a category constructed from the given one using the aforementioned metric. In the algebro-geometric setting, this gives us a way of producing new semiorthogonal decompositions on various small triangulated categories associated to a scheme, if we are given one. In the general setting, the work is related to that of Sun-Zhang, while its applications to algebraic geometry are related to the work of Bondarko and Kuznetsov-Shinder.

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arXiv2025

Descent and generation for noncommutative coherent algebras over schemes

Our work shows forms of descent, in the fppf, h and étale topologies, for strong generation of the bounded derived category of a noncommutative coherent algebra over a scheme. Even for (commutative) schemes this yields new perspectives. As a consequence we exhibit new examples where these bounded derived categories admit strong generators. We achieve our main results by leveraging the action of the scheme on the coherent algebra, allowing us to lift statements into the noncommutative setting. In particular, this leads to interesting applications regarding generation for Azumaya algebras.

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arXiv2025

Descending strong generation in algebraic geometry

We formalize the main approach for showing Zariski descent-type statements for strong generation of triangulated categories associated to algebro-geometric objects. This recovers various known statements in the literature. As applications we show that strong generation for the singularity category of a Noetherian separated scheme is Zariski local and obtain a strong generation result for the bounded derived category of a Noetherian concentrated algebraic stacks with finite diagonal.

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arXiv2025

Approximability and Rouquier dimension for noncommutative algebras over schemes

This work is concerned with approximability (à la Neeman) and Rouquier dimension for triangulated categories associated to noncommutative algebras over schemes. Amongst other things, we establish that the category of perfect complexes of a Noetherian quasi-coherent algebra over a separated Noetherian scheme is strongly generated if, and only if, there exists an affine open cover where the algebra has finite global dimension. As a consequence, we solve an open problem posed by Neeman. Further, as a first application, we study the existence of generators for Azumaya algebras.

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arXiv2024

Bounded $t$-structures, finitistic dimensions, and singularity categories of triangulated categories

Recently, Amnon Neeman settled a bold conjecture by Antieau, Gepner, and Heller regarding the relationship between the regularity of finite-dimensional noetherian schemes and the existence of bounded $t$-structures on their derived categories of perfect complexes. In this paper, using different methods, we prove some very general results about the existence of bounded $t$-structures on (not necessarily algebraic or topological) triangulated categories and their invariance under completion. We show that if the opposite category of an essentially small triangulated category has finite finitistic dimension in our sense, then the existence of a bounded t-structure on it forces it to be equal to its completion. We also prove a parallel result regarding the equivalence of all bounded t-structures on any intermediate triangulated category between the starting category and its completion. Our general treatment, when specialized to the case of schemes, immediately gives us Neeman's theorem as an application and significantly generalizes another remarkable theorem by Neeman about the equivalence of bounded $t$-structures on the bounded derived categories of coherent sheaves. When specialized to other cases like associative rings, nonpositive DG-rings, connective $\mathbb{E}_1$-rings, triangulated categories without models, etc., we get many other applications. Under mild finiteness assumptions, these results not only give a categorical obstruction (the singularity category in our sense) to the existence of bounded $t$-structures on a triangulated category, but also provide plenty of triangulated categories on which all bounded $t$-structures are equivalent. The strategy used in our treatment is introducing a new concept of finitistic dimension for triangulated categories and lifting $t$-structures along completions of triangulated categories.

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