Pat Lank

Compact approximation and descent for algebraic stacks
Jack Hall, Fei Peng, Alicia Lamarche
in preparation

Integral transforms on singularity categories for Noetherian schemes
Uttaran Dutta, Kabeer Manali Rahul
in preparation

Zariski descent for singularity categories
Timothy De Deyn, Kabeer Manali Rahul
in preparation

Compact objects detect big generators for weak approximable triangulated categories
Timothy De Deyn, Kabeer Manali Rahul
in preparation

Regular locus and singularity categories for noncommutative algebras over schemes
Timothy De Deyn, Kabeer Manali Rahul
in preparation

Derived characterizations for rational pairs à la Schwede-Takagi and Kollár-Kovács
Peter McDonald, Sridhar Venkatesh
arXiv version

Abstract This short note establishes derived characterizations for notions of rational pairs à la Schwede-Takagi and Kollár-Kovács. We use a concept of generation in triangulated categories, introduced by Bondal and Van den Bergh, to study these classes of singularities for pairs. One component of our work introduces rational pairs à la Kollár-Kovács for quasi-excellent schemes of characteristic zero, which gives a Kovács style splitting criterion and a Kovács-Schwede style cohomological vanishing result.

Descent and generation for noncommutative coherent algebras over schemes
Timothy De Deyn, Kabeer Manali Rahul
arXiv version

Abstract Our work investigates a form of descent, in the fppf and h topologies, for generation of triangulated categories obtained from noncommutative coherent algebras over schemes. In addition, also the behaviour of generation with respect to the derived pushforward of proper morphisms is studied. This allows us to exhibit many new examples where the associated bounded derived categories of coherent sheaves admit strong generators. We achieve our main results by combining Matthew's concept of descendability with Stevenson's tensor actions on triangulated categories, allowing us to generalize statements regarding generation into the noncommutative setting. In particular, we establish a noncommutative generalization of Aoki's result to Azumaya algebras over quasi-excellent schemes. Moreover, as a byproduct of the tensor action, we extend Olander's result on countable Rouquier dimension to the noncommutative setting for Azumaya algebras over derived splinters, and we extend a result of Ballard-Iyengar-Lank-Mukhopadhyay-Pollitz regarding strong generation for schemes of prime characteristic to the case of Azumaya algebras.

Classification and nonexistence for t-structures on derived categories of schemes
Alexander Clark, Kabeer Manali Rahul, Chris J. Parker
arXiv version

Abstract This work establishes new results on the classification of t-structures for many subcategories of the derived category of quasi-coherent sheaves on a Noetherian scheme. Our work makes progress in two different directions. On one hand, we provide an improvement of a result of Takahashi on t-structures, generalising it to the case of the bounded derived category of coherent sheaves on a quasi-compact CM-excellent scheme of finite Krull dimension. On the other hand, via independent techniques, we prove a variation of a recent result of Neeman which resolved a conjecture of Antieau, Gepner, and Heller.

Approximability and Rouquier dimension for noncommuative algebras over schemes
Timothy De Deyn, Kabeer Manali Rahul
arXiv version

Abstract 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.

Triangulated characterizations of singularities
Sridhar Venkatesh
arXiv version

Abstract This work presents a range of triangulated characterizations for important classes of singularities such as derived splinters, rational singularities, and Du Bois singularities. An invariant called 'level' in a triangulated category can be used to measure the failure of a variety to have a prescribed singularity type. We provide explicit computations of this invariant for reduced Nagata schemes of Krull dimension one and for affine cones over smooth projective hypersurfaces. Furthermore, these computations are utilized to produce upper bounds for Rouquier dimension on the respective bounded derived categories.

Closedness of the singular locus and generation for derived categories
Souvik Dey
arXiv version

Abstract This work is concerned with a relationship regarding the closedness of the singular locus of a Noetherian scheme and existence of classical generators in its category of coherent sheaves, associated bounded derived category, and singularity category. Particularly, we extend an observation initially made by Iyengar and Takahashi in the affine context to the global setting. Furthermore, we furnish an example a Noetherian scheme whose bounded derived category admits a classical generator, yet not every finite scheme over it exhibits the same property.

Dévissage for generation in derived categories
Souvik Dey
arXiv version

Abstract This note is concerned with generation in the derived category of bounded complexes with coherent cohomology over a Noetherian scheme. We demonstrate a flavor of `dévissage' by identifying two explicit collections of structure sheaves for closed subschemes that classically generate the bounded derived category. Amongst the two, one consists of those supported on the singular locus of the scheme. Moreover, building from the work of Aoki, we show the essential image of the derived pushforward along a proper surjective morphism of Noetherian schemes strongly generates the targets bounded derived category.

Approximation by perfect complexes detects Rouquier dimension
Noah Olander
arXiv version
accepted to Mosc. Math. J.

Abstract This work explores bounds on the Rouquier dimension in the bounded derived category of coherent sheaves on Noetherian schemes. By utilizing approximations, we exhibit that Rouquier dimension is inherently characterized by the number of cones required to build all perfect complexes. We use this to prove sharper bounds on Rouquier dimension of singular schemes. Firstly, we show Rouquier dimension doesn't go up along étale extensions and is invariant under étale covers of affine schemes admitting a dualizing complex. Secondly, we demonstrate that the Rouquier dimension of the bounded derived category for a curve, with a delta invariant of at most one at closed points, is no larger than two. Thirdly, we bound the Rouquier dimension for the bounded derived category of a (birational) derived splinter variety by that of a resolution of singularities.

A note on generation and descent for derived categories of noncommutative schemes
Anirban Bhaduri, Souvik Dey
arXiv version

Abstract This work demonstrates classical generation is preserved by the derived pushforward along the canonical morphism of a noncommutative scheme to its underlying scheme. There are intriguing examples illustrating this phenomenon, particularly from noncommutative resolutions, categorical resolutions, and homological projective duality. Additionally, we establish that the Krull dimension of a variety over a field is a lower bound for the Rouquier dimension of the bounded derived category associated with a noncommutative scheme on it. This is an extension of a classical result of Rouquier to the noncommutative context.

Descent conditions for generation in derived categories
J.Pure Appl. Algebra (2024)

Abstract This work establishes a condition that determines when strong generation in the bounded derived category of a Noetherian J-2 scheme is preserved by the derived pushforward of a proper morphism. Consequently, we can produce upper bounds on the Rouquier dimension of the bounded derived category, and applications concerning affine varieties are studied. In the process, a necessary and sufficient constraint is observed for when a tensor-exact functor between rigidly compactly generated tensor triangulated categories preserves strong ⊕-generators.

Strong generation for module categories
Souvik Dey, Ryo Takahashi
arXiv version

Abstract This article investigates strong generation within the module category of a commutative Noetherian ring. We establish a criterion for such rings to possess strong generators within their module category, addressing a question raised by Iyengar and Takahashi. As a consequence, this not only demonstrates that any Noetherian quasi-excellent ring of finite Krull dimension satisfies this criterion, but applies to rings outside this class. Additionally, we identify explicit strong generators within the module category for rings of prime characteristic, and establish upper bounds on Rouquier dimension in terms of classical numerical invariants for modules.

High Frobenius Pushforwards generate the bounded derived category
Matthew Ballard, Srikanth B. Iyengar, Alapan Mukhopadhyay, and Josh Pollitz
arXiv version

Abstract This work concerns generators for the bounded derived category of coherent sheaves over a noetherian scheme X of prime characteristic. The main result is that when the Frobenius map on X is finite, for any compact generator G of D(X) the Frobenius pushforward Fe∗G generates the bounded derived category whenever pe is larger than the codepth of X, an invariant that is a measure of the singularity of X. The conclusion holds for all positive integers e when X is locally complete intersection. The question of when one can take G=OX is also investigated. For smooth projective complete intersections it reduces to a question of generation of the Kuznetsov component.

Generation and dimension for derived categories
PhD thesis, 2024