arXiv subjects

Pat Lank

Publications and source records attributed to Pat Lank.

31 records · Page 2Linked to original sources

arXiv2025

Dévissage for generation in derived categories

We study a form of dévissage for generation in derived categories of Noetherian schemes. First, we extend a result of Takahashi from the affine context to the global setting, showing that the bounded derived category is classically generated by a perfect complex together with structure sheaves of closed subschemes supported on the singular locus. Second, we make an observation for how generation behaves under the derived pushforward of a proper surjective morphism between Noetherian schemes. These results enable us to explicitly identify strong generators for projective schemes with isolated singularities and for singular varieties over a perfect field.

math.AG↗

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.

math.AG↗

arXiv2025

Strong generation for module categories

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.

math.AC↗

arXiv2025

Simple criteria for higher rational singularities

This work establishes simple criteria for detecting higher rational singularities via the intersection Du Bois complex and the irrationality complex of a normal variety over the complex numbers.

math.AG↗

arXiv2025

Closedness of the singular locus and generation for derived categories

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.

math.AG↗

arXiv2025

Compact approximation and descent for algebraic stacks

This work focuses on approximation and generation for the derived category of complexes with quasi-coherent cohomology on algebraic stacks. Our methods establish that approximation by compact objects descends along covers that are quasi-finite and flat. This generalizes a result of Lipman--Neeman for schemes and extends a related result known for algebraic spaces. We also study the behavior of generation under the derived pushforward and pullback of a morphism between algebraic stacks.

math.AG↗

arXiv2025

Preservation for generation along the structure morphism of coherent algebras over a scheme

This work demonstrates classical generation is preserved by the derived pushforward along the structure morphism of a noncommutative coherent algebra to its underlying scheme. 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 coherent algebra on it. This is an extension of a classical result of Rouquier to the noncommutative context.

math.AG↗

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.

math.AG↗

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.

math.AG↗

arXiv2025

Triangulated characterizations of singularities

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.

math.AG↗

arXiv2025

Approximation by perfect complexes detects Rouquier dimension

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.

math.AG↗

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.

math.AG↗

arXiv2024

Descent conditions for generation in derived categories

This work establishes a condition that determines when strong generation in the bounded derived category of a Noetherian $J\textrm{-}2$ 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 $\oplus$-generators.

math.AG↗