Related papers: Re-framing the classification of ideals in noncomm…
This paper contains three related groupings of results. First, we consider a new notion of an admissible skein module of a surface associated to an ideal in a (non-semisimple) pivotal category. Second, we introduce the notion of a chromatic…
The representations of some Hopf algebras have curious behavior: Nonprojective modules may have projective tensor powers, and the variety of a tensor product of modules may not be contained in the intersection of their varieties. We explain…
Tannaka duality and its extensions by Lurie, Sch\"appi et al. reveal that many schemes as well as algebraic stacks may be identified with their tensor categories of quasi-coherent sheaves. In this thesis we study constructions of cocomplete…
The goal of this paper is to find a close to isomorphic presentation of 3-manifolds in terms of Hopf algebraic expressions. To this end we define and compare three different braided tensor categories that arise naturally in the study of…
We establish conditions on a family of coproduct-preserving tt-functors $f_i\colon \mathcal{T}\to \mathcal{T}_i$ between tt-categories with small coproducts, ensuring that the localizing tensor ideal generated by an object $x \in…
Classical approximation bases such as Chebyshev polynomials provide principled and interpretable representations, but their multivariate tensor-product constructions scale exponentially with dimension and impose axis-aligned structure that…
We define duality triples and duality pairs in compactly generated triangulated categories and investigate their properties. This enables us to give an elementary way to determine whether a class is closed under pure subobjects, pure…
Spectral variations pose a common challenge in analyzing hyperspectral images (HSI). To address this, low-rank tensor representation has emerged as a robust strategy, leveraging inherent correlations within HSI data. However, the spatial…
We develop the theory of costratification in the setting of relative tensor-triangular geometry, in the sense of Stevenson, providing a unified approach to classification results of Neeman and Benson--Iyengar--Krause, while laying the…
We discuss dualisable objects in minimal subcategories of compactly generated tensor triangulated categories, paying special attention to the derived category of a commutative noetherian ring. A cohomological criterion for detecting these…
We develop a new framework for noncommutative differential geometry based on double derivations. This leads to the notion of moment map and of Hamiltonian reduction in noncommutative symplectic geometry. For any smooth associative algebra…
We introduce Manifold tensor categories, which make precise the notion of a tensor category with a manifold of simple objects. A basic example is the category of vector spaces graded by a Lie group. Unlike classic tensor category theory,…
For a perfect field $k$ of characteristic $p>0$ and for a finite dimensional symmetric $k$-algebra $A$ K\"ulshammer studied a sequence of ideals of the centre of $A$ using the $p$-power map on degree 0 Hochschild homology. In joint work…
In the present survey paper, we present several new classes of Hochster's spectral spaces "occurring in nature", actually in multiplicative ideal theory, and not linked to or realized in an explicit way by prime spectra of rings. The…
We consider the problem of jointly modeling and clustering populations of tensors by introducing a high-dimensional tensor mixture model with heterogeneous covariances. To effectively tackle the high dimensionality of tensor objects, we…
We explore questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. We construct many classes of examples in which tensor powers of nonprojective modules are projective and tensor products of modules in…
In this work we classify the thick subcategories of the bounded derived category of dg modules over a Koszul complex on any list of elements in a regular ring. This simultaneously recovers a theorem of Stevenson when the list of elements is…
For a finite group $G$ and an arbitrary commutative ring $R$, Brou\'e has placed a Frobenius exact structure on the category of finitely generated $RG$-modules by taking the exact sequences to be those that split upon restriction to the…
A number of spectrum constructions have been devised to extract topological spaces from algebraic data. Prominent examples include the Zariski spectrum of a commutative ring, the Stone spectrum of a bounded distributive lattice, the Gelfand…
For a finite-dimensional gentle algebra, it is already known that the functorially finite torsion classes of its category of finite-dimensional modules can be classified using a combinatorial interpretation, called maximal non-crossing sets…