Related papers: Derived Gamma Geometry II: Stable $\infty$-Categor…

A discrete countable group G is matricially stable if the finite dimensional approximate unitary representations of G are perturbable to genuine representations in the point-norm topology. For large classes of groups G, we show that…

We use finite group topological lattice gauge theory, also known as the quantum double model, as a lens to explore a notion of topological order enriched by a non-invertible symmetry. For invertible symmetry enriched topological order,…

Building on structure observed in equivariant homotopy theory, we define an equivariant generalization of a symmetric monoidal category: a $G$-symmetric monoidal category. These record not only the symmetric monoidal products but also…

Let $(\mathcal{C}, \otimes)$ be a monoidal dg-category. We construct a complex controlling the deformation of the monoidal structure on $\mathcal{C}$ together with the deformation of the underlying dg-category itself. We show that in the…

We prove that the bounded derived category of coherent sheaves with proper support is equivalent to the category of locally-finite, cohomological functors on the perfect derived category of a quasi-projective scheme over a field. We…

We show that, for a Noetherian algebraic stack with quasi-affine diagonal $X$, the stable $\infty$-category of quasi-coherent sheaves on $X$ is dualizable if and only if the reduced identity component of the stabilizer of $X$ at every…

The level moduli space $A_g^{4,8}$ is mapped to the projective space by means of gradients of odd Theta functions, such a map turning out no to be injective in the genus 2 case. In this work a congruence subgroup $\Gamma$ is located between…

Let $G$ be a countable monoid and let $A$ be an Artinian group (resp. an Artinian module). Let $\Sigma \subset A^G$ be a closed subshift which is also a subgroup (resp. a submodule) of $A^G$. Suppose that $\Gamma$ is a finitely generated…

The main purpose of this paper is to describe various phenomena and certain constructions arising in the process of studying derived noncommutative schemes. Derived noncommutative schemes are defined as differential graded categories of a…

This study introduces a new unified structural framework for orbifold sigma models that incorporates twisted sectors, singularities, and smooth regions into a single algebraic object. Traditional approaches to orbifold theories often treat…

Using Quillen-Lurie deformation theory formalism we develop an obstruction theory for studying the stable $\infty$-category of modules over a given geometric $\infty$-stack. The obstruction theory studies the problem of lifting compact…

We generalise the techniques of semistable reduction for flat families of sheaves to the setting of the derived category $D^b(X)$ of coherent sheaves on a smooth projective three-fold $X$. Then we construct the moduli of PT-semistable…

We investigate the triangulated structure of stable monomorphism categories (filtered chain categories) over a Frobenius category. The high degree of symmetry of linear quivers leads to a plethora of semiorthogonal decompositions into…

In this paper we answer two long-standing questions in the classification of $G$-torsors on curves for an almost simple, simply connected algebraic group $G$ over the field of complex numbers. The first question is to give an intrinsic…

We introduce a notion of Gieseker stability for coherent sheaves on tame Deligne-Mumford stacks with projective moduli scheme and some chosen generating sheaf on the stack in the sense of Olsson and Starr \cite{MR2007396}. We prove that…

We prove that the bounded derived category of coherent sheaves on a quasicompact separated quasiexcellent scheme of finite dimension has a strong generator in the sense of Bondal-Van den Bergh. This extends a recent result of Neeman and is…

We observe that on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of characteristic $p> h$ (where $h$ is the Coxeter number), with a given (generalized) central character are…

We will generalize the projective model structure in the category of unbounded complexes of modules over a commutative ring to the category of unbounded complexes of quasi-coherent sheaves over the projective line. Concretely we will define…

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field. We exhibit slices of the representation theory of $\Lambda$ that are always classifiable in stringent geometric terms. Namely, we prove that, for any…

Families of stable curves of genus $\gamma$ over a smooth curve $C$ correspond to morphisms from $C$ to the moduli stack of stable curves $\bar{\cal M}_\gamma$. It is natural to compactify the corresponding moduli problem using stable maps…