Related papers: A curious example of two model categories and some…

Let $\mathfrak{g}$ be a semisimple complex Lie algebra, and let $W$ be a finite subgroup of $\mathbb{C}$-algebra automorphisms of the enveloping algebra $U(\mathfrak{g})$. We show that the derived category of $U(\mathfrak{g})^W$-modules…

Let $G$ be a finite group such that $\text{SL}(n,q)\subseteq G \subseteq \text{GL}(n,q)$ and $Z$ be a central subgroup of $G$. In this paper we determine the group $T(G/Z)$ consisting of the equivalence classes of endotrivial…

We obtain a presentation of the t-deformed Grothendieck ring of a quantum loop algebra of Dynkin type A, D, E. Specializing t at the the square root of the cardinality of a finite field F, we obtain an isomorphism with the derived Hall…

In this paper we study the homotopy theory of parameterized spectrum objects in the $\infty$-category of $(\infty, 2)$-categories, as well as the Quillen cohomology of an $(\infty, 2)$-category with coefficients in such a parameterized…

Special kinds of rank 2 vector bundles with (possibly irregular) connections on P^1 are considered. We construct an equivalence between the derived category of quasi-coherent sheaves on the moduli stack of such bundles and the derived…

This paper is a generalization of arXiv:0810.0808. We develop the de Rham homotopy theory of not necessarily nilpotent spaces, using closed dg-categories and equivariant dg-algebras. We see these two algebraic objects correspond in a…

We prove an analogue of the Gabriel--Quillen embedding theorem for exact $\infty$-categories, giving rise to a presentable version of Klemenc's stable envelope of an exact $\infty$-category. Moreover, we construct a symmetric monoidal…

In this paper we establish Koszul duality between dg categories and a class of curved coalgebras, generalizing the corresponding result for dg algebras and conilpotent curved coalgebras. We show that the normalized chain complex functor…

For a fixed finite group $Q$ and semi-simple finite dimensional algebra $S$, we examine an equivalence between strongly $Q$-graded algebras (extensions) with identity component $S$ and $S^1$-gerbes on action groupoids of $Q$ on the set of…

We give category-theoretic reformulations of stability, NIP, NTP, and non-dividing by observing that their characterisations in terms of indiscernible sequences are naturally expressed as Quillen lifting properties %(negation) of certain…

Let $\mathscr{A}$ and $\mathscr{B}$ be two connected cochain DG algebra such that $\mathscr{A}^{\#}=\mathscr{B}^{\#}$ and the cohomology rings $H(\mathscr{A})$ and $H(\mathscr{B})$ are isomorphic. We give examples to show that $\mathscr{A}$…

We give a combinatorial model structure to the category of, not necessarily conilpotent, differential graded (dg) cocommutative coalgebras and an $\infty$-category structure to the category of curved Lie algebras over an algebraically…

We deal with classes of prime ideals whose associated graded ring is isomorphic to the Rees algebra of the conormal module in order to describe the divisor class group of the Rees algebra and to examine the normality of the conormal module.

In this paper we use A-infinity modules to study the derived category of a finite dimensional algebra over an algebraically closed field. We study varieties parameterising A-infinity modules. These varieties carry an action of an algebraic…

We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context.…

From certain triangle functors, called non-negative functors, between the bounded derived categories of abelian categories with enough projective objects, we introduce their stable functors which are certain additive functors between the…

We show that a $KK$-equivalence between two unital $C^*$-algebras produces a correspondence between their DG categories of finitely generated projective modules which is a $\mathbf{K}_*$-equivalence, where $\mathbf{K}_*$ is Waldhausen's…

We prove that on a certain class of smooth complex varieties (those with "affine even stratifications"), the category of mixed Hodge modules is "almost" Koszul: it becomes Koszul after a few unwanted extensions are eliminated. We also give…

We study Maurer-Cartan moduli spaces of dg algebras and associated dg categories and show that, while not quasi-isomorphism invariants, they are invariants of strong homotopy type, a natural notion that has not been studied before. We…

In this paper, we consider the classification of irreducible ${\bf Z}$- and ${\bf Z}^2$-graded modules with finite dimensional homogeneous subspaces over the Virasoro-like algebra. We first prove that such a module is a uniformly bounded…