Related papers: Tilting Theory and Functor Categories III. The Map…

In this paper we study categories of tilting modules. Our starting point is the tilting modules for a reductive algebraic group G in positive characteristic. Here we extend the main result in [8] by proving that these tilting modules form a…

We develop the theory of Hopf bimodules for a finite rigid tensor category C. Then we use this theory to define a distinguished invertible object D of C and an isomorphism of tensor functors ?^{**} and D tensor ^{**}? tensor D^{-1}. This…

If H is a Hopf algebra with bijective antipode and \alpha, \beta \in Aut_{Hopf}(H), we introduce a category_H{\cal YD}^H(\alpha, \beta), generalizing both Yetter-Drinfeld and anti-Yetter-Drinfeld modules. We construct a braided T-category…

We show that differential calculus (in its usual form, or in the general form of topological differential calculus) can be fully imdedded into a functor category (functors from a small category of anchord tangent algebras to anchored sets).…

We introduce the notion of AIR tilting subcategories of extended hearts of $t$-structures on a triangulated category associated with silting subcategories. This notion generalizes $\tau_{[d]}$-tilting pairs of extended finitely generated…

If $k$ is a field, $A$ a finite dimensional $k$-algebra, then the simple $A$-modules form a simple minded collection in the derived category $\operatorname{D}^b( \operatorname{mod} A )$. Their extension closure is $\operatorname{mod} A$; in…

We begin the study of a tilting theory in certain truncated categories of modules $\mathcal G(\Gamma)$ for the current Lie algebra associated to a finite-dimensional complex simple Lie algebra, where $\Gamma = P^+ \times J$, $J$ is an…

This paper focuses on defining an analog of differential-graded triangular matrix algebra in the context of differential-graded categories. Given two dg-categories $\mathcal{U}$ and $\mathcal{T}$ and $M \in \text{DgMod}(\mathcal{U} \otimes…

We study rank functions on a triangulated category $\mathcal{C}$ via its abelianisation $\operatorname{mod}\mathcal{C}$. We prove that every rank function on $\mathcal{C}$ can be interpreted as an additive function on…

Let $\C$ be an $(n+2)$-angulated category with shift functor $\Sigma$ and $\X$ be a cluster-tilting subcategory of $\C$. Then we show that the quotient category $\C/\X$ is an $n$-abelian category. If $\C$ has a Serre functor, then $\C/\X$…

Let $\mathcal A$ be a Hom-finite abelian category with enough projectives. In this note, we show that any covariantly finite $\tau$-rigid subcategory is contained in a support $\tau$-tilting subcategory. We also show that support…

We show under suitable finiteness conditions that a functor between abelian categories induces a (not necessarily additive) map between their Grothendieck groups. This is related to the derived functors of Dold and Puppe, and generalizes a…

Let $C$ be an additive category with cokernels and let Mod($C$) be the category of additive functors from $C^{op}$ to the category Ab of abelian groups. Let mod($C$) be the full subcategory of Mod($C$) consisting of coherent functors. In…

In this article, we develop a theory of Grothendieck's six operations for derived categories in \'etale cohomology of Artin stacks, for both torsion and adic coefficients. We prove several desired properties of the operations, including the…

Let $K$ be a finite extension of $\mathbb{Q}_{p}$ and let $\Gamma$ be the Galois group of the cyclotomic extension of $K$. Fontaine's theory gives a classification of $p$-adic representations of $\mathrm{Gal}\left(\overline{K}/K\right)$ in…

The aim of this paper is to introduce tau-tilting theory, which completes (classical) tilting theory from the viewpoint of mutation. It is well-known in tilting theory that an almost complete tilting module for any finite dimensional…

We introduce the notion of Gabriel filter for a preadditive category C and we show that there is a bijective correspondence between Gabriel filters of C and hereditary torsion theories in the category of additive functors (C,Ab), obtaining…

Let ${\mathscr T}$ be a triangulated category. If $T$ is a cluster tilting object and $I = [ \operatorname{add} T ]$ is the ideal of morphisms factoring through an object of $\operatorname{add} T$, then the quotient category ${\mathscr T} /…

This paper brings together C*-algebras and algebraic topology in terms of viewing a C*-algebraic invariant in terms of a topological spectrum. E-theory, E(A,B), is a bivariant functor in the sense that is a cohomology functor in the first…

The codomain category of a generalized homology theory is the category of modules over a ring. For an abelian category A, an A-valued (generalized) homology theory is defined by formally replacing the category of modules with the category…