INTERNAL NATURAL TRANSFORMATIONS AND FROBENIUS ALGEBRAS IN THE DRINFELD CENTER

For ℳ and $$ \mathcal{N} $$ N finite module categories over a finite tensor category $$ \mathcal{C} $$ C , the category $$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$ ℛ ex C (ℳ, $$ \mathcal{N} $$ N ) of right exact module functors is a finite module category over the Drinfeld center $$ \mathcal{Z} $$ Z ($$ \mathcal{C} $$ C ). We study the internal Homs of this module category, which we call internal natural transformations. With the help of certain integration functors that map $$ \mathcal{C} $$ C -$$ \mathcal{C} $$ C -bimodule functors to objects of $$ \mathcal{Z} $$ Z ($$ \mathcal{C} $$ C ), we express them as ends over internal Homs and define horizontal and vertical compositions. We show that if ℳ and $$ \mathcal{N} $$ N are exact $$ \mathcal{C} $$ C -modules and $$ \mathcal{C} $$ C is pivotal, then the $$ \mathcal{Z} $$ Z ($$ \mathcal{C} $$ C )-module $$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$ ℛ ex C (ℳ, $$ \mathcal{N} $$ N ) is exact. We compute its relative Serre functor and show that if ℳ and $$ \mathcal{N} $$ N are even pivotal module categories, then $$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$ ℛ ex C (ℳ, $$ \mathcal{N} $$ N ) is pivotal as well. Its internal Ends are then a rich source for Frobenius algebras in $$ \mathcal{Z} $$ Z ($$ \mathcal{C} $$ C ).

Modified Traces and the Nakayama Functor

We organize the modified trace theory with the use of the Nakayama functor of finite abelian categories. For a linear right exact functor Σ on a finite abelian category ${\mathscr{M}}$ M , we introduce the notion of a Σ-twisted trace on the class $\text {Proj}({\mathscr{M}})$ Proj ( M ) of projective objects of ${\mathscr{M}}$ M . In our framework, there is a one-to-one correspondence between the set of Σ-twisted traces on $\text {Proj}({\mathscr{M}})$ Proj ( M ) and the set of natural transformations from Σ to the Nakayama functor of ${\mathscr{M}}$ M . Non-degeneracy and compatibility with the module structure (when ${\mathscr{M}}$ M is a module category over a finite tensor category) of a Σ-twisted trace can be written down in terms of the corresponding natural transformation. As an application of this principal, we give existence and uniqueness criteria for modified traces. In particular, a unimodular pivotal finite tensor category admits a non-zero two-sided modified trace if and only if it is spherical. Also, a ribbon finite tensor category admits such a trace if and only if it is unimodular.

Ring Constructions and Generation of the Unbounded Derived Module Category

We consider the smallest triangulated subcategory of the unbounded derived module category of a ring that contains the injective modules and is closed under set indexed coproducts. If this subcategory is the entire derived category, then we say that injectives generate for the ring. In particular, we ask whether, if injectives generate for a collection of rings, do injectives generate for related ring constructions, and vice versa. We provide sufficient conditions for this statement to hold for various constructions including recollements, ring extensions and module category equivalences.

On band modules and $$\tau $$-tilting finiteness

In this paper, motivated by a $$\tau $$ τ -tilting version of the Brauer-Thrall Conjectures, we study general properties of band modules and their endomorphisms in the module category of a finite dimensional algebra. As an application we describe properties of torsion classes containing band modules. Furthermore, we show that a special biserial algebra is $$\tau $$ τ -tilting finite if and only if no band module is a brick. We also recover a criterion for the $$\tau $$ τ -tilting finiteness of Brauer graph algebras in terms of the Brauer graph.

Recollements, comma categories and morphic enhancements

For each recollement of triangulated categories, there is an epivalence between the middle category and the comma category associated with a triangle functor from the category on the right to the category on the left. For a morphic enhancement of a triangulated category $\mathcal {T}$ , there are three explicit ideals of the enhancing category, whose corresponding factor categories are all equivalent to the module category over $\mathcal {T}$ . Examples related to inflation categories and weighted projective lines are discussed.