A $k$-linear triangulated category without a model

Abstract

In this paper we give an example of a triangulated category, linear over a field of characteristic zero, which does not carry a DG-enhancement. The only previous examples of triangulated categories without a model have been constructed by Muro, Schwede and Strickland. These examples are however not linear over a field. 1991 Mathematics Subject Classification. 13D09, 18E30, 14A22.

The first obstructions to enhancing a triangulated category

“…For this, we need a triangulated category without enhancements. There are few known examples, essentially those in [MSS07], their non-commutative analogues [Dim09], and a new recent family of examples over the rationals [RB18]. These new examples do not have an A ∞ -enhancement but they do have an A 6 -enhancement, so they have a universal Massey product, and therefore the first obstructions described in this paper vanish.…”

“…If so, how many essentially different ones? There are remarkable results on the existence and uniqueness of enhancements for certain triangulated categories [LO10,CS15] as well as examples which do not admit any enhancement [MSS07,Dim09,RB18]. There are even examples with essentially different algebraic enhancements over a field [Kaj13] and some others where the first question remains open [Ami07].…”

A refined derived Torelli theorem for Enriques surfaces

“…It should be noted that the dg approach to semiorthogonal decompositions discussed in this section was put in an even more general categorical setting in [38] (see, in particular, [38,Section 9.3]).…”

Compatibility of t-Structures in a Semiorthogonal Decomposition

“…At the moment, no answer has been found. As a matter of fact, it is incredibly hard to study the general case of triangulated categories; indeed, the definition of an exceptional object requires the category to be K-linear, with K a field, and the only known example of non-algebraic K-linear triangulated category is studied in [23].…”