Gluing Silting Objects

Abstract

Recent results by Keller and Nicolás and by Koenig and Yang have shown bijective correspondences between suitable classes of t-structures and cot-structures with certain objects of the derived category: silting objects. On the other hand, the techniques of gluing (co-)t-structures along a recollement play an important role in the understanding of derived module categories. Using the above correspondence with silting objects, we present explicit constructions of gluing of silting objects, and, furthermore, we answer the question of when the glued silting is tilting.

Deleting or adding arrows of a bound quiver algebra and Hochschild (co)homology

“…As proved in [35], the idempotent e = e 2 + e 3 provides a stratifying ideal ΛeΛ. We assert that ΛeΛ is a strongly stratifying ideal.…”

“…Moreover B = f Λf = k, thus the last requirement of Definition 2.11 holds. For the sake of completeness we next check that Tor A n (M, N ) = 0 for n > 0, after [35]. Consider the projective left A-modules…”

Silting Complexes and Partial Tilting Modules

“…Unfortunately, Rickard [5] also gave a simple counter-example to the Complement Question for partial tilting complexes. Moreover, it was also shown in [6] that the counter-example by Rickard and Schofield [4] is also a counter-example to this question.…”

Support $\tau $-tilting modules and recollements

“…Recently, gluing techniques with respect to a recollement of triangulated or abelian categories have been investigated; for instance, for a given recollement of triangulated categories, Chen [8] glued cotorsion pairs in a recollement; Liu, Vitória and Yang presented the construction of gluing of silting objects [17] in a recollement, which follows from the bijection between the equivalence classes of silting objects and of bounded co-t-structures whose co-hearts are additively generated by one object (see [21]). For a recollement of abelian categories, Parra and Vitória [22] glued some basic properties of abelian categories (well-poweredness, Grothendieck's axioms, existence of a generator) in a recollement.…”