On derived equivalences of categories of sheaves over finite posets

Abstract

A finite poset X carries a natural structure of a topological space. Fix a field k, and denote by D b (X ) the bounded derived category of sheaves of finite dimensional k-vector spaces over X . Two posets X and Y are said to be derived equivalent if D b (X ) and D b (Y ) are equivalent as triangulated categories.We give explicit combinatorial properties of X which are invariant under derived equivalence; among them are the number of points, the Z-congruency class of the incidence matrix, and the Betti numbers. We also show that taking opposites and products preserves derived equivalence.For any closed subset Y ⊆ X , we construct a strongly exceptional collection in D b (X ) and use it to show an equivalence D b (X ) D b (A) for a finite dimensional algebra A (depending on Y ). We give conditions on X and Y under which A becomes an incidence algebra of a poset.We deduce that a lexicographic sum of a collection of posets along a bipartite graph S is derived equivalent to the lexicographic sum of the same collection along the opposite S op .This construction produces many new derived equivalences of posets and generalizes other well-known ones. As a corollary we show that the derived equivalence class of an ordinal sum of two posets does not depend on the order of summands. We give an example that this is not true for three summands.

Recollements and tilting objects

“…Let D b (X) = D b (Sh(X)) ∼ = D b (mod(KX)) be the bounded derived category. By [25], there exists a 'left' recollement of More examples of this kind, also covered by our construction, can be found in another article by Ladkani, [26]. …”

“…The exceptional objects he considered in this context [25] are also produced by our construction in Section 2.…”

“…Following Ladkani's notation, we let Sh(X ) be the category of sheaves over X with values in the category of finite-dimensional vector spaces over a field K . By [25] this is equivalent to the category mod(KX ) of finite-dimensional modules over the incidence algebra KX . Let D b (X) = D b (Sh(X)) ∼ = D b (mod(KX)) be the bounded derived category.…”

Derived Equivalences of Triangular Matrix Rings Arising from Extensions of Tilting Modules

“…Another aspect of this question was addressed by considering examples of triangular matrix algebras of a simple form, such as incidence algebras of posets [14]. In this paper we extend the results of [14] to general triangular matrix rings.…”

Dualities between cellular sheaves and cosheaves

“…The nerve is also a poset, so composing ι Let [C, D] denote the category whose objects are functors F : C → D and whose morphisms are natural transformations. The following is a simple modification of an equivalence known to experts, which is normally stated for sheaves [13].…”