On derived equivalences of categories of sheaves over finite posets

A finite poset XX carries a natural structure of a topological space. Fix a field kk, and denote by Db(X)Db(X) the bounded derived category of sheaves of finite dimensional kk-vector spaces over XX. Two posets XX and YY are said to be derived equivalent   if Db(X)Db(X) and Db(Y)Db(Y) are equivalent as triangulated categories.We give explicit combinatorial properties of XX which are invariant under derived equivalence; among them are the number of points, the ZZ-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⊆XY⊆X, we construct a strongly exceptional collection in Db(X)Db(X) and use it to show an equivalence Db(X)≃Db(A)Db(X)≃Db(A) for a finite dimensional algebra AA (depending on YY). We give conditions on XX and YY under which AA becomes an incidence algebra of a poset.We deduce that a lexicographic sum of a collection of posets along a bipartite graph SS is derived equivalent to the lexicographic sum of the same collection along the opposite Sop.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.