On Derived Equivalences of Categories of Sheaves over Finite Posets

A finite poset X carries a natural structure of a topological space. Fix a field k, and denote by D(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(X) and D(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(X) and use it to show an equivalence D(X) ' D(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. 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.