On Derived Equivalences of Categories of Sheaves Over Finite Posets

Details On Derived Equivalences of Categories of Sheaves Over Finite Posets On Derived Equivalences of Categories of Sheaves Over Finite Posets

2006-10-23

arxiv.org/abs/math/0610685v1

Journal of Pure and Applied Algebra 212 (2008), 435-451.

Mathematics

Representation Theory

20 pages

Scientific paper

10.1016/j.jpaa.2007.06.005

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 a poset 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. Then we construct, for any closed subset Y of X, a strongly exceptional collection in D(X) and use it to show an equivalence between D(X) and the bounded derived category of 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 is derived equivalent to the lexicographic sum of the same collection along the opposite graph. 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.

Affiliated with

Also associated with

No associations

Rating

On Derived Equivalences of Categories of Sheaves Over Finite Posets does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On Derived Equivalences of Categories of Sheaves Over Finite Posets, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On Derived Equivalences of Categories of Sheaves Over Finite Posets will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-277288