# Checking Koszulness for incidence algebras of posets via Sage

In theorem 1.6. in the article https://www.sciencedirect.com/science... there is the characterisation that the incidence algebra kP over the field k of a given graded poset P is Koszul if and only if every open intervall (x,y) in P is Cohen-Macaulay over the field k.

My first question is whether one can check for a given bounded (meaning it has a global maximum and a global minimum) and graded poset P whether it is Koszul using Sage. Im especially interested in the cases where k is the rational number or the field with 3 elements.

My second question is wheter it is possible to check whether a given incidence algebra kP of a bounded poset is quadratic (this does not depend on the field k), which means that the quiver algebra kQ/I isomorphic to kP has admissible relations I where the relations are quadratic (so it contains only commutativity relations of length 2).

Thanks for any help.

You can compute by yourself the poset homology of every open interval..