## How to recognize a finite dimensional algebra is Koszul or quadratic?

I have a family of finite dimensional algebras that are directed quasihereditary. I think they might be Koszul algebras and I am wondering what approaches there are to check Koszulness or even quadraticity. I know the quivers of these algebras and can compute Ext^n between simple modules for all n, but I do not have a quiver presentation. I know that there are paths of length 2 and of higher lengths between all vertices of the quiver with nonvanishing Ext^2 so I cannot prove or eliminate quadraticity for trivial reasons. Any thoughts?

I should add that I do not have explicit minimal projective resolutions of the simple modules.

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I think this is honestly too vague to get useful answers. How are you computing Ext^n of all the simples without finding minimal projective resolutions? Are your algebras graded, or are you looking for a grading that makes things Koszul? – Ben Webster Jul 20 2011 at 18:38
I am looking for a grading. I was hoping to use the grading coming from path length. I compute the Ext^n using classifying spaces. My algebras are monoid algebras. We can show that Ext between simple modules can be computed as the cohomology of certain submonoids with coefficients in some nice module. This in turns out to be the cohomology of a certain finite category with coefficients in the field. This we compute by using Quillen's theorem A to get to a nice simplicial complex. So basically we have no nice resolution. If anything we are implicitly using bar resolutions. – Benjamin Steinberg Jul 21 2011 at 1:34
Can you share a typical example of your algebras? – Mariano Suárez-Alvarez Jul 21 2011 at 20:54