Koszulness

[Francesco Navarra]

Hi everyone, sorry for disturbing you. I have a question, hope someone can help me. I'd like to check if a graded k-algebra R is Koszul, but I can't compute the minimal graded free R-resolution of the ground field k to see if it is linear or not. How can I check it? Is there a command or a procedure that I missed?

Sorry for disturbing you and many thanks in advice for any advice!

[Michael DeBellevue]

Do you have an explicit description of the algebra?  If so, the best you could hope for is that it has a quadratic Groebner basis, which you could check by running "gens gb ideal R".  Another computation you could try is "betti res(coker vars R, LengthLimit => n)" for any n of your choosing.  If the result is concentrated on a single row then the resolution is linear for n steps.  Unfortunately neither of these methods are exhaustive, since there are Koszul algebras without quadratic Groebner basis, and algebras which fail to be Koszul yet have linear resolution for n-many steps, where n is arbitrary.  So there is no command "isKoszul" because such an algorithm has not yet been discovered.

[Francesco Navarra]

Dear Michael,

thank you very much for your answer! 

Yes, I have an explicit description of the algebra that I'd like to study, which is the Rees Algebra of a squarefree borel ideal. Basically, what I wanted to check is if there exists a Rees Algebra of a squarefree Borel ideal with more than two squarefree generators that is not Koszul. The case when a squarefree Borel ideal is generated by two squarefree monomials looks that can be trivially obtained from the same arguments done in The Rees algebra of a two-Borel ideal is Koszul