ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Tue, 06 Oct 2020 18:13:12 +0200Checking Koszulness for incidence algebras of posets via Sagehttps://ask.sagemath.org/question/53723/checking-koszulness-for-incidence-algebras-of-posets-via-sage/In theorem 1.6. in the article https://www.sciencedirect.com/science/article/pii/S0001870810000538?via%3Dihub 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.Sun, 04 Oct 2020 23:07:09 +0200https://ask.sagemath.org/question/53723/checking-koszulness-for-incidence-algebras-of-posets-via-sage/Comment by FrédéricC for <p>In theorem 1.6. in the article <a href="https://www.sciencedirect.com/science/article/pii/S0001870810000538?via=ihub">https://www.sciencedirect.com/science...</a> 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.</p>
<p>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.</p>
<p>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).</p>
<p>Thanks for any help.</p>
https://ask.sagemath.org/question/53723/checking-koszulness-for-incidence-algebras-of-posets-via-sage/?comment=53747#post-id-53747You can compute by yourself the poset homology of every open interval..Tue, 06 Oct 2020 18:13:12 +0200https://ask.sagemath.org/question/53723/checking-koszulness-for-incidence-algebras-of-posets-via-sage/?comment=53747#post-id-53747