Three simplicial resolutions (Q2895432)

Let \(S\) be the polynomial ring in \(n\) variables on an arbitrary field \(k\) and \(I\subset X\) a monomial ideal. Finding a minimal resolution of \(S/I\) can be achieved by a computer. Giving an abstract (mathematical), easily manageable description is not so straightforward. This paper surveys some solutions given in the litterature, using a simplicial approach: the Taylor resolution (which is not minimal in general), the Scarf complex, and the Lyubeznik resolution which is intermediate. A new result states that the Scarf complex of an ideal \(I\) is the intersection of all its minimal resolutions.NEWLINENEWLINEFor the entire collection see [Zbl 1237.13005].