A projective resolution for the Fomin-Kirillov algebra $\mathrm{FK}(4)$

In this article we show that, given a quadratic algebra satisfying some assumptions, which we call having a resolving datum, one can construct a projective resolution of the trivial module which is obtained as iterated cones of Koszul complexes, and this projective resolution is minimal under some further assumptions. We observe that many examples of quadratic algebras studied so far have a resolving datum, and that the (minimal) projective resolutions constructed for all of them in the literature are an example of our construction. The second main result of the article is that the Fomin-Kirillov algebra $\mathrm{FK}(4)$ of index $4$ has a resolving datum.