The Taylor resolution over a skew polynomial ring

Abstract

Let [Formula: see text] be a field and let [Formula: see text] be a monomial ideal in the polynomial ring [Formula: see text]. In her thesis, Taylor introduced a complex which provides a finite free resolution for [Formula: see text] as a [Formula: see text]-module. Later, Gemeda constructed a differential graded structure on the Taylor resolution. More recently, Avramov showed that this differential graded algebra admits divided powers. We generalize each of these results to monomial ideals in a skew polynomial ring [Formula: see text]. Under the hypothesis that the skew commuting parameters defining [Formula: see text] are roots of unity, we prove as an application that as [Formula: see text] varies among all ideals generated by a fixed number of monomials of degree at least two in [Formula: see text], there is only a finite number of possibilities for the Poincaré series of [Formula: see text] over [Formula: see text] and for the isomorphism classes of the homotopy Lie algebra of [Formula: see text] in cohomological degree larger or equal to two.