The Taylor resolution over a skew polynomial ring

Abstract

Let \(\Bbbk\) be a field and let \(I\) be a monomial ideal in the polynomial ring \(Q=\Bbbk[x_1,\ldots,x_n]\). In her thesis, Taylor introduced a complex which provides a finite free resolution for \(Q/I\) as a \(Q\)-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 \(R\). Under the hypothesis that the skew commuting parameters defining \(R\) are roots of unity, we prove as an application that as \(I\) varies among all ideals generated by a fixed number of monomials of degree at least two in \(R\), there is only a finite number of possibilities for the Poincar\'{e} series of \(\Bbbk\) over \(R/I\) and for the isomorphism classes of the homotopy Lie algebra of \(R/I\) in cohomological degree larger or equal to two.