The Taylor resolution over a skew polynomial ring

TL;DR

This paper extends the Taylor resolution and its algebraic structures from polynomial rings to skew polynomial rings, showing finiteness results for Poincaré series and homotopy Lie algebras under certain conditions.

Contribution

It generalizes the Taylor resolution, differential graded structure, and divided powers to monomial ideals in skew polynomial rings, and proves finiteness results for invariants when parameters are roots of unity.

Findings

Finite possibilities for Poincaré series of  over R/I.

Finite isomorphism classes of homotopy Lie algebra in higher degrees.

Generalization of algebraic structures to skew polynomial rings.

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.