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 Poincaré 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2109.00111
 Bibcode:
 2021arXiv210900111F
 Keywords:

 Mathematics  Rings and Algebras;
 Mathematics  Commutative Algebra;
 Mathematics  Quantum Algebra;
 16E05;
 16E45;
 16E40