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arXiv · 2109.00111

The Taylor resolution over a skew polynomial ring

Luigi Ferraro·Desiree Martin·W. Frank Moore
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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.

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Luigi Ferraro, Desiree Martin, W. Frank Moore. 2021-08-31. The Taylor resolution over a skew polynomial ring. https://arxiv.org/abs/2109.00111

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