Abstract
We extend the notion of monogenic extension to the noncommutative setting, and we study the Hochschild cohomology ring of such an extension. As an application we complete the computation of the cohomology ring of the rank one Hopf algebras begun in [S.M. Burciu, S.J. Witherspoon, Hochschild cohomology of smash products and rank one Hopf algebras, math.RA/0608762, 2006].
Highlights
Assume there is a central element g1 ∈ G such that χ(g1) is a primitive n-th root of 1
The aim of this paper is to solve this problem. We carry out this task by computing the cohomology ring of the monogenic extensions K[x, α]/ f of a separable k-algebra K and noting that each rank one Hopf algebras is such an extension
We fix a commutative ring k with 1, an associative k-algebra K, which we do not assume to be commutative, and a k-algebra endomorphism α of K, and we consider the Ore extension B = K[x, α], namely the algebra generated by K and x subject to the relations xλ = α(λ)x for all λ ∈ K
Summary
We fix the general terminology and notation used in the following, and establish some basic formulas. Let K, α, B, f and A be as in he introduction. From the condition Bf = f B and the fact that f monic, it follows that. Xn−1} is a left K-basis of the algebra A. M, we let M⊗ denote the quotient M/[M, K], where [M, K] is the k-module generated by the commutators mλ − λm with λ ∈ K and m ∈ M. Given a k-algebra extension C/K, let Cα2r := Cαr ⊗ C, where. Composing with the canonical projection Bα2 → A2α we obtain a well-defined derivation. On A2α the following equality holds for all 0 ≤ i ≤ n − 1:.
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