On the monoidal invariance of the cohomological dimension of Hopf algebras

Abstract

We discuss the question of whether the global dimension is a monoidal invariant for Hopf algebras, in the sense that if two Hopf algebras have equivalent monoidal categories of comodules, then their global dimensions should be equal. We provide several positive new answers to this question, under various assumptions of smoothness, cosemisimplicity or finite dimension. We also discuss the comparison between the global dimension and the Gerstenhaber-Schack cohomological dimension in the cosemisimple case, obtaining equality in the case the latter is finite. One of our main tools is the new concept of twisted separable functor.Résumé. Nous étudions la question de l'invariance monoïdale de la dimension globale des algèbres de Hopf : si deux algèbres de Hopf ont des catégories de comodules monoïdalement équivalentes, ont-elles même dimension globale ? Nous apportons plusieurs nouvelles réponses positives dans les cas d'algèbres de Hopf lisses, cosemisimples ou de dimension finie. Nous comparons également la dimension globale et la dimension cohomologique de Gerstenhaber-Schack dans le cas cosemisimple. Un outil important pour obtenir ces divers résultats est la nouvelle notion de foncteur séparable twisté.

Free resolutions for free unitary quantum groups and universal cosovereign Hopf algebras

“…In particular, for 𝐹 = 𝐾 * 𝐾 = diag(𝑞 −1 , 1, 𝑞), 𝑞 ∈ ℝ + , it turns out that the second Hochschild cohomology of Pol(𝑈 + 𝐾 ) ≅ (𝐹) with trivial coefficients is one-dimensional; this was the missing piece in [20], where the second Hochschild cohomology of all other free unitary quantum groups was determined. Theorem B also yields explicitly for every generic matrix 𝐹 a (one-dimensional) bimodule 𝑀 with H 3 ((𝐹), 𝑀) ≠ 0, which confirms that the cohomological dimension of (𝐹) is three; a fact Bichon conjectured in [11,Remark 5.15] and proved by more abstract considerations in [12,Theorem 8.1].…”

“…In particular, for $$F = K^*K = \operatorname{diag}(q^{-1},1,q)$$, $$q\in \mathbb {R}^+$$, it turns out that the second Hochschild cohomology of $$\operatorname{Pol}(U_{K}^+)\cong \mathcal {H}(F)$$ with trivial coefficients is one‐dimensional; this was the missing piece in [20], where the second Hochschild cohomology of all other free unitary quantum groups was determined. Theorem B also yields explicitly for every generic matrix $$F$$ a (one‐dimensional) bimodule $$M$$ with $$\mathrm{H}^3(\mathcal {H}(F),M)\ne 0$$, which confirms that the cohomological dimension of $$\mathcal {H}(F)$$ is three; a fact Bichon conjectured in [11, Remark 5.15] and proved by more abstract considerations in [12, Theorem 8.1]. Theorem Let $$F\in \operatorname{GL}_n(\mathbb {C})$$ be generic.…”

Faithful flatness of Hopf algebras over coideal subalgebras with a conditional expectation

“…Theorem 4.3 therefore ensures that if A has bijective antipode, then A * w A s (n) is projective as a left and right A * n -module. This was shown and used in the proof of [4,Theorem 8.4], using [7], under the additional assumption that k = C and that A is a compact Hopf algebra.…”

“…(3) Pointed Hopf algebras are free over their Hopf subalgebras: this was shown by Radford [14]. (4) The celebrated Nichols-Zoeller theorem [12] ensures that finite-dimensional Hopf algebras are free over their Hopf subalgebras (5) Chirvasitu [6] has shown that if H is cosemisimple, then it is faithfully flat over its Hopf subalgebras. The case when A ⊂ H is only a coideal subalgebra is also of high interest in view of quotient theory, but here the situation is not that we can reasonably expect that faithful flatness holds, since even in the commutative case it is easy to find natural coideal subalgebras over which the Hopf algebra is not faithfully flat (for example k[x, y] ⊂ O(SL 2 (k)).…”