On the monoidal invariance of the cohomological dimension of Hopf algebras (Q2143630)

Every algebra considered in this review is assumed over an algebraically closed field \(k\). For a Hopf algebra \(A\), it is known that \(\mathrm{r.gldim}(A) = \mathrm{pd}_A(k_\varepsilon) = \mathrm{cd}(A) = \mathrm{l.gldim}(A) = \mathrm{pd}_A(_\varepsilon k)\), where \(\mathrm{r.gldim}(A)\) and \(\mathrm{l.gldim}(A)\) denote the respective right and left global dimensions of \(A\), \(\mathrm{cd}(A)\) denotes the cohomological dimension of \(A\), \(\mathrm{pd}_A(k_\varepsilon)\) and \(\mathrm{pd}_A(_\varepsilon k)\) denote the projective dimensions respective of right and left trivial \(A\)-modules. A Hopf algebra \(A\) is smooth if \(k_\varepsilon\) or \(_\varepsilon k\) has a finite resolution by finitely generated projective modules. In the paper under review, the author discusses the question of whether the global dimension is a monoidal invariant for Hopf algebras in the sense that if Hopf algebras \(A\), \(B\) have equivalent monoidal categories of comodules, then \(\mathrm{cd}(A) = \mathrm{cd}(B)\), and provides several positive answers to this question under various assumptions of smoothness, cosemisimplicity or finite dimension. More precisely, let \(A\), \(B\) be Hopf algebras that have equivalent linear tensor categories of comodules. Then \(\mathrm{cd}(A) = \mathrm{cd}(B)\) if one of the following condition holds: (1) \(A\), \(B\) are smooth and have bijective antipodes (Theorem 8); (2) \(A\), \(B\) are cosemisimple and \(\mathrm{cd}(A)\), \(\mathrm{cd}(B)\) are finite (Theorem 24); (3) \(A\), \(B\) are finite-dimensional and \(\mathrm{char}k = 0\) or \(\mathrm{char}k > d^{\varphi (d)/2}\) where \(d=\mathrm{dim}(A)\) (Theorem 45); (4) \(A\), \(B\) are finite-dimensional and \(A^{\ast}\) is unimodular (Theorem 45). Finally, the author compares the global dimension and the Gerstenhaber-Schack cohomological dimension in the cosemisimple case, obtaining equality in the case the latter is finite. More precisely, let \(A\) be a cosemisimple Hopf algebra. If \(\mathrm{cd_{GS}}(A)\), the Gerstenhaber-Schack cohomological dimension of \(A\), is finite, then \(\mathrm{cd}(A) = \mathrm{cd_{GS}}(A)\) (Theorem 31). As a consequence of this theorem, the author proves a weak form of Theorem 24. Namely, let \(A\), \(B\) be Hopf algebras that have equivalent linear tensor categories of comodules. If \(A\), \(B\) are cosemisimple and \(\mathrm{cd_{GS}}(A)\) is finite, then \(\mathrm{cd}(A) = \mathrm{cd}(B)\) (Corollary 32). The paper continues [the author, Compos. Math. 149, No. 4, 658--678 (2013; Zbl 1365.16016); Doc. Math. 21, 955--986 (2016; Zbl 1385.16029); Publ. Mat., Barc. 62, No. 2, 301--330 (2018; Zbl 1427.16023)] and [\textit{X. Wang} et al., Pac. J. Math. 290, No. 2, 481--510 (2017; Zbl 1405.16015)].