Dieudonné module structures for ungraded and periodically graded Hopf rings.
There are several different theories of Dieudonné modules, used to classify different classes of group schemes over perfect fields of positive characteristic. Hence, there are different types of Dieudonné modules for different classes of Hopf algebras. In the work under review, the author discusses many of the different categorical equivalences between Dieudonné modules and Hopf algebras. Additionally, the study is extended to Hopf rings and their equivalence to Dieudonné rings (which, as one might imagine, are Dieudonné modules with additional structure). Let \(H\) be a graded, connected, bicommutative (a.k.a. commutative and cocommutative, or Abelian) Hopf algebra over the finite field \(\mathbb{F} _p\). Then its corresponding Dieudonné module is \(D_*H=\{\Hom(H(n),H)\} _{n\geq 1}\), where \(H(n)=\mathbb{F}_p[x_0,\dots,x_k]\) where \(n\) strongly divides \(p^k\), with comultiplication defined in such a way as to make all Witt polynomials primitive. The Frobenius and Verschiebung on \(H(n)\) induce maps \(F\) and \(V\) on \(D_*H\) with the properties that \(FV=VF=p\) and for each \(x\in D_*H\) there is an \(n\geq 0\) such that \(V^nx=0\). Thus \(D_*H\) is a graded module over \(R:=\mathbb{F}_p[F,V]/(FV-p)\), and one can view \(D_*\) as a covariant functor between these Hopf algebras and graded \(R\)-modules which induces an equivalence of categories. Let \(U_*\) be the inverse correspondence. By generalizing to the category of Hopf rings \(\{H_n\}_{n\in\mathbb{Z}}\) where each \(H_n\) is a Hopf algebra of the type above the equivalence generalizes to Dieudonné rings by putting additional structure on the \(R\)-modules. Indeed, the necessary multiplications \(\circ_{ij}\colon H_i\otimes H_j\to H_{i+j}\) induce products \(D_*(H_i)\otimes D_*(H_j)\to D_*(H_{i+j})\) in Hopf algebras, and it is shown that \(U_*D_*H=\{U_*D_*H_n\}_{n\in\mathbb{Z}}\) is a Hopf ring equivalent to \(H\). Similarly, given a Dieudonné ring \(\{M_n\}_{n\in\mathbb{Z}}\) the maps \(\circ_{ij}'\colon M_i\otimes M_j\to M_{i+j}\) induce maps \(U_*(M_i)\otimes U_*(M_j)\to U_*(M_{i+j})\) and \(D_*U_*M=D_*U_*M_n\) is equivalent to \(M\). (As a side note, the reader should note that the results above are misstated in paragraphs 6 and 7 of the introduction, where the author uses \(H\) instead of \(M\) in several easy-to-identify places.) When restricting to graded Hopf algebras which are group like in degree zero one has an equivalence with \(\mathbb{Z}_p\)-modules \(M=M_0\oplus M_c\) with maps \(V,F\colon M\to M\) such that \(M_c\) is a Dieudonné module as above, \(FV=VF=p\), and \(V\) is the identity on \(M_0\). This is extended to a correspondence on Hopf rings. Now suppose \(H\) is an ungraded bicommutative \(\mathbb{F}_p\)-Hopf algebra. Then \(DH:=\text{colim}_n\Hom(H(n),H)\) has the structure of an \(\mathbb{F}_p\)-module with actions \(F,V\) induced from the Frobenius and Verschiebung, respectively, giving \(DH\) a Dieudonné module structure. The category of ungraded bicommutative \(\mathbb{F}_p\)-Hopf algebras is shown to be equivalent to the category of \(\mathbb{F}_p\)-modules with endomorphisms \(F,V\) such that \(FV=VF=p\) and that for all \(x\in M\) we have \(V^nx=0\) for some \(n\geq 0\). This association also holds in the category of periodically graded Hopf algebras. Finally, suppose \(H\) is a bicommutative Hopf algebra over a commutative ring \(k\) such that \(H\cong H_0\otimes H_c\), where \(H_0\) is a group ring over \(k\) and \(H_c\) is an ungraded connected Hopf algebra. These Hopf algebras are shown to be equivalent to modules \(M_0\oplus M_c\) where \(V\) is the identity on \(M_0\). Overall, this paper does an effective job at describing the Dieudonné module / Dieudonné ring theories where \(H\mapsto M\) is a covariant functor.
scientific article; zbMATH DE number 5818577