ElsevierActa Medica Okayama0021-86934842017F-rationality of the ring of modular invariants207223ENMitsuyasuHashimotoDepartment of Mathematics, Okayama University Using the description of the Frobenius limit of modules over the ring of invariants under an action of a finite group on a polynomial ring over a field of characteristic p>0 developed by Symonds and the author, we give a characterization of the ring of invariants with a positive dual F-signature. Combining this result and Kemper's result on depths of the ring of invariants under an action of a permutation group, we give an example of an F-rational, but non-F-regular ring of invariants under the action of a finite group.No potential conflict of interest relevant to this article was reported.Cambridge University PressActa Medica Okayama0027-76302262017Canonical and n-canonical modules of a Noetherian algebra165203ENMitsuyasuHashimotoDepartment of Mathematics, Okayama University We define canonical and -canonical modules of a module-finite algebra over a Noether commutative ring and study their basic properties. Using -canonical modules, we generalize a theorem on -syzygy by Araya and Iima which generalize a well-known theorem on syzygies by Evans and Griffith. Among others, we prove a noncommutative version of Aoyamafs theorem which states that a canonical module descends with respect to a flat local homomorphism.No potential conflict of interest relevant to this article was reported.ElsevierActa Medica Okayama0001-87083052017The asymptotic behavior of Frobenius direct images of rings of invariants144164ENMitsuyasuHashimotoDepartment of Mathematics, Okayama UniversityPeterSymondsbUniversity of Manchester We define the Frobenius limit of a module over a ring of prime characteristic to be the limit of the normalized Frobenius direct images in a certain Grothendieck group. When a finite group acts on a polynomial ring, we calculate this limit for all the modules over the twisted group algebra that are free over the polynomial ring; we also calculate the Frobenius limit for the restriction of these to the ring of invariants. As an application, we generalize the description of the generalized F-signature of a ring of invariants by the second author and Nakajima to the modular case.No potential conflict of interest relevant to this article was reported.Taylor & FrancisActa Medica Okayama0092-78724542016Equivariant class group. II. Enriched descent theorem15091532ENMitsuyasuHashimotoDepartment of Mathematics, Okayama University We prove a version of Grothendieckfs descent theorem on an eenrichedf principal fiber bundle, a principal fiber bundle with an action of a larger group scheme. Using this, we prove the isomorphisms of the equivariant Picard and the class groups arising from such a principal fiber bundle.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15665912017Higher-dimensional absolute versions of symmetric, Frobenius, and quasi-Frobenius algebras131140ENMitsuyasuHashimotoDepartment of Mathematics Faculty of Science, Okayama UniversityIn this paper, we define and discuss higher-dimensional and absolute versions of symmetric, Frobenius, and quasi-Frobenius algebras. In particular, we compare these with the relative notions defined by Scheja and Storch. We also prove the validity of codimension two-argument for modules over a coherent sheaf of algebras with a 2-canonical module, generalizing a result of the author.No potential conflict of interest relevant to this article was reported.ACADEMIC PRESS INC ELSEVIER SCIENCEActa Medica Okayama0021-86934592016Equivariant class group. I. Finite generation of the Picard and the class groups of an invariant subring76108ENMitsuyasuHashimotoOkayama UniversityThe purpose of this paper is to define equivariant class group of a locally Krull scheme (that is, a scheme which is locally a prime spectrum of a Krull domain) with an action of a flat group scheme, study its basic properties, and apply it to prove the finite generation of the class group of an invariant subring.
In particular, we prove the following.
Let k be a field, G a smooth k-group scheme of finite type, and X a quasi-compact quasi-separated locally Krull G-scheme. Assume that there is a k-scheme Z of finite type and a dominant k -morphism Z¨XZ¨X. Let ƒÓ:X¨YƒÓ:X¨Y be a G -invariant morphism such that OY¨(ƒÓ⁎OX)GOY¨(ƒÓ⁎OX)G is an isomorphism. Then Y is locally Krull. If, moreover, Cl(X)Cl(X) is finitely generated, then Cl(G,X)Cl(G,X) and Cl(Y)Cl(Y) are also finitely generated, where Cl(G,X)Cl(G,X) is the equivariant class group.
In fact, Cl(Y)Cl(Y) is a subquotient of Cl(G,X)Cl(G,X). For actions of connected group schemes on affine schemes, there are similar results of Magid and Waterhouse, but our result also holds for disconnected G. The proof depends on a similar result on (equivariant) Picard groups.No potential conflict of interest relevant to this article was reported.Springer SingaporeActa Medica Okayama0251-41844032015Classification of the Linearly Reductive Finite Subgroup Schemes of SL2527534ENMitsuyasuHashimotoDepartment of Mathematics, Okayama UniversityWe classify the linearly reductive finite subgroup schemes G of SL2=SL(V) over an algebraically closed field k of positive characteristic, up to conjugation. As a corollary, we prove that such G is in one-to-one correspondence with an isomorphism class of two-dimensional F-rational Gorenstein complete local rings with the coefficient field k by the correspondence G↦((SymV)G) ˆ.No potential conflict of interest relevant to this article was reported.