start-ver=1.4
cd-journal=joma
no-vol=484
cd-vols=
no-issue=
article-no=
start-page=207
end-page=223
dt-received=
dt-revised=
dt-accepted=
dt-pub-year=2017
dt-pub=20170815
dt-online=
en-article=
kn-article=
en-subject=
kn-subject=
en-title=
kn-title=F-rationality of the ring of modular invariants
en-subtitle=
kn-subtitle=
en-abstract=
kn-abstract= 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.
en-copyright=
kn-copyright=

en-aut-name=HashimotoMitsuyasu
en-aut-sei=Hashimoto
en-aut-mei=Mitsuyasu
kn-aut-name=
kn-aut-sei=
kn-aut-mei=
aut-affil-num=1
ORCID=

affil-num=1
en-affil=Department of Mathematics, Okayama University
kn-affil=
en-keyword=F-rational
kn-keyword=F-rational
en-keyword=F-regular
kn-keyword=F-regular
en-keyword=Dual F-signature
kn-keyword=Dual F-signature
en-keyword=Frobenius limit
kn-keyword=Frobenius limit
END

start-ver=1.4
cd-journal=joma
no-vol=226
cd-vols=
no-issue=
article-no=
start-page=165
end-page=203
dt-received=
dt-revised=
dt-accepted=
dt-pub-year=2017
dt-pub=201706
dt-online=
en-article=
kn-article=
en-subject=
kn-subject=
en-title=
kn-title=Canonical and n-canonical modules of a Noetherian algebra
en-subtitle=
kn-subtitle=
en-abstract=
kn-abstract= 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 Aoyama’s theorem which states that a canonical module descends with respect to a flat local homomorphism.
en-copyright=
kn-copyright=

en-aut-name=HashimotoMitsuyasu
en-aut-sei=Hashimoto
en-aut-mei=Mitsuyasu
kn-aut-name=
kn-aut-sei=
kn-aut-mei=
aut-affil-num=1
ORCID=

affil-num=1
en-affil=Department of Mathematics, Okayama University
kn-affil=
END

start-ver=1.4
cd-journal=joma
no-vol=305
cd-vols=
no-issue=
article-no=
start-page=144
end-page=164
dt-received=
dt-revised=
dt-accepted=
dt-pub-year=2017
dt-pub=20170110
dt-online=
en-article=
kn-article=
en-subject=
kn-subject=
en-title=
kn-title=The asymptotic behavior of Frobenius direct images of rings of invariants
en-subtitle=
kn-subtitle=
en-abstract=
kn-abstract= 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.
en-copyright=
kn-copyright=

en-aut-name=HashimotoMitsuyasu
en-aut-sei=Hashimoto
en-aut-mei=Mitsuyasu
kn-aut-name=
kn-aut-sei=
kn-aut-mei=
aut-affil-num=1
ORCID=

en-aut-name=SymondsbPeter
en-aut-sei=Symondsb
en-aut-mei=Peter
kn-aut-name=
kn-aut-sei=
kn-aut-mei=
aut-affil-num=2
ORCID=

affil-num=1
en-affil=Department of Mathematics, Okayama University
kn-affil=

affil-num=2
en-affil=University of Manchester
kn-affil=
en-keyword=Frobenius direct image
kn-keyword=Frobenius direct image
en-keyword=Hilbert–Kunz multiplicity
kn-keyword=Hilbert–Kunz multiplicity
en-keyword=F-signature
kn-keyword=F-signature
en-keyword=Frobenius limit
kn-keyword=Frobenius limit
END

start-ver=1.4
cd-journal=joma
no-vol=45
cd-vols=
no-issue=4
article-no=
start-page=1509
end-page=1532
dt-received=
dt-revised=
dt-accepted=
dt-pub-year=2016
dt-pub=20161007
dt-online=
en-article=
kn-article=
en-subject=
kn-subject=
en-title=
kn-title=Equivariant class group. II. Enriched descent theorem
en-subtitle=
kn-subtitle=
en-abstract=
kn-abstract= We prove a version of Grothendieck’s descent theorem on an ‘enriched’ 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.
en-copyright=
kn-copyright=

en-aut-name=HashimotoMitsuyasu
en-aut-sei=Hashimoto
en-aut-mei=Mitsuyasu
kn-aut-name=
kn-aut-sei=
kn-aut-mei=
aut-affil-num=1
ORCID=

affil-num=1
en-affil=Department of Mathematics, Okayama University
kn-affil=
en-keyword=Class group
kn-keyword=Class group
en-keyword=descent theory
kn-keyword=descent theory
en-keyword=Picard group
kn-keyword=Picard group
en-keyword=principal fiber bundle
kn-keyword=principal fiber bundle
END

start-ver=1.4
cd-journal=joma
no-vol=59
cd-vols=
no-issue=1
article-no=
start-page=131
end-page=140
dt-received=
dt-revised=
dt-accepted=
dt-pub-year=2017
dt-pub=201701
dt-online=
en-article=
kn-article=
en-subject=
kn-subject=
en-title=
kn-title=Higher-dimensional absolute versions of symmetric, Frobenius, and quasi-Frobenius algebras
en-subtitle=
kn-subtitle=
en-abstract=
kn-abstract=In 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.
en-copyright=
kn-copyright=

en-aut-name=HashimotoMitsuyasu
en-aut-sei=Hashimoto
en-aut-mei=Mitsuyasu
kn-aut-name=
kn-aut-sei=
kn-aut-mei=
aut-affil-num=1
ORCID=

affil-num=1
en-affil=Department of Mathematics Faculty of Science, Okayama University
kn-affil=
en-keyword=canonical module
kn-keyword=canonical module
en-keyword=symmetric algebra
kn-keyword=symmetric algebra
en-keyword=Frobenius algebra
kn-keyword=Frobenius algebra
en-keyword=quasi-Frobenius algebra
kn-keyword=quasi-Frobenius algebra
en-keyword=n-canonical module
kn-keyword=n-canonical module
END

start-ver=1.4
cd-journal=joma
no-vol=459
cd-vols=
no-issue=
article-no=
start-page=76
end-page=108
dt-received=
dt-revised=
dt-accepted=
dt-pub-year=2016
dt-pub=20160801
dt-online=
en-article=
kn-article=
en-subject=
kn-subject=
en-title=
kn-title=Equivariant class group. I. Finite generation of the Picard and the class groups of an invariant subring
en-subtitle=
kn-subtitle=
en-abstract=
kn-abstract=The 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 φ:X→Yφ:X→Y be a G  -invariant morphism such that OY→(φ⁎OX)GOY→(φ⁎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.
en-copyright=
kn-copyright=

en-aut-name=HashimotoMitsuyasu
en-aut-sei=Hashimoto
en-aut-mei=Mitsuyasu
kn-aut-name=橋本光靖
kn-aut-sei=橋本
kn-aut-mei=光靖
aut-affil-num=1
ORCID=

affil-num=1
en-affil=Okayama University
kn-affil=岡山大学大学院自然科学研究科
en-keyword=Invariant theory
kn-keyword=Invariant theory
en-keyword=Class group
kn-keyword=Class group
en-keyword=Picard group
kn-keyword=Picard group
en-keyword=Krull ring
kn-keyword=Krull ring
END

start-ver=1.4
cd-journal=joma
no-vol=40
cd-vols=
no-issue=3
article-no=
start-page=527
end-page=534
dt-received=
dt-revised=
dt-accepted=
dt-pub-year=2015
dt-pub=201509
dt-online=
en-article=
kn-article=
en-subject=
kn-subject=
en-title=
kn-title=Classification of the Linearly Reductive Finite Subgroup Schemes of SL2
en-subtitle=
kn-subtitle=
en-abstract=
kn-abstract=We 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.
en-copyright=
kn-copyright=

en-aut-name=HashimotoMitsuyasu
en-aut-sei=Hashimoto
en-aut-mei=Mitsuyasu
kn-aut-name=橋本光靖
kn-aut-sei=橋本
kn-aut-mei=光靖
aut-affil-num=1
ORCID=

affil-num=1
en-affil=Department of Mathematics, Okayama University
kn-affil=岡山大学大学院自然科学研究科
en-keyword=Group scheme
kn-keyword=Group scheme
en-keyword=Kleinian singularity
kn-keyword=Kleinian singularity
en-keyword=Invariant theory
kn-keyword=Invariant theory
END