arXiv subjects

Mitsuyasu Hashimoto

Publications and source records attributed to Mitsuyasu Hashimoto.

At least 19 recordsLinked to original sources

arXiv2025

Acyclicity test of complexes modulo Serre subcategories using the residue fields

Let $R$ be a commutative noetherian ring, and let $\mathscr{S}$(resp. $\mathscr{L}$) be a Serre(resp. localizing) subcategory of the category of $R$-modules. If $\Bbb F$ is an unbounded complex of $R$-modules Tor-perpendicular to $\mathscr{S}$ and $d$ is an integer, then $\HH{i\geqslant d}{S\otimes_R \Bbb F}$ is in $\mathscr{L}$ for each $R$-module $S$ in $\mathscr{S}$ if and only if $\HH{i\geqslant d}{k(\fp)\otimes_R \Bbb F}$ is in $\mathscr{L}$ for each prime ideal $\fp$ such that $R/\fp$ is in $\mathscr{S}$, where $k(\fp)$ is the residue field at $\fp$. As an application, we show that for any $R$-module $M$, $\Tor_{i\geqslant 0}^R(k(\fp),M)$ is in $\mathscr{L}$ for each prime ideal $\fp$ such that $R/\fp$ is in $\mathscr{S}$ if and only if $\Ext^{i \geqslant 0}_R(S,M)$ is in $\mathscr{L}$ for each cyclic $R$-module $S$ in $\mathscr{S}$. We also obtain some new characterizations of regular and Gorenstein rings in the case of $\mathscr{S}$ consists of finite modules with supports in a specialization-closed subset $V(I)$ of $\Spec R$.

math.AC↗

arXiv2023

Frobenius representation type for invariant rings of finite groups

Let $V$ be a finite rank vector space over a perfect field of characteristic $p>0$, and let $G$ be a finite subgroup of $\operatorname{GL}(V)$. If $V$ is a permutation representation of $G$, or more generally a monomial representation, we prove that the ring of invariants $(\operatorname{Sym}V)^G$ has finite Frobenius representation type. We also construct an example with $V$ a finite rank vector space over the algebraic closure of the function field ${\mathbb{F}_3}(t)$, and $G$ an elementary abelian subgroup of $\operatorname{GL}(V)$, such that the invariant ring $(\operatorname{Sym}V)^G$ does not have finite Frobenius representation type.

math.AC↗

arXiv2023

The symmetry of finite group schemes, Watanabe type theorem, and the $a$-invariant of the ring of invariants

Let $k$ be a field, and $G$ be a $k$-group scheme of finite type. Let $G_{\mathrm{ad}}$ be the $k$-scheme $G$ with the adjoint action of $G$. We call $\lambda_{G,G}=H^0(\mathop{\mathrm{Spec}} k,e^*(\omega_{G_{\mathrm{ad}}}))$ the Knop character of $G$, where $e:\mathop{\mathrm{Spec}} k\rightarrow G_{\mathrm{ad}}$ is the unit element, and $\omega_{G_{\mathrm{ad}}}$ is the $G$-canonical module. We prove that $\lambda_{G,G}$ is trivial in the following cases: (1) $G$ is finite, and $k[G]^*$ is a symmetric algebra; (2) $G$ is finite and \'etale; (3) $G$ is finite and constant; (4) $G$ is smooth and connected reductive; (5) $G$ is abelian; (6) $G$ is finite, and the identity component $G^\circ$ of $G$ is linearly reductive; (7) $G$ is finite and linearly reductive. Let $V$ be a small $G$-module of dimension $n<\infty$. We assume that $\lambda_{G,G}$ is trivial. Let $H=\Bbb G_m$ be the one-dimensional torus, and let $V$ be of degree one as an $H$-module so that $S=\mathop{\mathrm{Sym}} V^*$ is a $\tilde G$-algebra generated by degree one elements, where $\tilde G=G\times H$. We set $A=S^G$. Then we have (i) $\omega_A\cong\omega_S^G$ as $(H,A)$-modules; (ii) $a(A)\leq -n$ in general, where $a(A)$ denotes the $a$-invariant. Moreover, the following are equivalent: (1) The action $G\rightarrow\mathrm{GL}(V)$ factors through $\mathrm{SL}(V)$; (2) $\omega_S\cong S(-n)$ as $(\tilde G,S)$-modules; (3) $\omega_S\cong S$ as $(G,S)$-modules; (4) $\omega_A\cong A(-n)$ as $(H,A)$-modules; (5) $A$ is quasi-Gorenstein; (6) $A$ is quasi-Gorenstein and $a(A)=-n$; (7) $a(A)=-n$. This partly generalizes recent results of Liedtke--Yasuda arXiv:2304.14711v2 and Goel--Jeffries--Singh arXiv:2306.14279v1.

math.AC↗

arXiv2023

Indecomposability of graded modules over a graded ring

Let $R=\bigoplus_{i\geq 0}R_i$ be a Noetherian commutative non-negatively graded ring such that $(R_0,\mathfrak{m}_0)$ is a Henselian local ring. Let $\mathfrak{m}$ be its unique graded maximal ideal $\mathfrak{m}_0+\bigoplus_{i>0}R_i$. Let $T$ be a module-finite (non-commutative) graded $R$-algebra. Let $T\mathop{\mathrm{grmod}}$ denote the category of finite graded left $T$-modules, and $M\in T\mathop{\mathrm{grmod}}$. Then the following are equivalent: (1) $\hat M$ is an indecomposable $\hat T$-module, where $\widehat{(-)}$ denotes the $\mathfrak{m}$-adic completion; (2) $M_{\mathfrak{m}}$ is an indecomposable $T_{\mathfrak{m}}$-module; (3) $M$ is an indecomposable $T$-module; (4) $M$ is indecomposable as a graded $T$-module. As a corollary we prove that for two finite graded left $T$-modules $M$ and $N$, the following are equivalent: (1) If $M=M_1\oplus\cdots\oplus M_s$ and $N=N_1\oplus\cdots\oplus N_t$ are decompositions into indecomposable objects in $T\mathop{\mathrm{grmod}}$, then $s=t$, and there exist some permutation $\sigma\in \frak S_s$ and integers $d_1,\ldots,d_s$ such that $N_i\cong M_{\sigma i}(d_i)$, where $-(d_i)$ denotes the shift of degree; (2) $M\cong N$ as $T$-modules; (3) $M_{\mathfrak{m}}\cong N_{\mathfrak{m}}$ as $T_{\mathfrak{m}}$-modules; (4) $\hat M\cong \hat N$ as $\hat T$-modules. As an application, we compare the FFRT property of rings of characteristic $p$ in the graded sense and in the local sense.

math.AC↗

arXiv2023

Hochster-Eagon type theorem for Serre's $(S_n)$ condition

Let $(A,\mathfrak m)\rightarrow (B,\mathfrak n)$ be a pure homomorphism between Noetherian commutative rings. If $B/\mathfrak m B$ is an Artinian ring, then we have $\dim A=\dim B$ and $\mathop{\mathrm{depth}} A\geq \mathop{\mathrm{depth}} B$. Using this version of Hochster-Eagon theorem, we prove the following: Let $A\rightarrow B$ be a pure homomorphism between Noetherian commutative rings. Assume that the fiber ring $\kappa(\mathfrak p)\otimes_A B$ is Artinian for each $\mathfrak p\in\mathop{\mathrm{Spec}} A$, and $B$ satisfies Serre's $(S_n)$ condition. Then $A$ also satisfies Serre's $(S_n)$ condition. In particular, if a finite group $G$ acts on $B$ and the order $|G|$ of $G$ is invertible in $B$, and if $B$ is Noetherian with the $(S_n)$ condition, then the ring of invariants $A=B^G$ also satisfies the $(S_n)$ condition.

math.AC↗

arXiv2023

Generalized $F$-signatures of the rings of invariants of finite group schemes

Let $k$ be a perfect field of prime characteristic $p$, $G$ a finite group scheme over $k$, and $V$ a finite-dimensional $G$-module. Let $S=\mathop{\mathrm{Sym}}V$ be the symmetric algebra with the standard grading. Let $M$ be a $\Bbb Q$-graded $S$-finite $S$-free $(G,S)$-module, and $L$ be its $S$-reflexive graded $(G,S)$-submodule. Assume that the action of $G$ on $V$ is small in the sense that there exists some $G$-stable Zariski closed subset $F$ of $V$ of codimension two or more such that the action of $G$ on $V\setminus F$ is free. Generalizing the result of P. Symonds and the first author, we describe the Frobenius limit $\mathop{\mathrm{FL}}(L^G)$ of the $S^G$-module $L^G$. In particular, we determine the generalized $F$-signature $s(M,S^G)$ for each indecomposable gradable reflexive $S^G$-module $M$. In particular, we prove the fact that the $F$-signature $s(S^G)=s(S^G,S^G)$ equals $1/\dim k[G]$ if $G$ is linearly reductive (already proved by Watanabe--Yoshida, Carvajal-Rojas--Schwede--Tucker, and Carvajal-Rojas) and $0$ otherwise (some important cases has already been proved by Broer, Yasuda, Liedtke--Martin--Matsumoto).

math.AC↗

arXiv2016

Higher-dimensional absolute versions of symmetric, Frobenius, and quasi-Frobenius algebras

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.

math.RA↗

arXiv2015

F-rationality of the ring of modular invariants

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.

math.AC↗

arXiv2015

The asymptotic behavior of Frobenius direct images of rings of invariants

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.

math.AC↗

arXiv2015

Canonical and $n$-canonical modules on a Noetherian algebra

We define canonical and $n$-canonical modules on a module-finite algebra over a Noether commutative ring and study their basic properties. Using $n$-canonical modules, we generalize a theorem on $(n,C)$-syzygy by Araya and Iima which generalize a well-known theorem on syzygies by Evans and Griffith. Among others, we prove a non-commutative version of Aoyama's theorem which states that a canonical module descends with respect to a flat local homomorphism. We also prove the codimension two-argument for modules over a coherent sheaf of algebras with a $2$-canonical module, generalizing a result of the author.

math.RA↗

arXiv2015

Generalized F-signature of invariant subrings

It is known that a certain invariant subring $R$ has finite $F$-representation type. Thus, we can write the $R$-module ${}^eR$ as a finite direct sum of finitely many $R$-modules. In such a decomposition of ${}^eR$, we pay attention to the multiplicity of each direct summand. For the multiplicity of free direct summand, there is the notion of $F$-signature defined by C. Huneke and G. Leuschke and it characterizes some singularities. In this paper, we extend this notion to non free direct summands and determine the explicit values of them.

math.AC↗

arXiv2015

Equivariant class group. III. Almost principal fibrer bundles

As a formulation of 'codimension-two arguments' in invariant theory, we define a (rational) almost principal bundle. It is a principal bundle off closed subsets of codimension two or more. We discuss the behavior of the category of reflexive modules over locally Krull schemes, the category of the coherent sheaves which satisfy Serre's condition $(S'_2)$ over Noetherian $(S_2)$ schemes with dualizing complexes, the class group, the canonical module, the Frobenius pushforwards, and global $F$-regularity, of a rational almost principal bundle. We give examples of finite group schemes, multisection rings, surjectively graded rings, and determinantal rings, and give unified treatment and new proofs to known results in invariant theory, algebraic geometry, and commutative algebra, and generalize some of them. In particular, we generalize the result on the canonical module of the multisection ring of a sequence of divisors by Kurano and the author. We also give a new proof of a generalization of Thomsen's result on the Frobenius pushforwards of the structure sheaf of a toric variety.

math.AG↗

arXiv2014

$G$-prime and $G$-primary $G$-ideals on $G$-schemes

Let $G$ be a flat finite-type group scheme over a scheme $S$, and $X$ a noetherian $S$-scheme on which $G$-acts. We define and study $G$-prime and $G$-primary $G$-ideals on $X$ and study their basic properties. In particular, we prove the existence of minimal $G$-primary decomposition and the well-definedness of $G$-associated $G$-primes. We also prove a generalization of Matijevic-Roberts type theorem. In particular, we prove Matijevic-Roberts type theorem on graded rings for $F$-regular and $F$-rational properties.

math.AC↗

arXiv2014

Equivariant class group. II. Enriched descent theorem

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.

math.AC↗

arXiv2014

Classification of the linearly reductive finite subgroup schemes of $SL_2$

We classify the linearly reductive finite subgroup schemes $G$ of $SL_2=SL(V)$ 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 $G\mapsto ((\mathop{\mathrm{Sym}} V)^G)\,\hat{~}$.

math.AC↗

arXiv2013

Equivariant class group. I. Finite generation of the Picard and the class groups of an invariant subring

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 dominating $k$-morphism $Z\rightarrow X$. Let $φ:X\rightarrow Y$ be a $G$-invariant morphism such that $\mathcal O_Y\rightarrow (φ_*\mathcal O_X)^G$ is an isomorphism. Then $Y$ is locally Krull. If, moreover, $\Cl(X)$ is finitely generated, then $\Cl(G,X)$ and $\Cl(Y)$ are also finitely generated, where $\Cl(G,X)$ is the equivariant class group. In fact, $\Cl(Y)$ is a subquotient of $\Cl(G,X)$. 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.

math.AC↗

arXiv2012

$F$-finiteness of homomorphisms and its descent

Let $p$ be a prime number. We define the notion of $F$-finiteness of homomorphisms of $\mathbb F_p$-algebras, and discuss some basic properties. In particular, we prove a sort of descent theorem on $F$-finiteness of homomorphisms of $\mathbb F_p$-algebras. As a corollary, we prove the following. Let $g:B\to C$ be a homomorphism of Noetherian $\mathbb F_p$-algebras. If $g$ is faithfully flat reduced, and $C$ is $F$-finite, then $B$ is $F$-finite. This is a generalization of Seydi's result on excellent local rings of characteristic $p$.

math.AC↗

arXiv2011

The canonical module of a Cox ring

In this paper, we shall describe the graded canonical module of a Noetherian multi-section ring of a normal projective variety. In particular, in the case of the Cox ring, we prove that the graded canonical module is a graded free module of rank one with the shift of degree $K_X$. We shall give two kinds of proofs. The first one utilizes the equivariant twisted inverse functor developed by the first author. The second proof is down-to-earth, that avoids the twisted inverse functor.

math.AG↗