arXiv subjects

Ryo Kanda

Publications and source records attributed to Ryo Kanda.

At least 19 recordsLinked to original sources

arXiv2026

A Grothendieck category with a noetherian generator and exact products that is not a module category

We construct a Grothendieck category that has a noetherian generator, satisfies AB4*, and is not equivalent to a module category. This gives a negative answer to Djament's problem. The category is obtained as a Gabriel quotient of a module category over the endomorphism ring appearing in the work of Herbera, P\v{r}\'{\i}hoda, and Wiegand. The proof relies on the trace criterion established by Martini, Parra, Saor\'{\i}n, and Virili.

math.CT↗

arXiv2024

Module-theoretic approach to dualizable Grothendieck categories

We prove that every dualizable Grothendieck category whose dual is again a Grothendieck category satisfies Grothendieck's conditions Ab6 and Ab4*, by taking a module-theoretic approach based on the Gabriel-Popescu embedding. Combining this with a result by Stefanich, we conclude that the class of dualizable linear cocomplete categories is precisely the class of linear Grothendieck category satisfying Ab6 and Ab4*. This provides a complete answer to a modified conjecture on the dualizability, originally posed by Brandenburg, Chirvasitu, and Johnson-Freyd.

math.CT↗

arXiv2022

The symplectic leaves for the elliptic Poisson bracket on projective space defined by Feigin-Odesskii and Polishchuk

This paper determines the symplectic leaves for a remarkable Poisson structure on $\mathbb{C}\mathbb{P}^{n-1}$ discovered by Feigin and Odesskii, and, independently, by Polishchuk. The Poisson bracket is determined by a holomorphic line bundle of degree $n \ge 3$ on a compact Riemann surface of genus one or, equivalently, by an elliptic normal curve $E\subseteq\mathbb{C}\mathbb{P}^{n-1}$. The symplectic leaves are described in terms of higher secant varieties to $E$.

math.AG↗

arXiv2021

Modular properties of elliptic algebras

Fix a pair of relatively prime integers $n>k\ge 1$, and a point $(\eta\,|\,\tau)\in\mathbb{C}\times\mathbb{H}$, where $\mathbb{H}$ denotes the upper-half complex plane, and let ${{a\;\,b}\choose{c\,\;d}}\in\mathrm{SL}(2,\mathbb{Z})$. We show that Feigin and Odesskii's elliptic algebras $Q_{n,k}(\eta\,|\,\tau)$ have the property $Q_{n,k}\big(\frac{\eta}{c\tau+d}\,\big\vert\,\frac{a\tau+b}{c\tau+d}\big)\cong Q_{n,k}(\eta\,|\,\tau)$. As a consequence, given a pair $(E,\xi)$ consisting of a complex elliptic curve $E$ and a point $\xi\in E$, one may unambiguously define $Q_{n,k}(E,\xi):=Q_{n,k}(\eta\,|\,\tau)$ where $\tau\in\mathbb{H}$ is any point such that $\mathbb{C}/\mathbb{Z}+\mathbb{Z}\tau\cong E$ and $\eta\in\mathbb{C}$ is any point whose image in $E$ is $\xi$. This justifies Feigin and Odesskii's notation $Q_{n,k}(E,\xi)$ for their algebras.

math.RA↗

arXiv2021

Flat cotorsion modules over Noether algebras

For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs' result for a commutative noetherian ring. As a consequence, we show that pointwise Matlis duality gives a bijective correspondence between the isoclasses of indecomposable injective left modules and the isoclasses of indecomposable flat cotorsion right modules. This correspondence is an explicit realization of Herzog's homeomorphism induced from elementary duality of Ziegler spectra.

math.RT↗

arXiv2020

Elliptic R-matrices and Feigin and Odesskii's elliptic algebras

The algebras $Q_{n,k}(E,\tau)$ introduced by Feigin and Odesskii as generalizations of the 4-dimensional Sklyanin algebras form a family of quadratic algebras parametrized by coprime integers $n>k\ge 1$, a complex elliptic curve $E$, and a point $\tau\in E$. The main result in this paper is that $Q_{n,k}(E,\tau)$ has the same Hilbert series as the polynomial ring on $n$ variables when $\tau$ is not a torsion point. We also show that $Q_{n,k}(E,\tau)$ is a Koszul algebra, hence of global dimension $n$ when $\tau$ is not a torsion point, and, for all but countably many $\tau$, it is Artin-Schelter regular. The proofs use the fact that the space of quadratic relations defining $Q_{n,k}(E,\tau)$ is the image of an operator $R_{\tau}(\tau)$ that belongs to a family of operators $R_{\tau}(z):\mathbb{C}^n\otimes\mathbb{C}^n\to\mathbb{C}^n\otimes\mathbb{C}^n$, $z\in\mathbb{C}$, that (we will show) satisfy the quantum Yang-Baxter equation with spectral parameter.

math.RA↗

arXiv2019

Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

The elliptic algebras in the title are connected graded $\mathbb{C}$-algebras, denoted $Q_{n,k}(E,\tau)$, depending on a pair of relatively prime integers $n>k\ge 1$, an elliptic curve $E$, and a point $\tau\in E$. This paper examines a canonical homomorphism from $Q_{n,k}(E,\tau)$ to the twisted homogeneous coordinate ring $B(X_{n/k},\sigma',\mathcal{L}'_{n/k})$ on the characteristic variety $X_{n/k}$ for $Q_{n,k}(E,\tau)$. When $X_{n/k}$ is isomorphic to $E^g$ or the symmetric power $S^gE$ we show the homomorphism $Q_{n,k}(E,\tau) \to B(X_{n/k},\sigma',\mathcal{L}'_{n/k})$ is surjective, that the relations for $B(X_{n/k},\sigma',\mathcal{L}'_{n/k})$ are generated in degrees $\le 3$, and the non-commutative scheme $\mathrm{Proj}_{nc}(Q_{n,k}(E,\tau))$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$, respectively. When $X_{n/k}=E^g$ and $\tau=0$, the results about $B(X_{n/k},\sigma',\mathcal{L}'_{n/k})$ show that the morphism $\Phi_{|\mathcal{L}_{n/k}|}:E^g \to \mathbb{P}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.

math.AG↗

arXiv2019

Projective discrete modules over profinite groups

We show that the category of discrete modules over an infinite profinite group has no non-zero projective objects and does not satisfy Ab4*. We also prove the same types of results in a generalized setting using a ring with linear topology.

math.RA↗

arXiv2019

Finite quotients of powers of an elliptic curve

Let $E$ be an elliptic curve. When the symmetric group $\Sigma_{g+1}$ of order $(g+1)!$ acts on $E^{g+1}$ in the natural way, the subgroup $E_0^{g+1}$, consisting of those $(g+1)$-tuples whose coordinates sum to zero, is stable under the action of $\Sigma_{g+1}$. It is isomorphic to $E^g$. This paper concerns the structure of the quotient variety $E^g/\Sigma$ when $\Sigma$ is a subgroup of $\Sigma_{g+1}$ generated by simple transpositions. In an earlier paper we observed that $E^g/\Sigma$ is a bundle over a suitable power, $E^N$, with fibers that are products of projective spaces. This paper shows that $E^g/\Sigma$ has an \'etale cover by a product of copies of $E$ and projective spaces with an abelian Galois group.

math.AG↗

arXiv2019

The characteristic variety for Feigin and Odesskii's elliptic algebras

This paper examines an algebraic variety that controls an important part of the structure and representation theory of the algebra $Q_{n,k}(E,\tau)$ introduced by Feigin and Odesskii. The $Q_{n,k}(E,\tau)$'s are a family of quadratic algebras depending on a pair of coprime integers $n>k\ge 1$, an elliptic curve $E$, and a point $\tau\in E$. It is already known that the structure and representation theory of $Q_{n,1}(E,\tau)$ is controlled by the geometry associated to $E$ embedded as a degree $n$ normal curve in the projective space $\mathbb P^{n-1}$, and by the way in which the translation automorphism $z\mapsto z+\tau$ interacts with that geometry. For $k\ge 2$ a similar phenomenon occurs: $(E,\tau)$ is replaced by $(X_{n/k},\sigma)$ where $X_{n/k}\subseteq\mathbb P^{n-1}$ is the characteristic variety of the title and $\sigma$ is an automorphism of it that is determined by the negative continued fraction for $\frac{n}{k}$. There is a surjective morphism $\Phi:E^g \to X_{n/k}$ where $g$ is the length of that continued fraction. The main result in this paper is that $X_{n/k}$ is a quotient of $E^g$ by the action of an explicit finite group. We also prove some assertions made by Feigin and Odesskii. The morphism $\Phi$ is the natural one associated to a particular invertible sheaf $\mathcal L_{n/k}$ on $E^g$. The generalized Fourier-Mukai transform associated to $\mathcal L_{n/k}$ sends the set of isomorphism classes of degree-zero invertible $\mathcal O_E$-modules to the set of isomorphism classes of indecomposable locally free $\mathcal O_E$-modules of rank $k$ and degree $n$. Thus $X_{n/k}$ has an importance independent of the role it plays in relation to $Q_{n,k}(E,\tau)$. The backward $\sigma$-orbit of each point on $X_{n/k}$ determines a point module for $Q_{n,k}(E,\tau)$.

math.RA↗

arXiv2018

Feigin and Odesskii's elliptic algebras

We study the elliptic algebras $Q_{n,k}(E,\tau)$ introduced by Feigin and Odesskii as a generalization of Sklyanin algebras. They form a family of quadratic algebras parametrized by coprime integers $n>k\geq 1$, an elliptic curve $E$, and a point $\tau\in E$. We consider and compare several different definitions of the algebras and provide proofs of various statements about them made by Feigin and Odesskii. For example, we show that $Q_{n,k}(E,0)$, and $Q_{n,n-1}(E,\tau)$ are polynomial rings on $n$ variables. We also show that $Q_{n,k}(E,\tau+\zeta)$ is a twist of $Q_{n,k}(E,\tau)$ when $\zeta$ is an $n$-torsion point. This paper is the first of several we are writing about the algebras $Q_{n,k}(E,\tau)$.

math.RA↗

arXiv2018

Non-exactness of direct products of quasi-coherent sheaves

For a noetherian scheme that has an ample family of invertible sheaves, we prove that direct products in the category of quasi-coherent sheaves are not exact unless the scheme is affine. This result can especially be applied to all quasi-projective schemes over commutative noetherian rings. The main tools of the proof are the Gabriel-Popescu embedding and Roos' characterization of Grothendieck categories satisfying Ab6 and Ab4*.

math.CT↗

arXiv2018

Extension groups between atoms in abelian categories

We introduce the extension groups between atoms in an abelian category. For a locally noetherian Grothendieck category, the localizing subcategories closed under injective envelopes are characterized in terms of those extension groups. We also introduce the virtual duals of the extension groups between atoms to measure the global dimension of the category. A new topological property of atom spectra is revealed and it is used to relate the projective dimensions of atoms with the Krull-Gabriel dimensions. As a byproduct of the topological observation, we show that there exists a spectral space that is not homeomorphic to the atom spectrum of any abelian category.

math.CT↗

arXiv2018

Construction of Grothendieck categories with enough compressible objects using colored quivers

We introduce a new method to construct a Grothendieck category from a given colored quiver. This is a variant of the construction used to prove that every partially ordered set arises as the atom spectrum of a Grothendieck category. Using the new method, we prove that for every finite partially ordered set, there exists a locally noetherian Grothendieck category such that every nonzero object contains a compressible subobject and its atom spectrum is isomorphic to the given partially ordered set.

math.RA↗

arXiv2017

Integrality of noetherian Grothendieck categories

We introduce the notion of integrality of Grothendieck categories as a simultaneous generalization of the primeness of noncommutative noetherian rings and the integrality of locally noetherian schemes. Two different spaces associated to a Grothendieck category yield respective definitions of integrality, and we prove the equivalence of these definitions using a Grothendieck-categorical version of Gabriel's correspondence, which originally related indecomposable injective modules and prime two-sided ideals for noetherian rings. The generalization of prime two-sided ideals is also used to classify locally closed localizing subcategories. As an application of the main results, we develop a theory of singular objects in a Grothendieck category and deduce Goldie's theorem on the existence of the quotient ring as its consequence.

math.RA↗

arXiv2017

Flat families of point schemes for connected graded algebras

We study truncated point schemes of connected graded algebras as families over the parameter space of varying relations for the algebras, proving that the families are flat over the open dense locus where the point schemes achieve the expected (i.e. minimal) dimension. When the truncated point scheme is zero-dimensional we obtain its number of points counted with multiplicity via a Chow ring computation. This latter application in particular confirms a conjecture of Brazfield to the effect that a generic two-generator, two-relator 4-dimensional Artin-Schelter regular algebra has seventeen truncated point modules of length six.

math.RA↗

arXiv2017

New Artin-Schelter regular and Calabi-Yau algebras via normal extensions

We introduce a new method to construct 4-dimensional Artin-Schelter regular algebras as normal extensions of (not necessarily noetherian) 3-dimensional ones. The method produces large classes of new 4-dimensional Artin-Schelter regular algebras. When applied to a 3-Calabi-Yau algebra our method produces a flat family of central extensions of it that are 4-Calabi-Yau, and all 4-Calabi-Yau central extensions having the same generating set as the original 3-Calabi-Yau algebra arise in this way. Each normal extension has the same generators as the original 3-dimensional algebra, and its relations consist of all but one of the relations for the original algebra and an equal number of new relations determined by "the missing one" and a tuple of scalars satisfying some numerical conditions. We determine the Nakayama automorphisms of the 4-dimensional algebras obtained by our method and as a consequence show that their homological determinant is 1. This supports the conjecture by Mori-Smith that the homological determinant of the Nakayama automorphism is 1 for all Artin-Schelter regular connected graded algebras. Reyes-Rogalski-Zhang proved this is true in the noetherian case.

math.QA↗

arXiv2015

Finiteness of the number of minimal atoms in Grothendieck categories

For a Grothendieck category having a noetherian generator, we prove that there are only finitely many minimal atoms. This is a noncommutative analogue of the fact that every noetherian scheme has only finitely many irreducible components. It is also shown that each minimal atom is represented by a compressible object.

math.RT↗