arXiv subjects

Ryu Tomonaga

Publications and source records attributed to Ryu Tomonaga.

6 recordsLinked to original sources

arXiv2026

Auslander correspondence for higher stable dg categories and cluster Morita theory

The notion of $d$-stable dg categories axiomatizes $d$-cluster tilting subcategories of stable dg categories. We establish an Auslander correspondence for $d$-stable dg categories: we characterize the $d$-stability of an additive connective dg category in terms of coherence, weak global dimension, and a duality on finitely presented modules. This gives a homological characterization of $d$-stability and reveals it as a twisted form of $(d+1)$-Calabi--Yau duality. For locally finite connective dg algebras, this interpretation becomes particularly transparent under Koszul duality, where $d$-stability corresponds to a shifted self-injectivity condition on the Koszul dual. Following the constructions of Amiot, Guo and Keller, for a $d$-stable dg category $M$, we introduce its $d$-cluster dg category $\mathcal C_{d,{\rm dg}}(M):=\operatorname{per}_{\rm dg}M/^\mathbb{L}\mathcal D^b_{\rm fp, dg}(M)$. Using our Auslander correspondence, we show that $\mathcal C_{d,{\rm dg}}(M)$ contains $M$ as a $d$-cluster tilting subcategory. In particular, every $d$-stable dg category can be realized as a $d$-cluster tilting subcategory of a stable dg category. We then develop cluster Morita theory: a pretriangulated dg category equipped with a $d$-cluster tilting subcategory $M$ is quasi-equivalent to $\mathcal C_{d,{\rm dg}}(M)$. Thus the connective dg structure of a cluster tilting subcategory determines its ambient dg category up to quasi-equivalence. As an application of cluster Morita theory, we prove a Morita-theoretic variant of Amiot's conjecture. More precisely, we establish a Calabi--Yau correspondence: for a locally finite $d$-stable dg category $M$ over a field, right $(d+1)$-Calabi--Yau structures on $\mathcal D^b_{\rm fp, dg}(M)$ are in bijection with right $d$-Calabi--Yau structures on $\mathcal C_{d,{\rm dg}}(M)$.

math.RT↗

arXiv2026

On silting mutations preserving global dimension

A $d$-silting object is a silting object whose derived endomorphism algebra has global dimension $d$ or less. We give an equivalent condition, which can be stated in terms of dg quivers, for silting mutations to preserve the $d$-siltingness under a mild assumption. Moreover, we show that this mild assumption is always satisfied by $ν_d$-finite algebras. As an application, we give counterexamples to the open question by Herschend--Iyama--Oppermann: the quivers of higher hereditary algebras are acyclic. Our examples consist of a $2$-representation tame algebra with a $2$-cycle, and a $3$-homogeneous $2$-representation finite algebra with a cycle.

math.RT↗

arXiv2026

Non-commutative crepant resolutions of toric singularities with divisor class group of rank one

We prove the existence and give a classification of toric non-commutative crepant resolutions (NCCRs) of Gorenstein toric singularities whose divisor class group has rank one. More precisely, such toric NCCRs are in bijection with non-trivial upper sets in a certain quotient of the divisor class group equipped with a natural partial order. This classification allows us to prove that all toric NCCRs of such toric singularities are connected by iterated Iyama--Wemyss mutations, and hence are derived equivalent to one another. We further give a dimer-model realization of this classification in the non-pyramidal case. More precisely, we construct periodic quivers with cuts on a $d$-dimensional torus, establish a cut-upper set correspondence, and prove that the resulting cut quiver with relations presents the corresponding toric NCCR. For $d=2$, this recovers the quiver-theoretic part of the usual dimer-model construction. In the appendix, we give an explicit formula for the volume of $d$-dimensional lattice polytopes with $d+2$ vertices. As an application, we verify Van den Bergh's conjectural equality, for Gorenstein toric singularities with divisor class group of rank one, between the number of indecomposable direct summands of a toric NCCR and the normalized volume of the corresponding lattice polytope.

math.RT↗

arXiv2026

Higher representation infinite algebras and toric Fano stacks of Picard number one or two

Tilting bundles translate geometry into non-commutative algebra via derived equivalences. We prove the existence of, and classify, $d$-tilting bundles consisting of line bundles on $d$-dimensional smooth toric Fano stacks of Picard number one or two. Their endomorphism algebras give natural examples of $d$-representation infinite algebras and are closely related to the derived McKay correspondence. The classification is motivated by dimer models: an internal perfect matching gives a positive grading on a dimer algebra, whose degree-zero part yields a $2$-representation infinite algebra. The algebras of type $\widetilde A$ introduced by Herschend--Iyama--Oppermann are higher-dimensional analogues of this construction in the simplex case. Applying the same principle to the next case leads to a new class of higher representation infinite algebras, which we call algebras of type $\widetilde A\widetilde A$. Upper sets provide a common framework for tilting bundles, toric non-commutative crepant resolutions (NCCRs), and cuts of higher-dimensional dimer-type quivers. In the Picard-number-one case, $d$-tilting bundles consisting of line bundles are parametrized by non-trivial upper sets in the Picard group, and their endomorphism algebras are precisely the algebras of type $\widetilde A$. In the Picard-number-two case, the upper-set construction becomes two-step: the first upper set determines the ambient toric NCCR, and the second selects an internal cut of its quiver. This classifies all such $d$-tilting bundles and realizes their endomorphism algebras precisely as algebras of type $\widetilde A\widetilde A$. Thus smooth toric Fano stacks of Picard number one and two serve as geometric models of algebras of type $\widetilde A$ and $\widetilde A\widetilde A$, respectively. Using these models, we also show that both classes are closed under $d$-APR tilts.

math.AG↗

arXiv2026

Weak del Pezzo surfaces are characterized by the existence of $2$-tilting bundles

Tilting bundles provide a fundamental bridge between algebraic geometry and representation theory. For a tilting bundle on a smooth proper $d$-dimensional variety, the global dimension of its endomorphism algebra is at least $d$, and the most meaningful case is when this lower bound is attained. Such a tilting bundle, called a $d$-tilting bundle, fits into the framework of the derived McKay correspondence and higher Auslander--Reiten theory. The first main result of this paper shows that the existence of such a bundle forces the variety to be weak Fano: more precisely, if a smooth proper $d$-dimensional variety admits a $d$-tilting bundle, then its anti-canonical bundle is semiample and big. As a consequence, the endomorphism algebra of a $d$-tilting bundle is $d$-representation tame, so the geometry naturally produces higher-dimensional analogues of extended Dynkin quivers. Second, we prove a converse in dimension two: every weak del Pezzo surface over an algebraically closed field admits a $2$-tilting bundle. Together, these results give an affirmative answer to a conjecture posed by Daniel Chan for the variety case: a smooth projective surface admits a $2$-tilting bundle if and only if it is a weak del Pezzo surface. As an application, we construct non-commutative crepant resolutions (NCCRs) of anti-canonical cones over Du Val del Pezzo surfaces. Such an NCCR is obtained as the $3$-Calabi--Yau completion of the endomorphism algebra of a $2$-tilting bundle on the corresponding weak del Pezzo surface. This extends the known construction for smooth del Pezzo surfaces to the Du Val case and places Du Val del Pezzo cones within the framework of the derived McKay correspondence via higher Auslander--Reiten theory.

math.AG↗

arXiv2025

Cohen-Macaulay representations of invariant subrings

We classify two-dimensional complete local rings $(R,\mathfrak{m},k)$ of finite Cohen-Macaulay type where $k$ is an arbitrary field of characteristic zero, generalizing works of Auslander and Esnault for algebraically closed case. Our main result shows that they are precisely of the form $R=l[[x_1,x_2]]^G$ where $l/k$ is a finite Galois extension and $G$ is a finite group acting on $l[[x_1,x_2]]$ as a $k$-algebra. In fact, $G$ can be linearized to become a subgroup of $GL_2(l)\rtimes{\rm Gal}(l/k)$. Moreover, we establish algebraic McKay correspondence in this general setting and completely describe its McKay quiver, which is often non-simply laced, as a quotient of another certain McKay quiver. Combining these results, we classify the quivers that may arise as the Auslander-Reiten quivers of two-dimensional Gorenstein rings of finite Cohen-Macaulay type of equicharacteristic zero. These are shown to be either doubles of (not necessarily simply-laced!) extended Dynkin diagrams or of type $\widetilde{A}_0$ or $\widetilde{CL}_n$ having loops. More generally, we consider higher dimensional $R=l[[x_1,\cdots,x_d]]^G\ (G\subseteq GL_d(l)\rtimes{\rm Gal}(l/k))$ and show they have non-commutative crepant resolutions (NCCRs). Furthermore, we explicitely determine the quivers of the NCCRs as quotients of another certain quivers. To accomplish these, we establish two results which are of independent interest. First, we prove the existence of $(d-1)$-almost split sequences for arbitrary $d$-dimensional Cohen-Macaulay rings having NCCR, even when their singularities are not isolated. Second, we give an explicit recipe to determine irreducible representations of skew group algebras $l*G$ in terms of those over the group algebras $lH$ where $H$ is the kernel of the action of $G$ on $l$.

math.AC↗