arXiv subjects
Publications and source records attributed to Özgür Esentepe.
For a one dimensional analytically unramified Cohen-Macaulay local ring $R$, the blowup algebra of the canonical ideal is a module finite birational extension. The conductor of this extension always contains the conductor of $R$. We study the case when there is equality. This is the case where $R$ is far from being almost Gorenstein. We study this property within the landscape of numerical semigroup rings and local Arf rings.
We study Penrose tilings of the plane $\mathbb{R}^2$ and nonperiodic infinite frieze patterns from the point of view of Cohen--Macaulay representation theory: Triangulations of the completed infinity-gon correspond to subcategories of the Frobenius category $\mathcal{C}_2=\mathrm{CM}_{\mathbb{Z}}(\mathbb{C}[x,y]/(x^2))$, the singularity category of the curve singularity of type $A_\infty$. We relate Penrose tilings to certain triangulations of the completed infinity-gon, and thus to the corresponding subcategories of $\mathcal{C}_2$. We then extend the cluster character of Paquette and Yıldırım for a triangulated category modelling said triangulations to our setting. This allows us to define nonperiodic infinite friezes patterns coming from triangulations of the completed infinity-gon and in particular from Penrose tilings.
Given a Cohen-Macaulay local ring, the cohomology annihilator ideal and the annihilator of the stable category of maximal Cohen-Macaulay modules are two ideals closely related both with each other and the singularities of the ring. Kimura recently showed that the two ideals agree up to radicals. In this article, we give a sufficient condition for the two ideals to be equal. As an application, we show that the cohomology annihilator ideal of a one dimensional analytically unramified almost Gorenstein complete local ring agrees with the conductor ideal.
In a pandemic era preprint, Dao showed showed two remarkable properties of Arf rings: under some mild conditions, they admit finitely many indecomposable reflexive modules up to isomorphism and every reflexive module is actually isomorphic to its own dual. In fact, the latter property characterises Arf rings. Arf rings are one dimensional rings and it is natural to wonder what happens in higher Krull dimension. In this paper, we investigate the self-dual property for commutative Noetherian local rings.
The Auslander-Reiten Conjecture for commutative Noetherian rings predicts that a finitely generated module is projective when certain Ext-modules vanish. But what if those Ext-modules do not vanish? We study the annihilators of these Ext-modules and formulate a generalisation of the Auslander-Reiten Conjecture. We prove this general version for high syzygies of modules over several classes of rings including analytically unramified Arf rings, 2-dimensional local normal domains with rational singularities, Gorenstein isolated singularities of Krull dimension at least 2 and more. We also prove results for the special case of the canonical module of a Cohen-Macaulay local ring. These results both generalise and also provide evidence for a version of the Tachikawa Conjecture that was considered by Dao-Kobayashi-Takahashi.
Given any commutative Noetherian ring $R$ and an element $x$ in $R$, we consider the full subcategory $\C(x)$ of its singularity category consisting of objects for which the morphism that is given by the multiplication by $x$ is zero. Our main observation is that we can establish a relation between $\C(x), \C(y)$ and $\C(xy)$ for any two ring elements $x$ and $y$. Utilizing this observation, we obtain a decomposition of the singularity category and consequently an upper bound on the dimension of the singularity category.
Using the theory of cohomology annihilators, we define a family of topologies on the set of isomorphism classes of maximal Cohen-Macaulay modules over a Gorenstein ring. We study compactness of these topologies.
We consider dominant dimension of an order over a Cohen-Macaulay ring in the category of centrally Cohen-Macaulay modules. There is a canonical tilting module in the case of positive dominant dimension and we give an upper bound on the global dimension of its endomorphism ring.
The aim of this paper is to study the theory of cohomology annihilators over commutative Gorenstein rings. We adopt a triangulated category point of view and study the annihilation of stable category of maximal Cohen-Macaulay modules. We prove that in dimension one the cohomology annihilator ideal and the conductor ideal coincide under mild assumptions. We present a condition on a ring homomorphism between Gorenstein rings which allows us to carry the cohomology annihilator of the domain to that of the codomain. As an application, we generalize the Milnor-Jung formula for algebraic curves to their double branched covers. We also show that the cohomology annihilator of a Gorenstein local ring is contained in the cohomology annihilator of its Henselization and in dimension one the cohomology annihilator of its completion. Finally, we investigate a relation between the cohomology annihilator of a Gorenstein ring and stable annihilators of its noncommutative resolutions.