arXiv subjects

Olgur Celikbas

Publications and source records attributed to Olgur Celikbas.

At least 19 recordsLinked to original sources

arXiv2026

Modules of minimal multiplicity over one-dimensional Cohen-Macaulay local rings

We study finitely generated modules of minimal multiplicity, a notion introduced by Puthenpurakal that extends the classical concept of minimal multiplicity from rings to modules. Our main result characterizes when trace ideals or reflexive ideals yield modules of minimal multiplicity over one-dimensional Cohen-Macaulay local rings. As a consequence, we show that a one-dimensional non-Gorenstein reduced local ring with a canonical module has minimal multiplicity if and only if its canonical module has minimal multiplicity as a module. We also construct several examples and compare them with Burch and Ulrich modules, highlighting cases where minimal multiplicity coincides with the Burch or Ulrich property.

math.AC↗

arXiv2025

Full-Trace Modules

Motivated by the definition of nearly Gorenstein rings, we introduce the notion of full-trace modules over commutative Noetherian local rings--namely, finitely generated modules whose trace equals the maximal ideal. We investigate the existence of such modules and prove that, over rings that are neither regular nor principal ideal rings, every positive syzygy module of the residue field is full-trace. Moreover, over Cohen-Macaulay rings, we study full-trace Ulrich modules--that is, maximally generated maximal Cohen-Macaulay modules that are full-trace. We establish the following characterization: a non-regular Cohen-Macaulay local ring has minimal multiplicity if and only if it admits a full-trace Ulrich module. Finally, for numerical semigroup rings with minimal multiplicity, we show that each full-trace Ulrich module decomposes as the direct sum of the maximal ideal and a module that is either zero or Ulrich.

math.AC↗

arXiv2025

On the test properties of the Frobenius endomorphism

In this paper, we prove two theorems concerning the test properties of the Frobenius endomorphism over commutative Noetherian local rings of prime characteristic $p$. Our first theorem generalizes a result of Funk-Marley on the vanishing of Ext and Tor modules, while our second theorem generalizes one of our previous results on maximal Cohen-Macaulay tensor products. In these earlier results, we replace $^{e}R$ with a more general module $^{e}M$, where $R$ is a Cohen-Macaulay ring, $M$ is a Cohen-Macaulay $R$-module with full support, and $^{e}M$ is the module viewed as an $R$-module via the $e$-th iteration of the Frobenius endomorphism. We also provide examples and present applications of our results, yielding new characterizations of the regularity of local rings.

math.AC↗

arXiv2024

On the vanishing of self extensions of even-periodic modules

In this paper we study rigid modules over commutative Noetherian local rings, establish new freeness criteria for certain periodic rigid modules, and extend several results from the literature. Along the way, we prove general Ext vanishing results over Cohen-Macaulay rings and investigate modules which have zero class in the reduced Grothendieck group with rational coefficients.

math.AC↗

arXiv2023

Depth formula for modules of finite reducing projective dimension

We prove that the depth formula holds for two finitely generated Tor-independent modules over Cohen-Macaulay local rings if one of the modules considered has finite reducing projective dimension (for example, if it has finite projective dimension, or the ring is a complete intersection). This generalizes a result of Bergh-Jorgensen which shows that the depth formula holds for two finitely generated Tor-independent modules over Cohen-Macaulay local rings if one of the modules considered has reducible complexity and certain additional conditions hold. Each module that has reducible complexity also has finite complexity and finite reducing projective dimension, but not necessarily vice versa. So a new advantage we have is that, unlike modules of reducible complexity, Betti numbers of modules of finite reducing projective dimension can grow exponentially.

math.AC↗

arXiv2023

Two theorems on the vanishing of Ext

We prove two theorems on the vanishing of Ext over commutative Noetherian local rings. Our first theorem shows that there are no Burch ideals which are rigid over non-regular local domains. Our second theorem reformulates a conjecture of Huneke-Wiegand in terms of the vanishing of Ext, and highlights its relation with the celebrated Auslander-Reiten conjecture. We also discuss several consequences of our results, for example, about the rigidity of the Frobenius endomorphism in prime characteristic p and a generalization of a result of Araya.

math.AC↗

arXiv2023

On a generalization of Ulrich modules and its applications

We study a modified version of the classical Ulrich modules, which we call $c$-Ulrich. Unlike the traditional setting, $c$-Ulrich modules always exist. We prove that these modules retain many of the essential properties and applications observed in the literature. Additionally, we reveal their significance as obstructions to Cohen-Macaulay properties of tensor products. Leveraging this insight, we show the utility of these modules in testing the finiteness of homological dimensions across various scenarios.

math.AC↗

arXiv2023

On the ubiquity of Arf rings

We introduce and develop the theory of weakly Arf rings, which is a generalization of Arf rings, initially defined by J. Lipman in 1971. We provide characterizations of weakly Arf rings and study the relation between these rings, the Arf rings, and the strict closedness of rings. Furthermore, we give various examples of weakly Arf rings that come from idealizations, fiber products, determinantal rings, and invariant subrings.

math.AC↗

arXiv2023

On the vanishing of (co)homology for modules admitting certain filtrations

We study the vanishing of (co)homology along ring homomorphisms for modules that admit certain filtrations, and generalize a theorem of O. Celikbas-Takahashi. Our work produces new classes of rigid and test modules, in particular over local rings of prime characteristic. It also gives applications in the study of torsion in tensor products of modules, for example, concerning the Huneke-Wiegand conjecture.

math.AC↗

arXiv2023

On the projective dimension of tensor products of modules

In this paper, we consider finitely generated modules over commutative Noetherian rings whose tensor products have finite projective dimension. We construct examples of modules of infinite projective dimension (and also of infinite Gorenstein dimension) whose tensor products nonetheless have finite projective dimension. Furthermore, we establish nontrivial conditions under which such examples cannot arise. For example, we prove that if the tensor product of two nonzero modules--at least one of which is totally reflexive--has finite projective dimension, then both modules in question must have finite projective dimension.

math.AC↗

arXiv2022

Some characterizations of local rings via reducing dimensions

In this paper we study homological dimensions of finitely generated modules over commutative Noetherian local rings, called reducing homological dimensions. We obtain new characterizations of Gorenstein and complete intersection local rings via reducing homological dimensions. For example, we extend a classical result of Auslander and Bridger, and prove that a local ring is Gorenstein if and only if each finitely generated module over it has finite reducing Gorenstein dimension. Along the way, we prove various connections between complexity and reducing projective dimension of modules.

math.AC↗

arXiv2022

On the reducing projective dimension of the residue field

In this paper we are concerned with certain invariants of modules, called reducing invariants, which have been recently introduced and studied by Araya-Celikbas and Araya-Takahashi. We raise the question whether the residue field of each commutative Noetherian local ring has finite reducing projective dimension and obtain an affirmative answer for the question for a large class of local rings. Furthermore, we construct new examples of modules of infinite projective dimension that have finite reducing projective dimension, and study several fundamental properties of reducing dimensions, especially properties under local homomorphisms of local rings.

math.AC↗

arXiv2022

Remarks on a conjecture of Huneke and Wiegand and the vanishing of (co)homology

In this paper we study a long-standing conjecture of Huneke and Wiegand which is concerned with the torsion submodule of certain tensor products of modules over one-dimensional local domains. We utilize Hochster's theta invariant and show that the conjecture is true for two periodic modules. We also make use of a result of Orlov and formulate a new condition which, if true over hypersurface rings, forces the conjecture of Huneke and Wiegand to be true over complete intersection rings of arbitrary codimension. Along the way we investigate the interaction between the vanishing of Tate (co)homology and torsion in tensor products of modules, and obtain new results that are of independent interest.

math.AC↗

arXiv2022

A study of Tate homology via the approximation theory with applications to the depth formula

In this paper we are concerned with absolute, relative and Tate Tor modules. In the first part of the paper we generalize a result of Avramov and Martsinkovsky by using the Auslander-Buchweitz approximation theory, and obtain a new exact sequence connecting absolute Tor modules with relative and Tate Tor modules. In the second part of the paper we consider a depth equality, called the depth formula, which has been initially introduced by Auslander and developed further by Huneke and Wiegand. As an application of our main result, we generalize a result of Yassemi and give a new sufficient condition implying the depth formula to hold for modules of finite Gorenstein and finite injective dimension.

math.AC↗

arXiv2021

An extension of a depth inequality of Auslander

In this paper, we consider a depth inequality of Auslander which holds for finitely generated Tor-rigid modules over commutative Noetherian local rings. We raise the question of whether such a depth inequality can be extended for $n$-Tor-rigid modules, and obtain an affirmative answer for 2-Tor-rigid modules that are generically free. Furthermore, in the appendix, we use Dao's eta function and determine new classes of Tor-rigid modules over hypersurfaces that are quotient of unramified regular local rings.

math.AC↗

arXiv2021

On a class of Burch ideals and a conjecture of Huneke and Wiegand

In this paper we are concerned with a long-standing conjecture of Huneke and Wiegand. We introduce a new class of ideals and prove thateach ideal from such class satisfies the conclusion of the conjecture in question. We also study the relation between the class of Burch ideals and that of the ideals we define, and construct several examples that corroborate our results.

math.AC↗

arXiv2021

On modules with finite reducing Gorenstein dimension

If $M$ is a nonzero finitely generated module over a commutative Noetherian local ring $R$ such that $M$ has finite injective dimension and finite Gorenstein dimension, then it follows from a result of Holm that $M$ has finite projective dimension, and hence a result of Foxby implies that $R$ is Gorenstein. We investigate whether the same conclusion holds for nonzero finitely generated modules that have finite injective dimension and finite reducing Gorenstein dimension, where the reducing Gorenstein dimension is a finer invariant than the classical Gorenstein dimension, in general.

math.AC↗

arXiv2021

On the depth and reflexivity of tensor products

In this paper we study the depth of tensor products of homologically finite complexes over commutative Noetherian local rings. As an application of our main result, we determine new conditions under which nonzero tensor products of finitely generated modules over hypersurface rings can be reflexive only if both of their factors are reflexive. A result of Asgharzadeh shows that nonzero symbolic powers of prime ideals in a local ring cannot have finite projective dimension, unless the ring in question is a domain. We make use of this fact in the appendix and consider the reflexivity of tensor products of prime ideals over hypersurface rings.

math.AC↗