arXiv subjects

Mohsen Asgharzadeh

Publications and source records attributed to Mohsen Asgharzadeh.

At least 19 recordsLinked to original sources

arXiv2026

Remarks on some Homological Problems regarding Infinite Integral Extensions

Let $R$ be an excellent local domain. $R$ is said to be $NBIM$ if $Tor_{i}^{R}(R^{+}, k) = 0$ for some $i\geq d:=\dim(R)$. Bhatt, Iyengar, and Ma ask if equi-characteristic zero $NBIM$ rings are regular. If $R$ is of positive characteristic, Asgharzadeh and Mahdavi conjecture that $Ext^{i}_{R}(k,R^{\infty}) = 0$ for some $i>d$ implies that $R$ is regular. It is an open question whether $R^{+}$ and $R^{\infty}$ are $\mathfrak{m}$-adically idealwise separated in positive characteristic, a condition from the `local criterion of flatness'. These are analogues of Kunz's theorem and intimately related to the homological conjectures and singularities in algebraic geometry. We apply a result of Avramov, Hochster, Iyengar, and Yao on contracting endomorphisms to make progress on the first two. We observe that it implies toric $NBIM$ rings are regular and solves the conjecture for $F$-pure rings. These improvements are inaccessible by previous techniques and give new and simple proofs of earlier results. In mixed characteristic, we show several linked results for perfectoid-pure rings. We show the third statement when there is $R\rightarrow S$ finite and flat on the punctured spectrum and $S$ is regular, this uses Cohen-Macaulayness of $S^{+}$.

math.AC↗

arXiv2026

Descent along flat composed with radicalization

We study the following general situation. Let $R\to S$ be a finite flat morphism of regular rings of zero (resp. prime) characteristic. For $P\in Spec R$, put $A=R/P, C=S/PS,$ and $B=S/\sqrt{PS}=C_{\mathrm{red}}.$ The basic question is whether a property of $B$ forces the same property of $A$. The main point is that ordinary finite-flat descent applies naturally to $A\to C$, whereas the passage $C\to C_{\mathrm{red}}$ may destroy nilpotent information.

math.AC↗

arXiv2026

Representing a ring as the endomorphism ring of an abelian group

We investigate Baer's realization problem for almost free abelian groups, focusing on the extent to which rings can be represented as endomorphism rings under strong freeness conditions. Building on earlier work that relied on additional set-theoretic principles such as the diamond or strong black boxes, we develop new methods that significantly weaken these assumptions. The main result is obtained in ZFC (assuming a mild cardinal arithmetic configuration): for a strong limit singular cardinal $μ$ with $μ^+ < 2^μ< 2^{μ^+}$, and for a wide class of cotorsion-free rings, we construct $μ^+$-free modules whose endomorphism rings are isomorphic to the given ring. This provides a substantial partial solution to a problem of Göbel and Trlifaj. The key innovation is the integration of $κ$-frame constructions with Shelah's Super Black Box, enabling a delicate diagonalization that eliminates nontrivial endomorphisms while preserving high degrees of freeness.

math.LO↗

arXiv2026

Annihilator of $Ext^i_R(R/I,R)$

We investigate the higher divisorial ideal $D(I) := Ann(Ext^g_R(R/I,R))$ associated to an ideal $I$ of grade $g$. Our main focus is the containment problem $ D(I) \subseteq \overline{I}$. We show that this inclusion holds for broad classes of ideals, including unmixed ideals of finite projective dimension over $3$-dimensional quasi-normal rings, parameter ideals in quasi-Gorenstein rings, and powers of perfect ideals under suitable homological conditions. Conversely, we construct explicit examples demonstrating the necessity of these hypotheses. We develop structural properties of $D(I)$, relating it to unmixed parts, reflexive closures, symbolic powers, Frobenius closure, and trace ideals. Applications include the rigidity property of homological annihilators, a criteria for the triviality of reflexive modules and vector bundles on punctured spectra, as well as new connections among annihilators of Ext, conductor ideals, and local cohomology.

math.AC↗

arXiv2026

Perfect closure detects injective dimension

Let $R$ be a noetherian local ring of prime characteristic $p$, and let $R^\infty$ denote its perfect closure. We prove that a finitely generated $R$-module $N$ has finite injective dimension if and only if $\operatorname{Ext}_R^i(R^\infty, N) = 0$ for all $i > 0$. As a Gorenstein counterpart, we show that finiteness of the Gorenstein injective (or projective) dimension of $R^\infty$ forces $R$ to be Gorenstein, and we relate this to a non-noetherian Cohen factorization of $R \to R^\infty$. Applications include preservation of Gorensteinness under weakly etale extensions, structural results on F-coherent and weakly F-nilpotent rings, along with the ascent of Frobenius closure of a parameter ideal to its powers. Finally, assuming $\operatorname{Ext}^{i}_{R}(\frac{R}{\mathfrak m},R^{\infty})=0$ for some $i>\dim(R)$, we show $R$ is regular. This has some applications.

math.AC↗

arXiv2026

Non-Noetherian Bass and Betti numbers

This paper investigates the vanishing and non-vanishing of Betti and Bass numbers for non-finitely generated modules. We prove that for $d$-dimensional Cohen--Macaulay local rings, every non-zero $\mathfrak{m}$-torsion module satisfies $β_d(M)\neq 0$, and we establish the Betti number behavior of the injective hull $E_R(k)$. We study tor-rigidity for $H^d_{\mathfrak{m}}(R)$. We also provide partial positive answers to Schoutens' question on whether the vanishing of some Betti number of a big Cohen--Macaulay algebra forces the Cohen--Macaulay property of $R$. For the absolute integral closure $R^+$, we establish both Tor and Ext results. On the Tor side, we prove that $β_i(R^+)=0$ for some $i>0$ implies regularity in a series cases. On the Ext side, we prove that $μ_i(R^+)=0$ for some $i> d$ forces regularity for Gorenstein domains of prime characteristic, and we obtain analogous results for graded normal domains of dimension $2$ and also for quotient and isolated singularities in any dimension. Also $μ_i(R^\infty)=0$ forces regularity for isolated singularity.

math.AC↗

arXiv2026

Homological Detection by Perfectoid Algebras

We first establish estimates for the projective and injective dimensions for quotients of perfectoid algebras by radical ideals. As applications, we obtain homological characterizations of modules of finite injective dimension, (big) Cohen--Macaulay modules, Gorenstein local rings, and regular local rings in mixed characteristic in terms of perfectoid algebras. These are mixed characteristic analogues of the corresponding characterizations via Frobenius morphisms in positive characteristic.

math.AC↗

arXiv2026

Notes on modules of finite injective dimension

Motivated by Bass' conjecture, we study finitely generated modules of finite injective dimension and the additional constraints they impose on the ambient ring. Beyond ensuring the Cohen--Macaulay property, the presence of such modules enforces further conditions on the ring, including reducedness, normality, being an integral domain, and various singularity conditions such as complete intersection, Gorenstein, and beyond. This continues to detect non-singularity as well. We also address the reflexivity (and also torsionlessness) of modules with finite injective dimension and show that this forces the ring to be quasi-normal. In the same vein, we investigate the injective dimension of tensor products and endomorphism rings. Finally, we study the behavior of R when high syzygies of ksurject onto a non-zero R-module of finite injective dimension.

math.AC↗

arXiv2026

A note on UFD

We investigate conditions under which height-one ideals are principal. As a representative case, let $R$ be a strongly normal, almost factorial, complete intersection local ring, and let $p$ be a prime ideal of height one. We show that if $depth(R/ p)\geq dim(R)-2,$ then $p$ is principal. As an immediate application, we use elementary local cohomology techniques to reprove the celebrated Auslander--Buchsbaum theorem, thereby providing a streamlined approach to certain results of Dao and shedding new light on a problem of Samuel. As a further consequence, we prove that local rings of multiplicity at most three are hypersurfaces. We also give an affirmative answer to a question of Braun concerning reflexive ideals of finite injective dimension. Finally, we show how the reflexive hull can be recovered from the ideal transform by a simple argument, thereby providing a simplified proof of a result of Hartshorne. Then, we compute ideal transform of tensor product $D_{m}(M\otimes_RN)$ in terms of $Hom_R(M^{*},N)$.

math.AC↗

arXiv2026

Finite support of tensor products

We determine the submodule of finite support of the tensor product of two modules M and N over a local ring and estimate its length in terms of $M$ and $N$. Also, we compute higher local cohomology modules of tensor products in a serial of nontrivial cases. As applications, we compute the depth of tensor powers and present some freeness criteria.

math.AC↗

arXiv2026

Remarks on modules of finite projective dimension

We investigate homological and depth-theoretic properties of finitely generated modules of finite projective dimension over Noetherian local rings. A central theme is the study of criteria for freeness and reflexivity derived from the torsion-freeness or reflexivity of tensor products of the form $ M \otimes_R M $ and $ M \otimes_R M^* $. Under mild homological assumptions, we prove that such properties of these tensor products impose strong structural constraints on $ M $, often forcing it to be free. These results generalize classical theorems of Auslander beyond the regular case. The second part of the paper is devoted to the dimension and support of Ext-modules, particularly $ \operatorname{Ext}^i_R(M, R) $ for critical values of $ i $, when $ M $ has finite projective dimension. We establish sharp bounds on their Krull dimensions, analyze their behavior for prime and equidimensional modules, and relate these findings to the grade conjecture and other homological conjectures, i.e., whenever $\operatorname{grade}(M) = \operatorname{ht}(\operatorname{Ann}(M))$ where Gdim$(M)<\infty$. We consider the problem that asks whenever is $ \pd_R(M \otimes_R N) = 1 $? Applications include new cases of a question of Jorgensen, which asks whether $ \operatorname{pd}(M) < i $ whenever $ \operatorname{Ext}^i_R(M, M) = 0 $ and $ M $ has finite projective dimension over a complete intersection ring. Finally, we examine the projective dimension of prime ideals in rings that fail chain conditions.

math.AC↗

arXiv2026

Failure of singular compactness for Hom

Assuming Gödel's axiom of constructibility $V=L$, we construct a $χ$-free abelian group $G$ of singular cardinality for some suitable cardinal $χ$ which is regular and uncountable, equipped with the property that for every nontrivial subgroup $G' \subseteq G$ of smaller cardinality, $Hom(G',\mathbb{Z}) \neq 0$, while $Hom(G,\mathbb{Z}) = 0$. This provides a consistent counterexample to the singular compactness of nontrivial duality with respect to the functor $Hom(-,\mathbb{Z})$.

math.GR↗

arXiv2026

On the initial Betti numbers

Let $R$ be a Cohen-Macaulay local ring possessing a canonical module. We compare the initial and terminal Betti numbers of modules in a series of nontrivial cases. We pay special attention to the Betti numbers of the canonical module. Also, we compute $β_0(ω_{\frac{R}{I}})$ in some cases, where $I$ is a product of two ideals.

math.AC↗

arXiv2026

Integral closure of 1-dimensional rings

We study certain properties of modules over 1-dimensional local integral domains. First, we examine the order of the conductor ideal and its expected relationship with multiplicity. Next, we investigate the reflexivity of certain colength-two ideals. Finally, we consider the freeness problem of the absolute integral closure of a DVR, and connect this to the reflexivity problem of $R^{\frac{1}{p^n}}$.

math.AC↗

arXiv2025

Quite free p-groups with trivial duality

We present a class of abelian groups that exhibit a high degree of freeness while possessing no non-trivial homomorphisms to a canonical free object. Unlike prior investigations, which primarily focused on torsion-free groups, our work broadens the scope to include groups with torsion. Our main focus is on p-groups, for which we formulate and prove the Trivial Duality Conjecture. Key tools in our analysis include the multi black box method and the application of specific homological properties of relative trees.

math.GR↗

arXiv2025

Algebrization of some complete modules

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $\widehat{R}$ its $\mathfrak{m}$-adic completion. We study the problem of determining when a finitely generated $\widehat{R}$-module arises from an $R$-module, i.e., when it is algebraic. We introduce and investigate the class of \emph{strongly algebraic} modules, those complete modules all of whose direct summands are algebraic. Our approach unifies and extends several known results of Levy--Odenthal, Weston, Peskine--Szpiro, Puthenpurakal, and several others, and provides new examples and homological criteria for algebrization. Applications include a computation of the Grothendieck group $G_0(R)$ in dimension one and new algebrization results for generalized Cohen--Macaulay modules and vector bundles along with a connection to local cohomology modules.

math.AC↗

arXiv2025

Naturality and Definability III

In this paper, we deal with the notions of naturality from category theory and definablity from model theory and their interactions. In this regard, we present three results. First, we show, under some mild conditions, that naturality implies definablity. Second, by using the reverse Easton iteration of Cohen forcing notions, we construct a transitive model of ZFC in which every uniformisable construction is weakly natural. Finally, we show that if F is a natural construction on a class K of structures which is represented by some formula, then it is uniformly definable without any extra parameters. Our results answer some questions by Hodges and Shelah.

math.LO↗

arXiv2025

Reflexivity revisited

We study some aspects of reflexive modules. For example, we search conditions for which reflexive modules are free or being very close to free modules.

math.AC↗