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Mohsen Asgharzadeh

Publications and source records attributed to Mohsen Asgharzadeh.

At least 37 records · Page 2Linked to original sources
arXiv2025

Vector space summands of lower syzygies

In this paper, we investigate problems concerning when the residue field $k$ of a local ring $(R,\frak m$, $k)$ appears as a direct summand of syzygy modules, from two perspectives. First, we prove that the following conditions are equivalent: (i) $k$ is a direct summand of second syzygies of all non-free finitely generated $R$-modules; (ii) $k$ is a direct summand of third syzygies of all non-free finitely generated $R$-modules; (iii) $k$ is a direct summand of $\frak m$. We also prove various consequences of these conditions. The second point of this article is to investigate for what artinian local rings $R$ the dual $E^*=Hom_R(E_R(k),R)$ of the injective envelope of the residue field, which is also a second syzygy, is a $k$-vector space. Using the notion of Eliahou-Kervaire resolution, we introduce a large class of artinian local rings that satisfy this condition.

math.AC↗
arXiv2024

Expressive Power of Infinitary Logic and Absolute co-Hopfianity

Recently, Paolini and Shelah have constructed absolutely Hopfian torsion-free abelian groups of any given size. In contrast, we show that this is not necessarily the case for absolutely co-Hopfian groups. We use the infinitary logic to show that there are no absolute co-Hopfian abelian groups above the first beautiful cardinal. An extension of this result to the category of modules over a commutative ring is given.

math.LO↗
arXiv2024

On groups well represented as automorphism groups of groups

Assuming Gödel's axiom of constructibility $\bold V=\bold L,$ we present a characterization of those groups $L$ for which there exist arbitrarily large groups $H$ such that $aut(H) \cong L$. In particular, we show that it suffices to have one such group $H$ such that the size of its center is bigger than $ 2^{|L |+\aleph_0}$.

math.GR↗
arXiv2024

Freeness criteria via vanishing of Tor

We investigate some aspects of the module $L$ equipped with the property that $Tor_1^R(L, F) =0$ implies that $F$ is free. This has some applications.

math.AC↗
arXiv2024

Weak normality, Gorenstein and Serre's conditions

We compute the associated prime ideals of the normalization modulo the ring, and establish connections between different types of generalizations (resp. specializations) of the normalization. This has some applications. For example, we present a characterization of quasi-normality, in the format that was conjectured by Vasconcelos. Also, we show in some cases, the normalization modulo the ring, is homologically reduced. This provides a partial answer to the conjecture of Matlis.

math.AC↗
arXiv2024

A note on cohomological vanishing theorems

We study $cd(M,N):=\sup\{j:H^j_{m}(M,N)\neq0\}$, and we prove the following over $AB$-rings: $cd(M,N)<\infty$ iff $cd(M, N)\leq2 dim R$. For locally free over the punctured spectrum, we present the better bound, namely $cd(M, N)<\infty$ iff $cd(M, N)\leq dim R,$ and show this is sharp for maximal Cohen-Macaulay, and prove that this detects freeness of $M$. We present some explicit examples to compute $cd(M, N)$. Now, suppose $R$ is only Cohen-Macaulay and of prime characteristic equipped with the Frobenius map $φ$. We show for some $n\gg 0$ that $cd(^{φ_n}R,M)<\infty$ iff $id_R(M)<\infty.$ This presents some criteria on regularity. Also, some vanishing results on $Ext^i_R(^φR,-)$ are given, where $(-)\in\{R,^φR\}$. We determine conditions under which the vanishing $Ext^i_R(^φR,-)$ of restricted many $i$-th, implies the vanishing of all.

math.AC↗
arXiv2023

Co-Hopfian and boundedly endo-rigid mixed abelian groups

For a given cardinal $λ$ and a torsion abelian group $K$ of cardinality less than $λ$, we present, under some mild conditions (for example $λ=λ^{\aleph_0}$), boundedly endo-rigid abelian group $G$ of cardinality $λ$ with $Tor(G)=K$. Essentially, we give a complete characterization of such pairs $(K, λ)$. Among other things, we use a twofold version of the black box. We present an application of the construction of boundedly endo-rigid abelian groups. Namely, we turn to the existing problem of co-Hopfian abelian groups of a given size, and present some new classes of them, mainly in the case of mixed abelian groups. In particular, we give useful criteria to detect when a boundedly endo-rigid abelian group is co-Hopfian and completely determine cardinals $λ> 2^{\aleph_{0}}$ for which there is a co-Hopfian abelian group of size $λ$.

math.LO↗
arXiv2023

Homology with the theme of Matlis

Matlis proved a lot of homological properties of the fraction field of an integral domain. In this paper, we simplify and extend some of them from 1-dimensional (resp. rank one) cases to the higher dimensional (resp. finite rank) cases. For example, we study the weakly co-torsion property of Ext$(-,\sim)$, and use it to present splitting criteria. These are equipped with several applications. For instance, we compute the projective dimension of $\widehat{R}$ and present some non-noetherian versions of Grothendieck's localization problem. We construct a new class of co-Hopfian modules and extend Matlis' decomposability problem to higher ranks. In particular, this paper deals with the basic properties of Matlis' quadric $(Q,Q/R, \widehat{R},\overset{\sim}R).$

math.AC↗
arXiv2022

Cohomological splitting, realization, and finiteness

We search for some splitting (resp. finiteness) criteria of a given module $M$ over a local ring $(R,\fm,k)$ in terms of the splitting (resp. finiteness) property of certain cohomological functors evaluated at $M$. In particular, we deal with the cohomological splitting question posted by Vasconcelos. We present a connection from our approach to the realization problem of Nunke. This is equipped with several applications. For instance, we recover some results of Jensen (and others) by applying simple methods. Additional applications, including a computation of the projective dimension of some injective modules, are given. This enables us to extend some results of Matlis (resp. Osofsky) on the projective dimension of $E_R(k)$ (resp. $\mathcal{Q}$) from Cohen-Macaulay rings (resp. regular rings) to non-Cohen-Macaulay (resp. Cohen-Macaulay) rings.

math.AC↗
arXiv2022

Surjectivity of some local cohomology map and the second vanishing theorem

The second vanishing theorem has a long history in the theory of local cohomology modules, which connects the vanishing of a complete regular local ring with a topological property of the punctured spectrum of the ring under some conditions. However, the case of complete ramified regular local rings is unresolved. In this paper, we give a partial answer to the second vanishing theorem in the ramified case. Our proof is inspired by the theory of surjective elements in the theory of local cohomology.

math.AC↗
arXiv2022

Graphs represented by Ext

This paper opens and discusses the question originally due to Daniel Herden, who asked for which graph $(μ,R)$ we can find a family $\{\mathbb G_α: α< μ\}$ of abelian groups such that for each $α,β\inμ$: $$Ext(\mathbb G_α, \mathbb G_β) = 0 \Longleftrightarrow(α,β) \in R.$$ In this regard, we present four results. First, we give a connection to Quillen's small object argument which helps $Ext$ vanishes and uses to present useful criteria to the question. Suppose $λ= λ^{\aleph_0}$ and $μ= 2^λ$. We apply Jensen's diamond principle along with the criteria to present $λ$-free abelian groups representing bipartite graphs. Third, we use a version of the black box to construct in ZFC, a family of $\aleph_1$-free abelian groups representing bipartite graphs. Finally, applying forcing techniques, we present a consistent positive answer for general graphs.

math.AC↗
arXiv2021

Kaplansky Test Problems for $R$-Modules in ZFC

Fix a ring $ R $ and look at the class of left $ R $-modules and naturally, we restrict ourselves to the case of rings such that this class is not too similar to the case $ R $ is a field. We shall solve Kaplansky test problems, all three of which say that we do not have decomposition theory, e.g., if the square of one module is isomorphic to the cube of another it does not follows that they are isomorphic. Our results are in the ordinary et theory, ZFC. For this, we look at bimodules, i.e., the structures which are simultaneously left $ R $-modules and right $ S $-modules with reasonable associativity, over a commutative ring $ T $ included in the centers of $ R $ and of $ S $. Eventually, we shall choose $ S $ to help solve each of the test questions. But first, we analyze what can be the smallest endomorphism ring of an $(R, S)$-bimodule. We construct such bimodules by using a simple version of the black box.

math.AC↗
arXiv2021

A note on the homology of differentials

We deal with the complete-intersection property of maximally differential ideals. Also, we connect the Gorenstein homology of derivations to the Gorenstein property of the base rings. These equipped with some applications.

math.AC↗
arXiv2021

Homological subsets of Spec

We recover some data of a module $M$ from the Ext-family $\{\Ext^i_R(M,R)\}$. In this regard, we investigate homological subsets of $\Spec(R)$, defined by the help of Ext-family. We extend Grothendieck's calculation of $\dim(\Ext^g_R(M,R))$. Also, we compute support and the set of all associated prime ideals of the Ext-family in a serial of nontrivial cases.

math.AC↗
arXiv2021

A note on Cohen-Macaulay descent

We investigate the behavior of Cohen-Macaulay defect undertaking tensor product with a perfect module. Consequently, we study the perfect defect of a module. As an application, we connect to associated prime ideals of tensor products.

math.AC↗
arXiv2020

A study of the cohomological rigidity property

In this paper, motivated by a work of Luk and Yau, and Huneke and Wiegand, we study various aspects of the cohomological rigidity property of tensor product of modules over commutative Noetherian rings. We determine conditions under which the vanishing of a single local cohomology module of a tensor product implies the vanishing of all the lower ones, and obtain new connections between the local cohomology modules of tensor products and the Tate homology. Our argument yields bounds for the depth of tensor products of modules, as well as criteria for freeness of modules over complete intersection rings. Along the way, we also give a splitting criteria for vector bundles on smooth complete intersections.

math.AC↗
arXiv2020

A note on a system of parameters

Let $u$ be in $\mathfrak{p}\in\mathrm{Assh}(R)$. We present several situations for which $(0 : u)$ is (not) in an ideal generated by a system of parameters. An application is given.

math.AC↗