arXiv subjects

Elham Mahdavi

Publications and source records attributed to Elham Mahdavi.

5 recordsLinked to original sources

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 $\beta_d(M)\neq 0$, 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 $\beta_i(R^+)=0$ for some $i>0$ implies regularity in a series cases. On the Ext side, we prove that $\mu_i(R^+)=0$ 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 $\mu_i(R^\infty)=0$ forces regularity for isolated singularity.

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↗

arXiv2022

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

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↗

arXiv2020

Covering theory, (mono)morphism categories and stable Auslander algebras

Let $\mathcal{A}$ be a locally bounded $k$-category and $G$ a torsion-free group of $k$-linear automorphisms of $\mathcal{A}$ acting freely on the objects of $\mathcal{A},$ and $F:\mathcal{A}\rightarrow \mathcal{B}$ is a Galois functor. We extend naturally the push-down functor $F_{\lambda}$ to the functor $\rm{H}\rm{F}_{\lambda}:\rm{H}(\rm{mod}\mbox{-} \mathcal{A})\rightarrow \rm{H}(\rm{mod}\mbox{-} \mathcal{B})$, resp. $\mathcal{S} \rm{F}_{\lambda}:\mathcal{S}(\rm{mod}\mbox{-} \mathcal{A})\rightarrow \mathcal{S}(\rm{mod}\mbox{-} \mathcal{B})$, between the corresponding morphism categories, resp. monomorphism categories, of $\rm{mod}\mbox{-} \mathcal{A}$ and $\rm{mod}\mbox{-} \mathcal{B}$. Under some additional conditions, we show that $\rm{H}(\rm{mod}\mbox{-}\mathcal{A})$, resp. $\mathcal{S}( \rm{mod}\mbox{-}\mathcal{A})$, is locally bounded if and only if $\rm{H}(\rm{mod}\mbox{-} \mathcal{B})$, resp. $\mathcal{S}(\rm{mod}\mbox{-}\mathcal{B})$, is of finite representation type. As an application, we show that the stable Auslander algebra of a representation-finite selfinjective algebra $\Lambda$ is again representation-finite if and only if $\Lambda$ is of Dynkin type $\mathbb{A}_{n}$ with $n\leqslant 4$.

math.RT↗