Trace ideal and annihilator of Ext and Tor of regular fractional ideals, and some applications

Given a commutative Noetherian ring $R$ with total ring of fractions $Q(R)$, and a finitely generated $R$-submodule $M$ of $Q(R)$, we prove an equality between trace ideal, and certain annihilator of Ext and Tor of $M$. As a consequence, we answer in one-dimensional local analytically unramified case, a question raised by the present author and R. Takahashi. As another application, we give an alternative proof of a recent result of \"O. Esentepe that for one-dimensional analytically unramified Gorenstein local rings, the cohomology annihilator of Iyengar and Takahashi coincides with the conductor ideal.Comment: Comments are welcome

Transcriptional regulation of ATF4 is critical for controlling the Integrated Stress Response during eIF2 phosphorylation

Indiana University-Purdue University Indianapolis (IUPUI)In response to different environmental stresses, phosphorylation of eIF2 (eIF2P) represses global translation coincident with preferential translation of ATF4. ATF4 is a transcriptional activator of the integrated stress response, a program of gene expression involved in metabolism, nutrient uptake, anti-oxidation, and the activation of additional transcription factors, such as CHOP/GADD153, that can induce apoptosis. Although eIF2P elicits translational control in response to many different stress arrangements, there are selected stresses, such as exposure to UV irradiation, that do not increase ATF4 expression despite robust eIF2P. In this study we addressed the underlying mechanism for variable expression of ATF4 in response to eIF2P during different stress conditions and the biological significance of omission of enhanced ATF4 function. We show that in addition to translational control, ATF4 expression is subject to transcriptional regulation. Stress conditions such as endoplasmic reticulum stress induce both transcription and translation of ATF4, which together enhance expression of ATF4 and its target genes in response to eIF2P. By contrast, UV irradiation represses ATF4 transcription, which diminishes ATF4 mRNA available for translation during eIF2∼P. eIF2P enhances cell survival in response to UV irradiation. However, forced expression of ATF4 and its target gene CHOP leads to increased sensitivity to UV irradiation. In this study, we also show that C/EBPβ is a transcriptional repressor of ATF4 during UV stress. C/EBPβ binds to critical elements in the ATF4 promoter resulting in its transcriptional repression. The LIP isoform of C/EBPβ, but not the LAP version is regulated following UV exposure and directly represses ATF4 transcription. Loss of the LIP isoform results in increased ATF4 mRNA levels in response to UV irradiation, and subsequent recovery of ATF4 translation, leading to enhanced expression of its target genes. Together these results illustrate how eIF2P and translational control, combined with transcription factors regulated by alternative signaling pathways, can direct programs of gene expression that are specifically tailored to each environmental stress

Ulrich split rings

A local Cohen--Macaulay ring is called Ulrich-split if any short exact sequence of Ulrich modules split. In this paper we initiate the study of Ulrich split rings. We prove several necessary or sufficient criteria for this property, linking it to syzygies of the residue field and cohomology annihilator. We characterize Ulrich split rings of small dimensions. Over complex numbers, $2$-dimensional Ulrich split rings, which are normal and have minimal multiplicity, are precisely cyclic quotient singularities with at most two indecomposable Ulrich modules up to isomorphism. We give several ways to construct Ulrich split rings, and give applications on detecting projective/injective modules via vanishing of $\operatorname{Ext}$

Strong generation and (co)ghost index for module categories

This work focuses on notions of generation and (co)ghost index in the module category of a Noetherian commutative ring. In particular, a sufficiency condition is established for the existence of strong generators in the module category of a Noetherian ring. As a result, this provides an affirmative answer to a question posed by Iyengar and Takahashi. Furthermore, the techniques developed provide upper bounds on the Rouquier dimension and rank respectively for the singularity category and category of maximal Cohen-Macaulay modules. Additionally, a local-to-global principle for (co)ghost index in the module category is investigated, and explicit computations are provided.Comment: V1, comments welcome; v2 consists of additional coauthors and massive improvements to results; v3, no changes, add grant informatio

Exact subcategories, subfunctors of Ext⁡\operatorname{Ext}, and some applications

Let $(\mathcal{A},\mathcal{E})$ be an exact category. We establish basic results that allow one to identify sub(bi)functors of $\operatorname{Ext}_{\mathcal{E}}(-,-)$ using additivity of numerical functions and restriction to subcategories. We also study a small number of these new functors over commutative local rings in details, and find a range of applications from detecting regularity to understanding Ulrich modules.Comment: To appear in Nagoya Mathematical Journal. This arXiv version contains one additional result (than the Journal Version), namely Proposition 5.1.2

On a generalized Auslander-Reiten conjecture

We study the symmetric Auslander condition (SAC) which is equivalent to the generalized Auslander-Reiten condition (GARC). First, we affirmatively answer a question posed by Celikbas and Takahashi, that is, the equivalence of (SAC) and the symmetric Auslander condition for modules with constant rank (SACC). As a corollary of the result, we also give the equivalence of (SAC) for $R$ and $R/xR$, where $x$ is an $R$-regular element. Secondly, we explore (SAC) among ring homomorphisms $R \to S$ of local rings. We prove that if $S$ satisfies (SAC) (resp. Auslander-Reiten conjecture), then $R$ also satisfies (SAC) (resp. Auslander-Reiten conjecture) if the flat dimension of $S$ over $R$ is finite. We also prove that $R$ satisfies (SAC) implies that $S$ satisfies (SAC) if $R$ is Gorenstein, $S=R/Q^\ell$, where $Q$ is generated by a regular sequence of $R$ and the length of the sequence is at least $\ell$. This is a consequence of more general results about Ulrich ideals proved in this paper. Applying these results to deteminantal rings and numerical semigroup rings, we provide new classes of rings satisfying (SAC).Comment: Substantial reorganization. Comments are welcome