Integrally closed rings in birational extensions of two-dimensional regular local rings

Let $D$ be an integrally closed local Noetherian domain of Krull dimension 2, and let $f$ be a nonzero element of $D$ such that $fD$ has prime radical. We consider when an integrally closed ring $H$ between $D$ and $D_f$ is determined locally by finitely many valuation overrings of $D$. We show such a local determination is equivalent to a statement about the exceptional prime divisors of normalized blow-ups of $D$, and, when $D$ is analytically normal, this property holds for $D$ if and only if it holds for the completion of $D$. This latter fact, along with MacLane's notion of key polynomials, allows us to prove that in some central cases where $D$ is a regular local ring and $f$ is a regular parameter of $D$, then $H$ is determined locally by a single valuation. As a consequence, we show that if $H$ is also the integral closure of a finitely generated $D$-algebra, then the exceptional prime ideals of the extension $H/D$ are comaximal. Geometrically, this translates into a statement about intersections of irreducible components in the closed fiber of the normalization of a proper birational morphism.Comment: 32 pp., to appear in Math. Proc. Camb. Phil. So

Star-Invertibility and tt-finite character in Integral Domains

Let $A$ be an integral domain. We study new conditions on families of integral ideals of $A$ in order to get that $A$ is of $t$-finite character (i.e., each nonzero element of $A$ is contained in finitely many $t$-maximal ideals). We also investigate problems connected with the local invertibility of ideals.Comment: 16 page

Invertibility of ideals in prüfer extensions

Using the general approach to invertibility for ideals in ring extensions given by Knebush and Zhang in [9], we investigate about connections between faithfully atness and invertibility for ideals in rings with zero divisors

INTEGRALLY CLOSED RINGS IN BIRATIONAL EXTENSIONS OF TWO-DIMENSIONAL REGULAR LOCAL RINGS

Abstract. Let D be an integrally closed local Noetherian domain of Krull dimension 2, and let f be a nonzero element of D such that f D has prime radical. We consider when an integrally closed ring H between D and D f is determined locally by finitely many valuation overrings of D. We show such a local determination is equivalent to a statement about the exceptional prime divisors of normalized blow-ups of D, and, when D is analytically normal, this property holds for D if and only if it holds for the completion of D. This latter fact, along with MacLane's notion of key polynomials, allows us to prove that in some central cases where D is a regular local ring and f is a regular parameter of D, then H is determined locally by a single valuation. As a consequence, we show that if H is also the integral closure of a finitely generated D-algebra, then the exceptional prime ideals of the extension H/D are comaximal. Geometrically, this translates into a statement about intersections of irreducible components in the closed fiber of the normalization of a proper birational morphism

APOLLO 11 Project, Consortium in Advanced Lung Cancer Patients Treated With Innovative Therapies: Integration of Real-World Data and Translational Research

Introduction: Despite several therapeutic efforts, lung cancer remains a highly lethal disease. Novel therapeutic approaches encompass immune-checkpoint inhibitors, targeted therapeutics and antibody-drug conjugates, with different results. Several studies have been aimed at identifying biomarkers able to predict benefit from these therapies and create a prediction model of response, despite this there is a lack of information to help clinicians in the choice of therapy for lung cancer patients with advanced disease. This is primarily due to the complexity of lung cancer biology, where a single or few biomarkers are not sufficient to provide enough predictive capability to explain biologic differences; other reasons include the paucity of data collected by single studies performed in heterogeneous unmatched cohorts and the methodology of analysis. In fact, classical statistical methods are unable to analyze and integrate the magnitude of information from multiple biological and clinical sources (eg, genomics, transcriptomics, and radiomics). Methods and objectives: APOLLO11 is an Italian multicentre, observational study involving patients with a diagnosis of advanced lung cancer (NSCLC and SCLC) treated with innovative therapies. Retrospective and prospective collection of multiomic data, such as tissue- (eg, for genomic, transcriptomic analysis) and blood-based biologic material (eg, ctDNA, PBMC), in addition to clinical and radiological data (eg, for radiomic analysis) will be collected. The overall aim of the project is to build a consortium integrating different datasets and a virtual biobank from participating Italian lung cancer centers. To face with the large amount of data provided, AI and ML techniques will be applied will be applied to manage this large dataset in an effort to build an R-Model, integrating retrospective and prospective population-based data. The ultimate goal is to create a tool able to help physicians and patients to make treatment decisions. Conclusion: APOLLO11 aims to propose a breakthrough approach in lung cancer research, replacing the old, monocentric viewpoint towards a multicomprehensive, multiomic, multicenter model. Multicenter cancer datasets incorporating common virtual biobank and new methodologic approaches including artificial intelligence, machine learning up to deep learning is the road to the future in oncology launched by this project

Divisorial prime ideals of Int(D) when D is a Krull-type domain

Let D be a domain with quotient field K. The ring of integer-valued polynomials over D is Int(D) := { f E K[S];f( D) C D) . We describe the divisorial prime ideals of Int(D) when D is a domain of Krull-type and, in particular, when D is also a d-ring

When the semistar operation ⋆~\tilde{\star} is the identity

We study properties of integral domains in which it is given a semistar operation such that ˜ is the identity. In particular, we put attention to the case = v, where v is the divisorial closure