In this paper we extend one direction of Fröberg's theorem on a combinatorial classification of quadratic monomial ideals with linear resolutions. We do this by generalizing the notion of a chordal graph to higher dimensions with the introduction of d-chorded and orientably-d-cycle-complete simplicial complexes. We show that a certain class of simplicial complexes, the d-dimensional trees, correspond to ideals having linear resolutions over fields of characteristic 2 and we also give a necessary combinatorial condition for a monomial ideal to be componentwise linear over all fields.

In this paper we give a necessary and sufficient combinatorial condition for a monomial ideal to have a linear resolution over fields of characteristic 2. We also give a new proof of Fröberg's theorem over fields of characteristic 2.

In this paper we introduce the notion of a d-dimensional cycle which is a homological generalization of the idea of a graph cycle to higher dimensions. We examine both the combinatorial and homological properties of this structure and use these results to describe the relationship between the combinatorial structure of a simplicial complex and its simplicial homology. In particular, we show that over any field of characteristic 2 the existence of nonzero d-dimensional homology corresponds exactly to the presence of a d-dimensional cycle in the simplicial complex. We also show that d-dimensional cycles which are orientable give rise to non-zero simplicical homology over any field.

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IntroductionSurvivors of acute pancreatitis (AP) have shorter overall survival and increased incidence of new-onset cardiovascular, respiratory, liver and renal disease, diabetes mellitus and cancer compared with the general population, but the mechanisms that explain this are yet to be elucidated. Our aim is to characterise the precise nature and extent of organ dysfunction following an episode of AP.Methods and analysisThis is an observational prospective cohort study in a single centre comprising a University hospital with an acute and emergency receiving unit and clinical research facility. Participants will be adult patient admitted with AP. Participants will undergo assessment at recruitment, 3 months and 3 years. At each time point, multiple biochemical and/or physiological assessments to measure cardiovascular, respiratory, liver, renal and cognitive function, diabetes mellitus and quality of life. Recruitment was from 30 November 2017 to 31 May 2020; last follow-up measurements is due on 31 May 2023. The primary outcome measure is the incidence of new-onset type 3c diabetes mellitus during follow-up. Secondary outcome measures include: quality of life analyses (SF-36, Gastrointestinal Quality of Life Index); montreal cognitive assessment; organ system physiological performance; multiomics predictors of AP severity, detection of premature cellular senescence. In a nested cohort within the main cohort, individuals may also consent to multiparameter MRI scan, echocardiography, pulmonary function testing, cardiopulmonary exercise testing and pulse-wave analysis.Ethics and disseminationThis study has received the following approvals: UK IRAS Number 178615; South-east Scotland Research Ethics Committee number 16/SS/0065. Results will be made available to AP survivors, caregivers, funders and other researchers. Publications will be open-access.Trial registration numbersClinicalTrials.gov Registry (NCT03342716) and ISRCTN50581876; Pre-results.