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We use the notion of universal extension in a linear abelian category to study extensions of variations of mixed Hodge structure and convergent and overconvergent isocrystals. The results we obtain apply, for example, to prove the exactness of some homotopy sequences for these categories and to study |$F$| -able isocrystals. [ABSTRACT FROM AUTHOR]

Background: Few data are available on the prognosis of older patients who received corticosteroids for COVID-19. We aimed to compare the in-hospital mortality of geriatric patients hospitalized for COVID-19 who received corticosteroids or not.Methods: We conducted a multicentric retrospective cohort study in 15 acute COVID-19 geriatric wards in the Paris area from March to April 2020 and November 2020 to May 2021. We included all consecutive patients aged 70 years and older who were hospitalized with confirmed COVID-19 in these wards. Propensity score and multivariate analyses were used.Results: Of the 1 579 patients included (535 received corticosteroids), the median age was 86 (interquartile range 81-91) years, 56% of patients were female, the median Charlson Comorbidity Index (CCI) was 2.6 (interquartile range 1-4), and 64% of patients were frail (Clinical Frailty Score 5-9). The propensity score analysis paired 984 patients (492 with and without corticosteroids). The in-hospital mortality was 32.3% in the matched cohort. On multivariate analysis, the probability of in-hospital mortality was increased with corticosteroid use (odds ratio [OR] = 2.61 [95% confidence interval (CI) 1.63-4.20]). Other factors associated with in-hospital mortality were age (OR = 1.04 [1.01-1.07], CCI (OR = 1.18 [1.07-1.29], activities of daily living (OR = 0.85 [0.75-0.95], oxygen saturation < 90% on room air (OR = 2.15 [1.45-3.17], C-reactive protein level (OR = 2.06 [1.69-2.51], and lowest lymphocyte count (OR = 0.49 [0.38-0.63]). Among the 535 patients who received corticosteroids, 68.3% had at least one corticosteroid side effect, including delirium (32.9%), secondary infections (32.7%), and decompensated diabetes (14.4%).Conclusions: In this multicentric matched-cohort study of geriatric patients hospitalized for COVID-19, the use of corticosteroids was significantly associated with in-hospital mortality. [ABSTRACT FROM AUTHOR]

We give a proof without use of perfectoid geometry of the following vanishing theorem of Scholze: for |$X\subset \mathbb{P}^n$| a projective scheme of dimension |$d$| over an algebraically closed characteristic |$0$| field, and |$X_r$| the inverse image of |$X$| via the map that assigns |$(x_0^{p^r}: \dots : x_n^{p^r})$| to the homogeneous coordinates |$(x_0:\ldots :x_n)$|⁠ , the induced map |$H^i(X, {{\mathbb{F}}}_p)\to H^i(X_r, {{\mathbb{F}}}_p)$| on étale cohomology dies for |$i>d$| and |$r$| large. Our proof holds in characteristic |$\ell \neq p$| as well. [ABSTRACT FROM AUTHOR]

We prove that the vanishing of the functoriality morphism for the étale fundamental group between smooth projective varieties over an algebraically closed field of characteristic p > 0 forces the same property for the fundamental groups of stratifications. [ABSTRACT FROM AUTHOR]

Over a connected geometrically unibranch scheme X of finite type over a finite field, we show, with purely geometric arguments, finiteness of the number of irreducible Qℓ-lisse sheaves, with bounded rank and bounded ramification in the sense of Drinfeld, up to twist by a character of the finite field. On X smooth, with bounded ramification in the sense of [7, Definition 3.6], this is Deligne's theorem [7, Theorem 1.1], the proof of which uses the whole strength of [3] and [6]. We also generalize Deligne's theorem [7, Theorem 1.1] from X smooth to X normal, using Deligne's theorem (not reproving it) and a few more geometric arguments. [ABSTRACT FROM AUTHOR]

Let X be a nonsingular projective variety over an algebraically closed field k of characteristic 0. If , then for any ample line bundle H on X, any semistable bundle E on X with all Chern classes 0 is trivial. Over , this a consequence of the fact that is a finitely generated group and hence any representation into a linear group is residually finite. We prove an analog of this theorem in characteristic p>0. Semistable bundles with vanishing Chern classes are replaced by Nori semistable bundles, that is those which are semistable of degree 0 on any curve mapping to X. The étale fundamental group is replaced by Nori’s fundamental group scheme πN(X) [Nori, M. “The fundamental group-scheme.” Proceedings of the Indian Academy of Sciences, Mathematical Sciences 91 no. 2 (1982), 73–122], which is the profinite completion of the tensor automorphism group scheme πS(X) of Nori semistable bundles [Langer, A. “On the S-Fundamental group scheme II,” (2009): preprint] studied by Langer [“On the S-Fundamental group scheme,” preprint arXiv/0905.4600;“On the S-Fundamental group scheme II,” (2009): preprint]. [ABSTRACT FROM PUBLISHER]

If X is an abelian variety over a field, and L ∈ Pic(X), we know that the degree of the 0-cycle Lg is divisible by g!. As a 0-cycle, it is not, even over a field of cohomological dimension 1, but we show that over a finite field, there is some hope. [ABSTRACT FROM AUTHOR]