Control of the non-geometrically integral reductions

In this paper, for a geometrically integral projective scheme, we will give an upper bound of the product of the norms of its non-geometrically integral reductions over an arbitrary number field. For this aim, we take the adelic viewpoint on this subject. Résumé (Contrôle des réductions non-géométriquement intègres) Dans cet article, pour un schéma projectif géométriquement intègre, on donnera une majoration du produit des norms de ses réductions non-géométriquement intègre sur un corps de nombres arbitraire. Pour le but, on prend le point de vue adélique autour de ce sujet.

Differentiability of Relative Volumes Over an Arbitrary Non-Archimedean Field

Given an ample line bundle $L$ on a geometrically reduced projective scheme defined over an arbitrary non-Archimedean field, we establish a differentiability property for the relative volume of two continuous metrics on the Berkovich analytification of $L$, extending previously known results in the discretely valued case. As applications, we provide fundamental solutions to certain non-Archimedean Monge–Ampère equations and generalize an equidistribution result for Fekete points. Our main technical input comes from determinant of cohomology and Deligne pairings.

Hochschild cohomology related to graded down-up algebras with weights (1,n)

Let [Formula: see text] be a graded down-up algebra with [Formula: see text] and [Formula: see text], and let [Formula: see text] be the Beilinson algebra of [Formula: see text]. If [Formula: see text], then a description of the Hochschild cohomology group of [Formula: see text] is known. In this paper, we calculate the Hochschild cohomology group of [Formula: see text] for the case [Formula: see text]. As an application, we see that the structure of the bounded derived category of the noncommutative projective scheme of [Formula: see text] is different depending on whether (10) [Formula: see text] [Formula: see text] is zero or not. Moreover, it turns out that there is a difference between the cases [Formula: see text] and [Formula: see text] in the context of Grothendieck groups.

Moduli spaces of shtukas

This chapter examines the moduli spaces of mixed-characteristic local G-shtukas and shows that they are representable by locally spatial diamonds. These will be the mixed-characteristic local analogues of the moduli spaces of global equal-characteristic shtukas introduced by Varshavsky. It may be helpful to briefly review the construction in the latter setting. The ingredients are a smooth projective geometrically connected curve X defined over a finite field Fq and a reductive group G/Fq. Each connected component is a quotient of a quasi-projective scheme by a finite group. From there, it is possible to add level structures to the spaces of shtukas, to obtain a tower of moduli spaces admitting an action of the adelic group. The cohomology of these towers of moduli spaces is the primary means by which V. Lafforgue constructs the “automorphic to Galois” direction of the Langlands correspondence for G over F.

On Ratliff-Rush closure of modules

In this paper, we introduce the notion of Ratliff-Rush closure of modules and explore whether the condition of the Ratliff-Rush closure coincides with the integral closure. The main result characterizes the condition in terms of the normality of the projective scheme of the Rees algebra. In conclusion, we shall give a criterion for the Buchsbaum Rees algebras.

Non-commutative algebraic geometry of semi-graded rings

In this paper, we introduce the semi-graded rings, which extend graded rings and skew Poincaré–Birkhoff–Witt (PBW) extensions. For this new type of non-commutative rings, we will discuss some basic problems of non-commutative algebraic geometry. In particular, we will prove some elementary properties of the generalized Hilbert series, Hilbert polynomial and Gelfand–Kirillov dimension. We will extend the notion of non-commutative projective scheme to the case of semi-graded rings and we generalize the Serre–Artin–Zhang–Verevkin theorem. Some examples are included at the end of the paper.

Pure Injective and Absolutely Pure Sheaves

We study two notions of purity in categories of sheaves: the categorical and the geometric. It is shown that pure injective envelopes exist in both cases under very general assumptions on the scheme. Finally, we introduce the class of locally absolutely pure (quasi-coherent) sheaves with respect to the geometrical purity, and characterize locally Noetherian closed subschemes of a projective scheme in terms of the new class.

The Augmented Base Locus in Positive Characteristic

Let L be a nef line bundle on a projective scheme X in positive characteristic. We prove that the augmented base locus of L is equal to the union of the irreducible closed subsets V of X such that L∣V is not big. For a smooth variety in characteristic 0, this was proved by Nakamaye using vanishing theorems.

Bigq-ample line bundles

A recent paper of Totaro developed a theory ofq-ample bundles in characteristic 0. Specifically, a line bundleLonXisq-ample if for every coherent sheaf ℱ onX, there exists an integerm0such thatm≥m0impliesHi(X,ℱ⊗𝒪(mL))=0 fori>q. We show that a line bundleLon a complex projective schemeXisq-ample if and only if the restriction ofLto its augmented base locus isq-ample. In particular, whenXis a variety andLis big but fails to beq-ample, then there exists a codimension-one subschemeDofXsuch that the restriction ofLtoDis notq-ample.