Related papers: On some permanence properties of (derived) splinte…

We prove a flatness result for entire nonlocal minimal graphs having some partial derivatives bounded from either above or below. This result generalizes fractional versions of classical theorems due to Bernstein and Moser. Our arguments…

We prove a generic flatness result for the cohomology of thickenings of a projective scheme that is smooth over a Noetherian domain containing a field of characteristic zero. Our study is motivated, in part, by a classical question in…

Let $f\colon Y \to X$ be a proper flat morphism of locally noetherian schemes. Then, the locus in $X$ over which $f$ is smooth is stable under generization. We prove that under suitable assumptions on the formal fibers of $X$, the same…

This chapter sets out preliminaries for the duality theory in later chapters. An underlying idea is that local cohomology functors are higher derived functors of colocalizations (a.k.a.~coreflections). Predominantly well-known facts about…

In this article, we study the behaviour of smooth algebra $R$ over local Noetherian local ring $A$. At first, we observe that for every $f\in R$, $R_f$ has finite length in the category of $D(R,A)$-module if dimension of $A$ is zero. This…

We define and study the regularity of distance maps on geodesically complete spaces with curvature bounded above. We prove that such a regular map is locally a Hurewicz fibration. This regularity can be regarded as a dual concept of…

Grothendieck Duality -- the theory of the twisted inverse image pseudofunctor (-)^! over a suitable category of scheme-maps -- can be developed concretely, with emphasis on explicit constructions, or abstractly, with emphasis on…

This a first step to develop a theory of smooth, etale and unramified morphisms between noetherian formal schemes. Our main tool is the complete module of differentials, that is a coherent sheaf whenever the map of formal schemes is of…

Let $f : X \longrightarrow Y$ be a proper and local complete intersection morphism of schemes. We prove that $\mathbb{R}f_{*}$ preserves perfect complexes, without any projectivity or noetherian assumptions. This provides a different proof…

Let R be a local Noetherian domain of positive characteristic. A theorem of Hochster and Huneke (1992) states that if R is excellent, then the absolute integral closure of R is a big Cohen-Macaulay algebra. We prove that if R is the…

Noether's theorem, which connects continuous symmetries to exact conservation laws, remains one of the most fundamental principles in physics and dynamical systems. In this work, we draw a conceptual parallel between two paradigms: the…

Symmetries are known to dictate important physical properties and can be used as a design principle in particular in wave physics, including wave structures and the resulting propagation dynamics. Local symmetries, in the sense of a…

We say an excellent local domain $(S,n)$ satisfies the vanishing conditions for maps of Tor, if for every $A\to R\to S$ with $A$ regular and $A\to R$ module-finite torsion-free extension, and every $A$-module $M$, the map $Tor^A_i(M, R)\to…

The goal of the present paper is to establish some kind of regularity of an energy minimizer map between Riemannian polyhedra. More precisely, we will show the h\"{o}lder continuity of local energy minimizers between Riemannian polyhedra…

A field extension $L/K$ of characteristic $p > 0$ is formally \'etale if and only if the relative Frobenius of $L/K$ is an isomorphism. Inspired by this classical result, we explore whether the formally \'etale property for a map $R \to S$…

We give several related versions of global Grothendieck Duality for unbounded complexes on noetherian formal schemes. The proofs, based on a non-trivial adaptation of Deligne's method for the special case of ordinary schemes, are reasonably…

We give an elementary proof of Grothendieck's non-vanishing Theorem: For a finitely generated non-zero module $M$ over a Noetherian local ring $A$ with maximal ideal $\m$, the local cohomology module $H^{\dim M}_{\m}(M)$ is non-zero.

In this paper, we give lower bounds for the homology of the fibers of a map to a manifold. Using new sheaf theoretic methods, we show that these lower bounds persist over whole open sets of the manifold, and that they are stable under…

We study Kaehlerian manifolds with Norden metric $g$ and develop the theory of their holomorphic hypersurfaces with constant totally real sectional curvatures. We prove a classification theorem for the holomorphic hypersurfaces of…

We introduce the notion of a separator for a morphism of schemes f:T\to S; in particular, it is universal among morphisms from T to separated S-schemes. A separator is a local isomorphism; this property conveys the intuition of gluing some…