Related papers: Higher Zariski Geometry

Let $R$ be a commutative ring with unity and $M$ be a left $R$-module. We define the secondary-like spectrum of $M$ to be the set of all secondary submodules $K$ of $M$ such that $Ann_R(soc(K))=\sqrt{Ann_R(K)}$, and we denote it by…

We study non-interacting electrons in disordered materials which exhibit a spectral gap, in each of the ten Altland--Zirnbauer symmetry classes, in all space dimensions. We define an appropriate space of Hamiltonians and a topology on it so…

In the present survey paper, we present several new classes of Hochster's spectral spaces "occurring in nature", actually in multiplicative ideal theory, and not linked to or realized in an explicit way by prime spectra of rings. The…

We show that compactly generated t-structures in the derived category of a commutative ring $R$ are in a bijection with certain families of compactly generated t-structures over the local rings $R_\mathfrak{m}$ where $\mathfrak{m}$ runs…

We formalize the main approach for showing Zariski descent-type statements for strong generation of triangulated categories associated to algebro-geometric objects. This recovers various known statements in the literature. As applications…

We develop the theory of topoi internal to an arbitrary $\infty$-topos $\mathcal B$. We provide several characterisations of these, including an internal analogue of Lurie's characterisation of $\infty$-topoi, but also a description in…

Let $\pi$ be a group satisfying the Farrell-Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincar\'e duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$…

The Zariski theorem says that for every hypersurface in a complex projective (resp. affine) space of dimension at least 3 and for every generic plane in the projective (resp. affine) space the natural embedding generates an isomorphism of…

There is a close relationship between the embedded topology of complex plane curves and the (group-theoretic) arithmetic of elliptic curves. In a recent paper, we studied the topology of some arrangements of curves which include a special…

For any compactly generated triangulated category we introduce two topological spaces, the shift-spectrum and the shift-homological spectrum. We use them to parametrise a family of thick subcategories of the compact objects, which we call…

Here we continue to characterize a recently introduced notion, le-modules $_{R}M$ over a commutative ring $R$ with unity \cite{Bhuniya}. This article introduces and characterizes Zariski topology on the set $Spec(M)$ of all prime submodule…

In this paper we prove the following theorem. Let $f$ be a dominant endomorphism of a smooth projective surface over an algebraically closed field of characteristic $0$. If there is no nonconstant invariant rational function under $f$, then…

This work constitutes the second part of a series of studies that aim to utilise tools from Hamiltonian mechanics to investigate the motion of an extended body in general relativity. The first part of this work [Refs. [1, 2]] constructed a…

Let R be an associative ring with identity. We study an elementary generalization of the classical Zariski topology, applied to the set of isomorphism classes of simple left R-modules (or, more generally, simple objects in a complete…

In this paper, we study geometric points in tensor triangular geometry. In doing so, we construct a counter-example to Balmer's Nerves of Steel conjecture using free constructions in higher Zariski geometry. We then go on to introduce and…

We present a unified framework demonstrating how the spinor complex Lorentz group SL(2,C)/Z\_2 is realized as a canonical subgroup within a four-dimensional complex Riemannian manifold. Building on the complex, holomorphic metric extension…

A typical large complex-structure limit for mirror symmetry consists of toric varieties glued to each other along their toric boundaries. Here we construct the mirror large volume limit space as a Weinstein symplectic manifold. We prove…

In this paper we study the embedded topology of reducible plane curves having a smooth irreducible component. In previous studies, the relation between the topology and certain torsion classes in the Picard group of degree zero of the…

A semigroup prime of a commutative ring $R$ is a prime ideal of the semigroup $(R,\cdot)$. One of the purposes of this paper is to study, from a topological point of view, the space $\scal(R)$ of prime semigroups of $R$. We show that, under…

This paper dualizes the setting of affine spaces as originally introduced by Diers for application to algebraic geometry and expanded upon by various authors, to show that the fundamental groups of pointed topological spaces appear as the…