Related papers: Under Spec Z

We define the derived category of a concrete category in a way which extends the usual definition of the derived category of a ring, and we prove that the bounded-below derived category of $\Spec \mathbb{M}_0$ (an approximation, used by…

In arXiv:2511.04191 we constructed schemes of objects in small categories which contained a set of basepoints with local representing (localizing) objects. Here we prove that the category $\cat{Rings}$ of associative rings with unit has a…

For the first time, we have introduced the concept of N-groups, N-semigroups, N-loops, and N-groupoids. We also define a mixed N-algebraic structure. The main aim of this book is to attract young mathematicians to this interesting field. It…

We describe a construction of cocyclic perturbations of the semigroup of shifts on the semiaxis by means of the theory of model spaces. It is shown that one can choose an inner function that determines the model space so that the elements…

In this work we prove that a semialgebraic set $M\subset{\mathbb R}^m$ is determined (up to a semialgebraic homeomorphism) by its ring ${\mathcal S}(M)$ of (continuous) semialgebraic functions while its ring ${\mathcal S}^*(M)$ of…

This is a survey about spectral sets, to appear in the second edition of Handbook of Linear Algebra (L. Hogben, ed.). Spectral sets and K-spectral sets, introduced by John von Neumann, offer a possibility to estimate the norm of functions…

Motivated by ideas from string theory and quantum field theory new invariants of knots and 3-dimensional manifolds have been constructed from complex algebraic structures such as Hopf algebras (Reshetikhin and Turaev), monoidal categories…

For an essentially small triangulated category $\mathcal{T}$, we introduce the notion of prime thick subcategories and define the spectrum of $\mathcal{T}$, which shares the basic properties with the spectrum of a tensor triangulated…

We develop a functorial framework for the ideal theory of commutative semirings using coherent frames and spectral spaces. Two central constructions-the radical ideal functor and the $k$-radical ideal functor-are shown to yield coherent…

We define a notion of a connectivity structure on an $\infty$-category, analogous to a $t$-structure but applicable in unstable contexts -- such as spaces, or algebras over an operad. This allows us to generalize notions of n-skeleta,…

We initiate a unified, axiomatic study of noncommutative algebras R whose prime spectra are, in a natural way, finite unions of commutative noetherian spectra. Our results illustrate how these commutative spectra can be functorially ``sewn…

Several known constructions relate initial degenerations of projective toric varieties and Grassmannians to regular subdivisions of appropriate point configurations. We define a general framework which allows for partial generalizations of…

Let $G=\Gamma(S)$ be a semigroup graph, i.e., a zero-divisor graph of a semigroup $S$ with zero element 0. For any adjacent vertices $x, y$ in $G$, denote $C(x,y)={z\in V(G) | N(z)={x,y}}$. Assume that in $G$ there exist two adjacent…

Let R be a commutative ring with identity and Specs(M) denote the set all second submodules of an R-module M. In this paper, we construct and study a sheaf of modules, denoted by O(N; M), on Specs(M) equipped with the dual Zariski topology…

For $G$ a connected, reductive group over an algebraically closed field $k$ of large characteristic, we use the canonical Springer isomorphism between the nilpotent variety of $\mathfrak{g}:=\mathrm{Lie}(G)$ and the unipotent variety of $G$…

In this paper, we develop basic results of algebraic geometry over abelian symmetric monoidal categories. Let $A$ be a commutative monoid object in an abelian symmetric monoidal category $(\mathbf C,\otimes,1)$ satisfying certain conditions…

The $D$-graded Proj construction provides a general framework for constructing schemes from rings graded by finitely generated abelian groups $D$, yet its properties and applications remain underdeveloped compared to the classical…

Let S and R be the rings of regular functions on affine algebraic varieties over a field of characteristic 0, R be embedded as a subring in S, and F : S --> S be an endomorphism such that F(R) subset R. Suppose that every ideal of height 1…

We define the spectrum of a tensor triangulated category $K$ as the set of so-called prime ideals, endowed with a suitable topology. In this very generality, the spectrum is the universal space in which one can define supports for objects…

This is a foundation for algebraic geometry, developed internal to the Zariski topos, building on the work of Kock and Blechschmidt. The Zariski topos consists of sheaves on the site opposite to the category of finitely presented algebras…